Unstable Operations in Generalized Cohomology J. Michael Boardman David Copeland Johnson W. Stephen Wilson January 1995 [To appear in Handbook of Algebraic Topology, ed. I. M. James, Elsevier (Amsterdam, 1995)] TABLE OF CONTENTS 1 Introduction 2 2 Cohomology operations 9 3 Group objects and E-cohomology 12 4 Group objects and E-homology 14 5 What is an additively unstable module? 17 6 Unstable comodules 22 7 What is an additively unstable algebra? 32 8 What is an unstable object? 38 9 Unstable, additive, and stable objects * *43 10 Enriched Hopf rings 47 11 The E-cohomology of a point 59 12 Spheres, suspensions, and additive operations * *61 13 Spheres, suspensions, and unstable operations * *63 14 Complex orientation and additive operations 66 15 Complex orientation and unstable operations 68 16 Examples for additive operations 70 17 Examples for unstable operations 78 18 Relations for additive BP -operations * *86 19 Relations in the Hopf ring for BP 92 20 Additively unstable BP -objects 102 21 Unstable BP -algebras 107 22 Additive splittings of BP -cohomology 111 23 Unstable splittings of BP -cohomology 115 Index of symbols 121 References 125 JMB, DCJ, WSW - 1 - 23 Feb 1995 Unstable cohomology operations 1 Introduction A multiplicative generalized cohomology theory E*(-) on spaces is represent* *ed by the spaces E_n of its -spectrum, as described in detail in [8, Thm. 3.17]. We d* *enote its coefficient ring by E*. Our five examples are ordinary cohomology H*(-; Fp* *), unitary cobordism MU*(-), Brown-Peterson cohomology BP *(-), complex K-theory KU*(-), and Morava K-theory K(n)*(-). (They were properly introduced in [8, x2]* *.) Recent work [25] shows that a sixth example, the cohomology theory P (n)*(-), a* *lso satisfies our hypotheses. We are interested in three kinds of cohomology operation: stable operations* *, which form the endomorphism ring E*(E; o) of E (in our notation) and were studied in * *[8]; unstable operations, defined on En(X) for spaces X and fixed n, which form E*(E* *_n); and additive unstable operations r on En(-) (that satisfy r(x+y) = r(x) + r(y)), which form the subset P E*(E_n). Since a stable operation restricts to an addi* *tive unstable operation on any degree, these are related by E*(E; o) --! P E*(E_n) E*(E_n) : Each of these is an E*-module in the usual way, by (r+s)(x) = r(x) + s(x) and (vr)(x) = v r(x) (for any v 2 E*). We can compose, (sr)(x) = (s Or)(x) = s(r(x)* *), whenever the sources and targets match. We can also multiply unstable operations together by (r ^ s)(x) = r(x)s(x). In the classical case E = H(F p), for which E*(E; o) is the Steenrod algebr* *a, it is true that: (a) every additive operation comes from a stable operation; (b) the * *addi- tive operations generate multiplicatively all the unstable operations. Our diff* *iculties stem from the fact that for MU and BP , both (a) and (b) are false. (See [27] * *for more discussion of the differences.) We propose to describe completely the alge* *braic structure that has to be present on an E*-module or E*-algebra to make it an un- stable object, with particular attention to the case E = BP . Our definitions l* *ead to structure theorems. Stable BP -operations have been available for quite some time and are well * *estab- lished. Less has been done with unstable BP -operations, owing to their complex* *ity, but we do have the work [4, 5] of Bendersky, Curtis, Davis, and Miller. The alg* *ebraic structure on an additively unstable module is described in [27] and (without pr* *oofs) in [6]. Our major task, therefore, is to set up precise algebraic descriptions of t* *he unstable structures we need on modules and algebras, along the lines of the stable struc* *tures in [8]. Part of the difficulty is that one is forced to work in the unfamiliar * *context of nonadditive operations; but the real problem turns out to be Thm. 9.4, that uns* *table modules (as distinct from unstable algebras) simply do not exist compatibly wit* *h our other objects! When we limit attention to the less exotic additive operations,* * this difficulty does not arise and we have both modules and algebras. In fact, there is a huge amount of data to be codified in an unstable algeb* *ra. The key idea is that given an E*-algebra M, we define (UM)k for each k as the set of all algebra homomorphisms E*(E_k) ! M; each such homomorphism may be thought of as a candidate for the values of all operations on a typical elem* *ent of Mk. Apparently merely a graded set, UM becomes an E*-algebra for suitable JMB, DCJ, WSW - 2 - 23 Feb 1995 x1. Introduction E, thanks to extra structure on the spaces E_n. Then an unstable structure on M is a homomorphism aeM : M ! UM of E*-algebras, which selects for each x 2 Mk the function aeM (x): E*(E_k) ! M; then we define r(x) = aeM (x)r. This is not enough; in order to compose operations correctly, it is necessary to know t* *he E-cohomology homomorphism r*: E*(E_m ) ! E*(E_k) induced by each operation r: Ek(-) ! Em (-). This extra structure makes the functor U a comonad, and (M; aeM ) a coalgebra over this comonad. We have a similar construction for add* *itive operations, and can compare with the stable constructions of [8]. This is our elegant but extremely terse answer, and we do not believe that * *it can be efficiently expressed without using comonads. But it does have the effe* *ct that the work consists largely of definitions. In x10, we translate this answe* *r into practical language, in the context of Hopf rings, that we can use for computati* *on. This includes Cartan formulae for r(x+y) as well as r(xy), and related formulae* * for r*(b*c) and r*(bOc) that we use to compute the induced E-homology homomorphism r*: E*(E_k) ! E*(E_m ) dual to r*. Landweber filtrations We recall that BP *= BP *(T ), the BP -cohomology of the one-point space T , is the polynomial ring Z(p)[v1; v2; v3: :]:, with deg(vn) =* * -2(pn-1) (under our degree conventions). It contains the well-known ideals In = (p; v1; v2; : :;:vn-1) BP * (1:1) for 0 n 1 (with the convention that I1 = (p; v1; v2; : :):, I1 = (p), and I0 * *= 0). The significance [8, Lemma 15.8] of In is that it is invariant under the ac* *tion of the stable operations on BP *(T ). Indeed, Landweber [15] and Morava [20] show* *ed that the In for 0 n < 1 are the only finitely generated invariant prime ideals* * in BP *. Landweber used this fact to show (see [16, Thm. 3:30] or [8, Thm. 15.11])* * that a stable (co)module M that is finitely presented as a BP *-module, including BP* * *(X) for any finite complex X, admits a finite filtration by invariant submodules 0 = M0 M1 M2 : : :Mm = M (1:2) in which each quotient Mi=Mi-1is generated (as a BP *-module) by a single eleme* *nt xi whose annihilator ideal Ann (xi) = Ini for some ni. Thus Mi=Mi-1~= BP *=Ini. The first unstable result on BP -cohomology, due to Quillen [22] (see Thm. * *20.2), was that for a finite complex X, BP *(X) is generated, as a BP *-module, by ele* *ments of non-negative degree. What started this project was the observation that if * *an unstable object M is generated by a single element x, there is an unstable oper* *ation (see Prop. 1.14 or the Remark following Cor. 20.9) that takes vnx to x, provided deg (x) is small enough; it follows that vnx 6= 0 and that M cannot be isomorph* *ic to BP *=In+1. The proof of Landweber's theorem depends on the concept of primitive elemen* *t in a comodule M. Given any x 2 M, there is the obvious homomorphism of BP *-modules f: BP *! M, defined by fv = vx. It is a morphism of stable modules if and only if x is primitive, and if so, we have the isomorphism BP *=Ann (x) ~=(BP *)x M* * of stable modules. An important example (see [8, Thm. 15.10]) is that the only non* *zero primitives in BP *=In, for n > 0, are the (images of the) elements vin, where * *2 Fp, JMB, DCJ, WSW - 3 - 23 Feb 1995 Unstable cohomology operations 6= 0, and i 0. For additive unstable operations, the appropriate definition * *of primitive becomes more restrictive. Theorem 1.3 (included in Thm. 20.10) Let M be the BP *-module generated by a single element x with Ann (x) = In, where n > 0. Then M admits an additively unstable module structure (as defined in x5) if and only if deg(x) f(n) - 2, a* *nd it is unique. The only nonzero primitive elements in M are those of the form vinx, where 2 Fp, and deg(vinx) f(n) if i > 0. Here, and everywhere, we need the numerical function |deg(vn)| 2(pn - 1) n-1 n-2 f(n) = ________ = _________= 2(p + p + : :+:p + 1) (1:4) p - 1 p - 1 for n > 0; it is reasonable to define also f(0) = 0. We use this result in Thm. 20.11 to construct a Landweber filtration (1.2) * *of an appropriate module M, including BP *(X) for any finite complex X, in which each quotient Mi=Mi-1has the form in Thm. 1.3 (or is BP *-free). Once our machinery * *is in working order, we are able to give a one-line proof of Thm. 20.3, the weak f* *orm of Quillen's theorem. In our main structure theorem, we do one better by allowing all unstable op- erations instead of only the additive ones. One complication is that the unsta* *ble analogue of Thm. 1.3 has to be stated for algebras only, owing to the nonexiste* *nce of unstable modules. Theorem 1.5 (stated precisely as Thm. 21.12) Let M be an unstable BP *-algebra such as BP *(X) for a finite complex X. Then M admits a filtration (1.2) by in- variant ideals Mi, in which each quotient Mi=Mi-1 is generated, as a BP *-modul* *e, by a single element xi such that Ann (xi) = Ini for some ni 0, where deg(xi) max (f(ni)-1; 0). Splittings of BP -cohomology Another application of our machinery yields idem- potent operations that split unstable BP -cohomology into indecomposable pieces. Such splittings were constructed in [26] by means of Postnikov systems. What is* * new is that explicit definitions of everything allow us to carry out computations. * *Our re- sults are logically independent of [26] and rely on it only to recognize the su* *mmands as known objects; nevertheless, it is a valuable guide as to what the summands * *look like and where to find them. In a sequel [9], two of the authors go on to apply the structure theorems of [25] to establish analogous (but slightly different) * *splitting theorems for the cohomology theory P (n)*(-), whose coefficient ring is BP *=In. For each n 0, we define the ideal Jn = (vn+1; vn+2; vn+3; : :): BP *: (1:6) In [26], Baas-Sullivan theory [2] was used to construct a cohomology * * the- ory BP *(-) having coefficients BP *=Jn ~= Z(p)[v1; v2; : :;:vn]. In part* *icular, BP <0>*(-) = H*(-; Z(p)). The desired splitting is Y j BP k(X) ~=BP k(X) BP k+2(p -1)(X) : (1:7) j>n JMB, DCJ, WSW - 4 - 23 Feb 1995 x1. Introduction The representing spectrum BP is (at least) a BP -module spectrum, and comes equipped with a canonical map of BP -module spectra that we shall call ss: BP ! BP . There is also a canonical map ss: BP ! BP whenever j > n. (Geometrically, BP allows more singularities than BP .) Everythin* *g we need to know about BP is contained in the commutative diagram vj ss BP__k+2(pj-1)______-BP__k __________BP__k- | | j3 | | j |ss |ss jj | | j ss (1:8) | | j |? v |? j j BP__k+2(pj-1)_____BP__k- of H-spaces and H-maps, where j > n. Although the cohomology theory BP *(-) may be unfamiliar, in the range of degrees of interest it is easily described in terms of BP -cohomology. It is c* *lear by construction that ss*: BP *(X) ! BP *(X) kills JnBP *(X). Theorem 1.9 Assume that k f(n+1), where n 0, and that X is finite- dimensional. Then ss induces a natural isomorphism of BP *-modules OE X j BP k(X) vjBP k+2(p -1)(X) ~=BP k(X) : (1:10) j>n We derive this below as an immediate consequence of Thm. 1.12. It is best p* *os- sible, as [26] shows that ss* is not surjective in general for k > f(n+1). Lemma 1.11 (included in_Lemma 22.1) Given k < f(n+1), where n 0, there is an H-space splitting n: BP__k! BP__kof ss: BP__k! BP__kwhich naturally embeds BP k(X) BP k(X) as a summand (as abelian groups). If also k f(n), the H-space BP__kdoes not decompose further. _ Remark The splittings n are not canonical or unique. The ideal Jn, unlike In* *, is in no way canonical, but depends on the choice of the polynomial generators of * *BP *. Although the BP -module structure of BP obviously depends on Jn, it follows* * from the Lemma that the resulting H-space structure_on BP__kis well defined. Even* * for fixed Jn, we find there are many choices for n, and no preferred choice is app* *arent. We establish Lemma 1.11 in x22 by constructing a suitable idempotent operat* *ion n on BP *(-). The second assertion implies that the first is best possible. We * *insert these splittings into diag. (1.8) to decompose BP -cohomology. Theorem 1.12 Assume n 0. Then: _ _ (a) For k < f(n+1), the injections n and vjO jfrom Lemma 1.11 induce the natural abelian group decomposition (1.7), which is maximal if k f(n); (b) For k = f(n+1), we have instead the natural short exact sequence of abel* *ian groups Y j ss* 0 --! BP k+2(p -1)(X) --! BP k(X) ----! BP k(X) --! 0; (1:13) j>n JMB, DCJ, WSW - 5 - 23 Feb 1995 Unstable cohomology operations where none of the groups decomposes_further naturally, and ss* admits a non- additive natural splitting n: BP k(X) ! BP k(X), so that we have eq. (1.7) * *as a bijection of sets. Remark The simplified description of BP <->-cohomology in Thm. 1.9 applies ev- erywhere (when X is finite-dimensional). These splittings definitely do not pre* *serve the BP *-module structure. We plan to return to this point in future work. Proof of Thm. 1.9 For finite-dimensional X, the sum in eq. (1.10) is in fact f* *inite. It is clear from eq. (1.7) or (1.13) that the sum contains Ker ss*. On the othe* *r hand, ss* is a homomorphism of BP *-modules which kills Jn. | Projection to the first factor of the product in eq. (1.7) yields an intere* *sting oper- ation 0 k0 r: BP k(X) --! BP k (X) BP (X); where k0= k+2(pn+1-1) = k+(p-1)f(n+1), which roughly has the effect of dividing 0 k0 by vn+1. Precisely, r(vn+1y) = y whenever y 2 BP k (X) BP (X). Given 0 k0 any x 2 BP k(X), we can put y = n+1x; then by Thm. 1.12(a), applied to BP (X), we have y x mod Jn+1. For convenience, we reindex. n-1) Proposition 1.14 If k pf(n), there is an operation r: BP k-2(p (X) ! BP k(X), which is additive if k < pf(n), with the property that given any eleme* *nt x 2 BP k(X), where X is finite-dimensional, there exists y such that y x mod JnBP * **(X) and r(vny) = y. | Equivalently, we can represent eq. (1.7) by the decomposition of spaces Y BP__k' BP__kx BP__k+2(pj-1): (1:15) j>n Theorem 1.16 Assume n 0. Then: (a) For k < f(n+1), we have the H-space decomposition (1.15), which is maxim* *al if k f(n); (b) For k = f(n+1), we have the fibration Y ss BP__k+2(pj-1)--!BP__k---! BP__k (1:17) j>n of H-spaces and H-maps, which admits a section (not an H-map), so that eq. (1.1* *5) holds as an equivalence of spaces (but not as H-spaces), and none of the spaces decomposes further as a product of spaces. (In other words, BP k(-) is represen* *ted by the right side of eq. (1.15), equipped with a different H-space multiplicati* *on.) We use Lemma 1.11 to prove parts (a) of Thms. 1.12 and 1.16 in x22. For par* *ts (b), the necessary idempotent n has to be nonadditive, and we construct it in x* *23. We need the full strength of our machinery just to prove that n is idempotent. History Our real motivation for this study is what is called the Johnson Quest* *ion, which is stated in [24, p. 745]. Rephrased as a conjecture, it is: JMB, DCJ, WSW - 6 - 23 Feb 1995 x1. Introduction Conjecture If x 6= 0 in BPn(X), where X is a space, then vinx 6= 0 for all * *i > 0. No counter-examples are known, although examples exist [13, 14, 24] where vjx = 0 for all j < n. It holds if x reduces nontrivially to homology, therefo* *re for n < 2p. We hoped to circumvent our lack of knowledge of unstable homology opera- tions by working instead with the rather better understood unstable BP -cohomol* *ogy operations and using the (not at all unstable) duality spectral sequence Ext**BP*(BP *(X); BP *) =) BP*(X) of Adams [1] (see also [12]). The reason for optimism is that if we substitute k(BP *=In) for BP *(X), a standard calculation shows that the only surviving Ext group is Extn = m (BP *=In), with m = f(n) - k - n; so that k f(n) - 1 implies -m n - 1, almost what we want. If we confine ourselves to additive operations, we obtain -m n - 2, off by one more. We can hope to work our way up from k(BP *=In) to a general BP *(X) by extension and the filtration (1.2). This is all grounds for our suspicion that for a geometric unstable algebra* *, i. e. M = BP *(X) for some space X, the bounds in Thm. 1.5 should be one better (thus giving us -m n in the above discussion). Again, there are no known counter- examples, although spaces are known which have deg(xi) = f(ni), thus showing th* *at the bounds cannot be improved by more than one. Recently, with the help of Mike Hopkins, a new approach to the Johnson Ques* *tion has been developed. It requires a much better understanding of the unstable spl* *ittings of BP . Now that we have so much explicit information on these splittings, this* * method of attack seems promising. Outline There are two main threads running through this work: the theory of additive unstable operations, which closely resembles the stable theory of [8],* * and the theory of all unstable operations, which is radically different. The comonad t* *ent is big enough to accommodate both, as well as the stable theory. We have kept the additive material in separate sections so that it can be read independently. In x2, we discuss several classes of cohomology operation. In xx3, 4, we st* *udy the E-(co)homology of group objects, in preparation for xx5, 7, where we study modu* *les and algebras from the additive point of view. In x6, we consider additive opera* *tions as linear functionals. In xx12, 14, we study suspensions and complex orientatio* *n. In x16, we present the additive structure for each of our five examples E. It turns out that much of the stable machinery does not extend to all unsta* *ble operations, because it relies too heavily on the bilinearity of tensor products* *. However, the approach in terms of comonads does work, and in x8 we develop the requisite comonad U. We also show in x9 that the corresponding comonad for unstable modul* *es does not exist and compare the various stable and unstable structures. In x10,* * we convert the categorical elegance into machinery we can use; specifically, cohom* *ology operations become linear functionals on Hopf rings. In Thm. 10.47, we display i* *n full detail the definition of an unstable algebra from this point of view. In xx11, 13, 15, we revisit the cohomology of a point, sphere, and complex * *pro- jective space C P 1 from this new Hopf ring point of view. These spaces alone y* *ield almost enough generators and relations to specify the Hopf rings for our five e* *xamples JMB, DCJ, WSW - 7 - 23 Feb 1995 Unstable cohomology operations E, as we discuss in detail in x17. The case E = KU is used to determine the str* *ucture of KU*(KU; o), as quoted in [8, x14]. From a sufficiently elevated perspective* *, the results of x17, the additive results of x16, and the stable results of [8] all * *fit into a grand master plan. In x20, we restrict attention to the case E = BP and use the additive opera* *tions to recover Quillen's theorem and prove Thm. 20.11. This relies on the relations developed in x18. In x21, we use nonadditive operations to improve Thm. 20.11 by one dimension to Thm. 21.12, which is Thm. 1.5. In x22, we construct additive idempotent operations n which yield the desir* *ed fac- torizations (1.7) in all except the top degree. In x23, we finish off Thms. 1.1* *2 and 1.16 by constructing nonadditive idempotent operations. To do this, it is necessary * *in x19 to develop the notion of a Hopf ring ideal. An index of symbols is included at the end. This work is also notable for what it does not contain. There are no spectr* *al se- quences, except implicitly in the references. There are no explicit Steenrod op* *erations, except in a few examples; in our wholesale approach, most individual operations* * never even acquire names. There are no formal indeterminates anywhere; the elements t* *hat are sometimes treated as such are really Chern classes x; but when xi= 0, we ca* *n no longer take the coefficients of xi. Notation We make heavy use of the notation and machinery developed in [8]. Topologically, we generally work in the homotopy category Ho of unbased spaces.* * For compatibility with the unstable notation, the E-cohomology and E-homology of a spectrum X are written E*(X; o) and E*(X; o). Algebraically, our most important categories are the categories FMod and FAlg of filtered E*-modules and algebr* *as. These and the other categories we need were introduced in [8, x6]. We make freq* *uent use of Yoneda's Lemma. All tensor products are taken over E* unless otherwise stated. For reasons discussed in [8], we always give cohomology E*(X) the profinite* * topol- ogy [8, Defn. 4.9], and complete it as in [8, Defn. 4.11] to E*(X)^ as necessar* *y. In contrast, the homology E*(X) is always discrete. Because we emphasize cohomolog* *y, we invariably assign the degree i to elements of Ei(X); this forces elements of* * Ei(X) to have degree -i. One theorem provides all the duality and K"unneth isomorphisms we need. Theorem 1.18 Assume that E*(X) is a free E*-module. Then we have: (a) d: E*(X) ~=DE*(X) in FMod , the strong duality homeomorphism; (b) E*(X xY ) ~=E*(X) E*(Y ), the K"unneth isomorphism in homology; (c) E*(X xY ) ~=E*(X) b E*(Y ) in FMod , the K"unneth homeomorphism in co- homology, provided E*(Y ) is also a free E*-module. Proof We collect Thms. 4.2, 4.14, and 4.19 from [8]. Indeed, (c) follows from* * (a) and (b). | Acknowledgements The genesis of this paper is that the last two authors had worked out much of the unstable BP structure theorems, without having a precise JMB, DCJ, WSW - 8 - 23 Feb 1995 x2. Cohomology operations definition of unstable algebra, when the first author supplied a suitable frame* *work, of which [7] is an early version. In fact, this is an oversimplification: the* * various contributions are more intermingled than this might suggest. In the proper cont* *ext, several of the proofs simplify significantly. We thank Martin Bendersky for poi* *nting out Lemma 19.32, which is vastly simpler than our previous treatment. The last two authors wish to thank the topologists at the University of Man* *chester and the Science and Engineering Research Council (SERC) for support during the summer of 1985 when this project got its start. The last author wishes to thank Miriam and Harold Levy for their hospitality and the peace they provided for the writing of the first draft. 2 Cohomology operations In this section, we consider several kinds of unstable cohomology operatio* *n. Yoneda's Lemma allows us to identify the following: (i)The cohomology operation r: Ek(-) ! Em (-); (ii)The cohomology class r = r(k) 2 Em (E_k); (2.1) (iii)The representing map r: E_k! E_m in Ho . We write any of these more succinctly as r: k ! m. We use all three interpretat* *ions. Some care is needed with degrees and signs, as (i) has degree m-k and (ii) has * *degree m, while (iii) has no degree at all. Based operations The following mild but useful condition can be interpreted ma* *ny ways. The space T is the one-point space. Definition 2.2 We call the operation r based if r(0) = 0 in E*(T ) = E*. Lemma 2.3 The following conditions on an operation r: k ! m are equivalent: (a) r(0) = 0 in E*(T ), i. e. r is a based operation; (b) For any based space (X; o), r restricts to the reduced operation r: Ek(X; o) --! Em (X; o); (2:4) (c) As a cohomology class, r 2 Em (E_k; o) Em (E_k); (d) The map r: E_k! E_m is (homotopically) based. Proof The short exact sequence [8, eq. (3.2)] shows that (a) and (c) are equiv* *alent, also that (a) implies (b); but (c) is the special case of (b) for k 2 Ek(E_k; o* *). Part (d) is just a restatement of (a). | Given any (good) pair of spaces (X; A), we can use (b) to make based operat* *ions r: k ! m act on relative cohomology as in [8, eq. (3.4)] by r m m Ek(X; A) = Ek(X=A; o) --! E (X=A; o) = E (X; A) : (2:5) Additive operations An additive operation r: k ! m is one that satis* *fies r(x+y) = r(x) + r(y) for any x; y 2 Ek(X). The universal example is X = E_kx E_k; with x = k x 1, y = 1 x k, x + y = k, (2:6) JMB, DCJ, WSW - 9 - 23 Feb 1995 Unstable cohomology operations which gives r(k) = rx1+1xr in E*(E_kxE_k). (The addition map k: E_kxE_k ! E_k was defined in [8, Thm. 3.6].) This allows us to recognize additive operations * *three ways. Proposition 2.7 The following conditions on an operation r: k ! m are equiva* *lent, and define the E*-submodule P E*(E_k) E*(E_k; o) E*(E_k): (a) The operation r: Ek(-) ! Em (-) is additive; (b) The class r 2 Em (E_k) satisfies *kr = p*1r + p*2r in Em (E_kxE_k), i. e. P E*(E_k) = Ker[*k- p*1- p*2: E*(E_k) --! E*(E_k x E_k)]; (2:8) (c) The map r: E_k! E_m is a morphism of group objects in Ho . | Corollary 2.9 Assume that E*(E_k) is a free E*-module. Then P E*(E_k) is com- plete Hausdorff and so an object of FMod . Proof In eq. (2.8), E*(E_k) and E*(E_kxE_k) are complete Hausdorff by Thm. 1.1* *8. | When E*(E_k) is free, the K"unneth homeomorphism for E*(E_kxE_k) makes E*(E_k) a completed Hopf algebra; then (b) agrees with the primitives in the se* *nse of [8, eq. (6.13)], completed. However, we need no hypotheses on E to define P E*(* *E_k). On some spaces, all operations are additive. Lemma 2.10 On the suspension X of any based space (X; o), we have r(x+y) = r(x) + r(y) in Em (X; o) for any based operation r: k ! m and any elements x; y* * 2 Ek(X; o). Proof By [8, Lemma 7.6(c)], r: Ek(X; o) ! Em (X; o) preserves the group struc- ture defined from the cogroup object X in Ho 0. By [8, Prop. 7.3], this struct* *ure coincides with the given E-cohomology addition. | Products of operations Given operations r: k ! m and s: k ! n, the product operation r ^ s: k ! m + n, defined by (r ^ s)x = (rx)(sx), corresponds to the cup product in E*(E_k), which may be constructed using the diagonal map : E_k! E_k x E_k. We often wish to neglect such operations; if r and s are additive, r* * ^ s is clearly not additive, but conveys no new information. The map , together with q: E_k! T , makes E_ka monoid object in the symmetr* *ic monoidal category (Ho op; x; T ). We therefore dualize eq. (2.8) and introduce* * the quotient E*-module QE*(E_k) = Coker[* - i*1- i*2: E*(E_k x E_k) --! E*(E_k)] (2:11) of "indecomposables" of E*(E_k), where i1 and i2 are the inclusions (using the * *base- point). (We shall not need a topology on this module.) When E*(E_k) is a free E*-module, we have by Thm. 1.18(c) a K"unneth homeomorphism for E*(E_kxE_k), and QE*(E_k) is the quotient of E*(E_k; o) by all finite (or infinite) sums of * *products of two based operations. Looping of operations On restriction to spaces, a stable operation r on E*(-; * *o) of degree h induces a sequence of additive operations rk: k ! k+h. It is clear * *from [8, fig. 2 in x9] that rk+1 determines rk. We generalize this construction to u* *nstable operations (but omit the sign, in order to make it a homomorphism of E*-modules* *). JMB, DCJ, WSW - 10 - 23 Feb 1995 x2. Cohomology operations Proposition 2.12 Given a based unstable operation r: k ! m, we can define the looped operation r: k-1 ! m-1 in any of three equivalent ways: (a) The operation that makes the diagram commute (with no sign), ~= ~= Ek-1(X) ________-Ek(S1xX; oxX) ______oEk((X+e); o) pp pp | | pp |r |r ppr | | (2:13) p? ||? ||? ~= ~= Em-1 (X) ________-Em (S1xX; oxX) ______oEme((X+ ); o) which we can express algebraically as (r)x = rx; (2:14) (b) The image of r under the E*-module homomorphism (-1)k-1f*k-1 : Em (E_k; o) --------! Em (E_k-1; o) ~=Em-1 (E_k-1; o) induced by the structure map fk-1: E_k-1 ! E_kof [8, Defn. 3.19]; (c) The map (-1)m-kr r: E_k-1' E_k --------! E_m ' E_m-1; where we use the right adjunct equivalences to fk-1 and fm-1 . Proof For a based space X, diag. (2.13) simplifies by naturality to ~= Ek-1(X; o) ______-Ek(X; o) pp ppr |r (2:15) ? ~ |? = Em-1 (X; o)______-Em (X; o) If we evaluate on the universal case k-1 2 Ek-1(E_k-1; o) by eq. (2.14), we find (r)k-1 = rk-1 = (-1)k-1rf*k-1k = (-1)k-1f*k-1r; which gives (b). Further, by [8, Lemma 3.21], the class (r)k-1 2 E*(E_k-1; o) corresponds, up to the sign (-1)m-1 , to the lower route in the square fk-1 E_k-1 _________E_k- | | | (r) || (-1)m-k |r in Ho (2:16) | |? fm-1 |? E_m-1 ________-E_m which therefore commutes up to sign. We take adjuncts of this to get (c). | We recall from [8, Defn. 9.3] the stabilization map oek: E_k! E of spectra. JMB, DCJ, WSW - 11 - 23 Feb 1995 Unstable cohomology operations Corollary 2.17 Ooe*k= oe*k-1: E*(E; o) ! E*(E_k-1; o): Proof Suppose the stable operation r 2 Eh(E; o) restricts to give the additive* * op- erations rk: k ! k + h and rk-1: k-1 ! k+h-1. By [8, eq. (9.8)], oe*kr = (-1)kh* *rk and oe*k-1r = (-1)(k-1)hrk-1. We compare diag. (2.16) with [8, eq. (9.2)] to s* *ee that rk = (-1)hrk-1. | Corollary 2.18 The loop construction in Prop. 2.12(b) factors as : E*(E_k; o) --! QE*(E_k) --! P E*(E_k-1) E*(E_k-1; o) : (2:19) Proof It is clear from Prop. 2.12(c), or from eq. (2.14) and Lemma 2.10, that * *r is always additive. The construction factors through QE*(E_k) by Prop. 2.12(b) and naturality of Q, since QE*(E_k-1) ~=E*(E_k-1; o). (Loosely, there are no produc* *ts in E*(X; o).) | These results allow us to rewrite the Milnor short exact sequence [8, eq. (* *9.7)] in the more useful form (which does not change any terms) 0 --! lim1P E*(E_k) --! E*(E; o) --! limP E*(E_k) --! 0 : (2:20) k k It remains true that the projection from E*(E; o) is an open map, and therefore* * a homeomorphism whenever it is a bijection. The kth component is the E*-module homomorphism oe*k: E*(E; o) --! P E*(E_k) E*(E_k) (2:21) induced by the stabilization map oek. It sends a stable operation r to the ind* *uced additive operation rk on Ek(-) (but with a sign; see [8, eq. (9.9)]). The factorization (2.19) raises two obvious questions: (a) Can every additive operation be delooped? (2.22) (b) Does r = 0 imply that r decomposes? Both hold precisely when we have an isomorphism : QE*(E_k) ~=P E*(E_k-1). We discuss this further in x4. 3 Group objects and E-cohomology Before we can discuss additive E-cohomology operations adequately, it is ne* *cessary to generalize x2. We extend Prop. 2.7 by defining the primitives P E*(X) for a* *ny group object X in the homotopy category Ho. Dually, we extend the definition of* * the indecomposables QE*(X) to any based space X. Coalgebra primitives We start from the definition (2.8) of P E*(E_k). Definition 3.1 Given any group object (or H-space) X in Ho , with multiplicat* *ion : X xX ! X, we define the E*-submodule P E*(X) of coalgebra primitives in E*(X) as P E*(X) = {x 2 E*(X) : *x = p*1x + p*2x in E*(X xX) }: JMB, DCJ, WSW - 12 - 23 Feb 1995 x3. Group objects and E-cohomology Remark As in Prop. 2.7(c), the class x 2 Ek(X) is primitive if and only if the associated map x: X ! E_kis a morphism of group objects in Ho . We note that P E*(X) is defined even if E*(X) is not a (completed) coalgebr* *a. Thus P E*(-): Gp(Ho )op ! Mod is a functor defined on the dual of the category* * of group objects in Ho . We topologize P E*(X) as a subspace of E*(X). If Y is another group object in Ho, we construct the product group object X* * x Y in the obvious way. The one-point space T is trivially a group object, and is t* *erminal in Gp(Ho ). Lemma 6.14 of [8] carries over to this situation. Lemma 3.2 For the product X x Y of two group objects X and Y in Ho , we have P E*(X xY ) ~=P E*(X) P E*(Y ) in FMod . Also, P E*(T ) = 0. In other words, the functor P E*(-) takes finite products in Gp(Ho ) to cop* *roducts (direct sums) in FMod . Remark No K"unneth formula is needed for this result. Proof We dualize the proof of [8, Lemma 6.11]. Let us write Z = X x Y for the product group object and !Y : T ! Y for the unit (or zero) map of Y . We note f* *irst that the maps j1 = 1X x!Y : X ~=X xT ! X xY = Z, j2: Y ! Z (defined similarly), p1: Z = X xY ! X, and p2: Z ! Y are all morphisms of group objects and therefore send primitives to primitives. Define the map f: Z = X x Y ~= (X xT ) x (T xY ) --! (X xY ) x (X xY ) = Z x Z using (1X x!Y ) x (!X x1Y ). Then Z Of = 1Z and PsOf = jsOps (for s = 1; 2), where Ps: Z xZ ! Z denotes the projection for Z. Any element z 2 P E*(Z) satisf* *ies *Zz = P1*z + P2*z, by definition. When we apply f*, we obtain z = p*1x + p*2y, * *where x = j*1z 2 E*(X) and y = j*2z 2 E*(Y ) must be primitive. Conversely, any primi* *tives x and y determine a primitive z by this formula. We have a homeomorphism because j*sand p*sare continuous. We compute P E*(T ) = {v 2 E* : v = v + v} = 0. | Since the unit map !: T ! X of X is a morphism of group objects, P E*(T ) =* * 0 implies that P E*(X) E*(X; o). The space E_k is more than just a group object. By [8, Cor. 7.8], we have * *the E*-module object n 7! E_nin Ho, on which v 2 Eh acts by the maps v: E_k! E_k+h that represent scalar multiplication by v. Clearly, v is additive. Lemma 3.3 Assume that E*(E_k) is Hausdorff for all k. Then: (a) We have the E*-module object n 7! P E*(E_n) in the ungraded category FMod op, with the action of v 2 Eh given by P (v)*: P E*(E_k+h) ! P E*(E_k); (b) The object in (a) is related to the stable E*-module object E*(E; o) of * *[8, Prop. 11.3] by the following diagram, which commutes up to sign for any v 2 Eh, (v)* E*(E; o) ________E*(E;-o) | | | ||oe*k(-1)hk+h ||oe*k (3:4) | | |? P(v)* |? P E*(E_k+h)______-P E*(E_k) JMB, DCJ, WSW - 13 - 23 Feb 1995 Unstable cohomology operations Proof In (a), the object n 7! E_nis in fact an E*-module object in Gp(Ho ). We* * apply [8, Lemma 7.6(a)] to the functor P E*(-); it preserves finite products by Lemma* * 3.2. For (b), we apply E-cohomology to diag. [8, eq. (9.8)], taking r = v. | Indecomposables Dually, we extend eq. (2.11) to any based space X by defining the quotient E*-module QE*(X) = Coker[* - i*1- i*2: E*(X xX) --! E*(X)] (3:5) of "indecomposables" of E*(X). (We shall not need a topology on it.) 4 Group objects and E-homology We dualize x2 by defining the indecomposables QE*(E_k) and primitives P E*(* *E_k) in E-homology. This will prove useful because E*(E_k) is usually smaller and m* *ore manageable than E*(E_k). As in x3, we need to handle more general X. However, some properties that were immediate in x2 become less intuitive and have to be proved. The structure map fk: E_k ! E_k+1 (see [8, Defn. 3.19]) of the spectrum E induces the important suspension homomorphism fk* E*(E_k) --! E*(E_k; o) ~=E*(E_k; o) ---! E*(E_k+1; o); (4:1) dual (apart from sign) to the looping in Prop. 2.12(b). Again, suspended eleme* *nts behave better. We dualize Lemma 2.10. Lemma 4.2 For any elements x; y 2 Ek(X; o), the induced E-homology homomor- phisms satisfy (x + y)* = x* + y*: E*(X; o) --! E*(E_k; o) : Proof By [8, Lemma 7.6(c)], E-homology induces a homomorphism Ho0(X; E_k) --! Mod (E*(X; o); E*(E_k; o)) of groups, where both group structures are induced by the cogroup structure on X in Ho 0. By [8, Prop. 7.3], they agree with the obvious group structures. | Indecomposables We dualize Defn. 3.1. Definition 4.3 Given any group object (or H-space) X in Ho , we define the E*- module QE*(X) of "indecomposables" of E*(X) as QE*(X) = Coker[* - p1*- p2*: E*(X xX) --! E*(X)]: It comes equipped with a canonical projection E*(X) ! QE*(X). When E*(X) is free, we have the K"unneth isomorphism Thm. 1.18(b)* * for E*(X xX) and this agrees with the usual definition for the algebra E*(X). We ne* *ed one easy example. Lemma 4.4 Let G be a discrete abelian group. Then QE*(G) ~= E* Z G as an E*-module. JMB, DCJ, WSW - 14 - 23 Feb 1995 x4. Group objects and E-homology Proof We recognize E*(G) as the group algebra of G over E*, with an E*-basis element [g] for each g 2 G. The correspondence we seek is induced by v[g] $ v * *g, and is well defined in both directions. | Lemma 3.2 dualizes without difficulty; again, no K"unneth formula is needed* *. Then we will be able to dualize Lemma 3.3. Lemma 4.5 For the product X x Y of two group objects X and Y in Ho , we have QE*(X xY ) ~=QE*(X) QE*(Y ). Also, QE*(T ) = 0. In other words, the functor QE*(-): Gp(Ho ) ! Mod preserves finite products. | We have an immediate application to the Hopf bundle. Lemma 4.6 Assume E has a complex orientation. Then the inclusion C P 1 ! Z x BU (see [8, eq. (5.8)]) defined by the Hopf line bundle over C P 1 induces* * an isomorphism of E*-modules ~= E*(C P 1) --! QE*(Z x BU) ~=E* QE*(BU) : Proof The second isomorphism comes from Lemmas 4.5 and 4.4. We compare Lem- mas 5.4 and 5.6 of [8]; the generators fii corrrespond, except that fi0 7! (1; * *0). | Lemma 4.7 For any ring spectrum E: (a) n 7! QE*(E_n) is an E*-module object in the ungraded category Mod of E*- modules; (b) The suspension (4.1) factors through QE*(E_k); (c) The stabilization oek*: E*(E_k; o) ! E*(E; o) factors through QE*(E_k). Proof The proof of (a) is like Lemma 3.3(a), except that we use the functor QE* **(-) and Lemma 4.5. For (c), we use oe*k = k to restate the universal example (2.6) as oekOk = oekOp1 + oekOp2: E_kx E_k--! E in Stab*. We apply E-homology to see that oek* factors as desired. Similarly for (b), ex* *cept that we use Lemma 4.2 with X = (E_kxE_k), x = p1, and y = p2. | Dually to the short exact sequence (2.20), we may use (b) and (c) to rewrit* *e [8, eq. (9.22)] in the more convenient form E*(E; o) = colimE*(E_k; o) = colimQE*(E_k) : (4:8) k k There is a multiplication, analogous to the stable multiplication on E*(E; * *o). Lemma 4.9 There is a bilinear multiplication QOE: QE*(E_k) QE*(E_m ) --! QE*(E_k+m ); which may be defined as a quotient of x OE* E*(E_k) E*(E_m ) --! E*(E_kxE_m ) --! E*(E_k+m ) : JMB, DCJ, WSW - 15 - 23 Feb 1995 Unstable cohomology operations Proof The only difficulty is to prove that QOE is well defined. We express th* *e dis- tributive law for the E*-algebra object n 7! E_nas the commutative square OEL E_k x E_kx E_m ______E_k+m-x E_k+m |fx1 |g (4:10) |? OE |? E_k x E_m ____________-E_k+m in which f = k, g = k+m , and OEL has the components OE O(p1 x 1) and OE O(p2 x* * 1). (Cohomologically, OEL represents the operation (x; y; z) 7! (xz; yz).) We deduc* *e the commutative diagram in homology x OEL* E*(E_kxE_k)E*(E_m ) ______-E*(E_kxE_kxE_m ) ________-E*(E_k+m xE_k+m ) |f 1 |(fx1) |g* | * | * | |? x |? OE |? * E*(E_k) E*(E_m ) __________-E*(E_k x E_m)_____________-E*(E_k+m ) (4:11) By Defn. 4.3, we have the exact sequence k*-p1*-p2* E*(E_k x E_k) ---------! E*(E_k) --! QE*(E_k) --! 0 : After tensoring with E*(E_m ), this remains exact. We note that diag. (4.10) an* *d hence diag. (4.11) also commute if we take f = p1 and g = p1, or f = p2 and g = p2. T* *hen diag. (4.11), with these three choices for f and g, shows that its bottom row i* *nduces a quotient pairing QE*(E_k) E*(E_m ) ! QE*(E_k+m ). A second similar step, on the right, uses this pairing to produce QOE. | Coalgebra primitives We also dualize eq. (3.5) in the obvious way. If X is a b* *ased space, we construct the E*-module homomorphism * - i1*- i2*: E*(X) --! E*(X x X) : (4:12) Definition 4.13 Given any based space X, we define the E*-submodule of coalge* *bra primitives P E*(X) = Ker[* - i1*- i2*] E*(X). Again, the definition is meaningful even without a K"unneth formula for E*(* *XxX). The companion result to Lemma 4.4 is elementary. Proposition 4.14 For any discrete based space X, we have P E*(X) = 0. | The suspension (4.1) factors, with the help of Lemma 4.7(b), as E*(E_k; o) --! QE*(E_k) --! P E*(E_k+1) E*(E_k+1; o) : (4:15) Again we ask whether QE*(E_k) ! P E*(E_k+1) is an isomorphism. Duality Under reasonable assumptions, the sequence (2.19) is dual to (4.15). O* *ne can see from Lemma 4.17 and x17 that this holds for each of our five examples E. Moreover, in each case there are isomorphisms QE*(E_k) ~=P E*(E_k+1) in (4.15),* * thus answering the questions (2.22) affirmatively. JMB, DCJ, WSW - 16 - 23 Feb 1995 x5. What is an additively unstable module? Lemma 4.16 Assume that E*(X) is a free E*-module. (a) If X is a group object in Ho (or an H-space), then d induces a homeomorp* *hism d: P E*(X) ~=DQE*(X) in FMod ; (b) If X is a based space and the image of the homomorphism (4.12) splits off both E*(X) and E*(X xX), then d induces a bijection d: QE*(X) ~=DP E*(X). Proof In (a), d induces the commutative diagram *-p*1-p*2 0 __________-P E*(X) _______-E*(X) ______________________-E*(X xX) |d |d |d |? |? D(*-p1*-p2*) |? 0 _________-DQE*(X) ______DE*(X)- ____________________-DE*(X xX) whose rows are exact by Defns. 3.1 and 4.3, because D automatically takes coker* *nels to kernels. Strong duality for X and X x X from Thm. 1.18 provides two homeomor- phisms d. The third d is therefore also a homeomorphism, because DQE*(X) has the subspace topology from DE*(X) by [8, Lemma 6.15(c)]. The proof of (b) is analogous, except that we assume the splittings to ensu* *re that the bottom row of the relevant diagram is (split) exact, use [8, Lemma 6.1* *5(a)] instead, and have no topology to check. | We clearly need information on when E*(E_k) is free. Lemma 4.17 For E = H(F p), BP , MU, K(n), or KU: (a) E*(E_k) and QE*(E_k) are free E*-modules for all k; (b) E*(E_k) and P E*(E_k) are complete Hausdorff for all k. Proof For E = H(F p) or K(n), all E*-modules are free and (a) is trivial. We consider the remaining three cases together. For odd k, E*(E_k) is an ex* *terior algebra over E* by [23] (for BP or MU) or [8, Cor. 5.12] (for KU, when E_k = U), and (a) is clear. For even k, we write E_k = Ek x E_0kas in [8, eq. (3.7)], where E_0kdenotes* * the zero component and Ek is treated as a discrete group. Then E*(E_0k) is a polyno* *mial algebra over E*, by [23] (for BP or MU) or [8, Lemma 5.6(c)] (for KU, when E_0k= BU), so that E*(E_0k) (and hence E*(E_k)) and QE*(E_0k) are free modules. To finish (a), we note that by Lemmas 4.5 and 4.4, QE*(E_k) = (E* Z Ek) QE*(E_0k) : The first summand is free, because Ek = Z (for KU), or is Z-free (for MU), or is Z (p)-free (for BP ). Part (b) is immediate from (a) by Thm. 1.18(a) and Cor. 2.9. | 5 What is an additively unstable module? In this section, we give various interpretations of what it means to have a* * module over the additive unstable operations on E-cohomology. All four stable answers * *in [8] generalize. JMB, DCJ, WSW - 17 - 23 Feb 1995 Unstable cohomology operations We recall from [8, Cor. 7.8] that each E_k is an abelian group object in Ho* * and therefore also in Gp(Ho ), and that n 7! E_nis an E*-module object in Ho, with * *v 2 Eh acting by the map v: E_k! E_k+h. From Prop. 2.7 we have the submodule P E*(E_k) of additive operations defined on Ek(-). We assume throughout that E*(E_k) is a free E*-module. Then by Cor. 2.9, P E*(E_k) is complete Hausdorff and an object of FMod . First Answer The additive operations r: k ! m act on E*(X) by composition O: P Em (E_k) x Ek(X) --! Em (X) (5:1) in Ho. We recover the stable action [8, eq. (10.1)] by using oe*k: E*(E; o) ! P* * E*(E_k). This composition is already biadditive. Given x 2 Ek(X) and v 2 Eh, the commutative square 1xv P Em (E_k+h) x Ek(X) ______P-Em (E_k+h) x Ek+h(X) | | |P(v)*x1 |O (5:2) | | |? |? O P Em (E_k) x Ek(X) ______________-Em (X) expresses the identity (r . v)x = rvx = r(vx) for operations r: k + h ! m. It s* *uggests that we should make the action (5.1) more closely resemble the stable action by introducing a formal shift and rewriting it with a tensor product as X : -kP Em (E_k) k Ek(X) --! Em (X) : (5:3) (Here, unlike [8], the action scheme is clearly visible: the notation k indicat* *es that the tensor product is to be formed using the two E*-actions indexed by k.) This approach was initiated in [27, x11]. However, it presents even more pr* *oblems than in the stable case, and we do not pursue it further here. Second Answer Our hypotheses ensure that P E*(E_k) is dual to QE*(E_k). We can convert the action of P E*(E_k) into a coaction Ek(X) --! E*(X) b QE*(E_k) : These are clearly not the components of an E*-module homomorphism, because the degree varies. In x6, as suggested by (5.3), we shall shift degrees by introducing Q(E)k*= kQE*(E_k), which will allow us to write the coaction as an E*-module homomor- phism with components aeX : Ek(X) --! E*(X) b Q(E)k* (5:4) and the same action scheme as stably. We shall construct a comultiplication Q(* * ) and counit Q(ffl) that make Q(E)**a coalgebra and allow us to interpret E*(X) a* *s a Q(E)**-comodule. Third Answer We write our Second Answer more functorially. Given any E*- module M, we construct the graded group A0M having the component (A0M)k = Mi biQ(E)ki= (M b Q(E)k*)k JMB, DCJ, WSW - 18 - 23 Feb 1995 x5. What is an additively unstable module? in degree k. In x6 we shall make A0M an E*-module. Then M Q( ) and M Q(ffl) define natural transformations 0: A0 ! A0A0 and ffl0: A0 ! I, which will make * *A0 a comonad in FMod and E*(X)^ an A0-coalgebra. Fourth Answer Still imitating the stable case, we eliminate all tensor product* *s by converting the First Answer to adjoint form. This will make everything very mu* *ch cleaner, evidence that this is the natural answer (although the Second Answer is undeniably convenient for computation). Any element x 2 Ek(X) may be regarded as a map x: X ! E_k, which induces the morphism x*: E*(E_k) ! E*(X)^ in FMod . Generally, given any object M in FMod , we define for each integer k the abelian group AkM = FMod (P E*(E_k); M) (5:5) of all continuous E*-module homomorphisms P E*(E_k) ! M. (There is no need to shift degrees.) Then we convert the action (5.1) to the coaction aeX : Ek(X) --! Ak(E*(X)^) = FMod (P E*(E_k); E*(X)^) (5:6) by defining aeX x = x*|P E*(E_k). We assemble the AkM, as k varies, to form the graded group AM with componen* *ts (AM)k = AkM, and the coactions aeX into the single homomorphism aeX : E*(X) ! A(E*(X)^) of graded groups of degree zero. The destabilization oe*k: E*(E; o) ! P E*(E_k) (see [8, Defn. 9.3]) induces AkM = FMod (P E*(E_k); M) --! FMod k(E*(E; o); M) = (SM)k; (5:7) if we also assume that E*(E; o) is Hausdorff. As k varies, we take these as the components of the stabilization natural transformation oeM: AM ! SM, of degree zero. It allows us to compare with the stable case. Theorem 5.8 Assume that E*(E_k) is a free E*-module for all k (as is true f* *or E = H(F p), BP , MU, KU, or K(n) by Lemma 4.17(a)). Then: (a) We can make the functor A, defined in eq. (5.5), a comonad in the catego* *ry FMod of complete Hausdorff filtered E*-modules; (b) If E*(E; o) is also Hausdorff, the stabilization oe: A ! S (defined in e* *q. (5.7)) is a morphism of comonads in FMod . The relevant definitions are now clear. Definition 5.9 An additively unstable (E-cohomology) module is an A-coalgebra in FMod , i. e. a complete Hausdorff filtered E*-module M equipped with a morph* *ism aeM : M ! AM in FMod that satisfies the coaction axioms [8, eq. (8.7)]. We t* *hen define the action of r 2 P Em (E_k) on x 2 Mk by rx = aeM (x)r 2 M (with no sig* *n). A closed submodule L M is called (additively unstably) invariant if aeM r* *estricts to give aeL: L ! AL. Then the quotient M=L inherits an additively unstable modu* *le structure. JMB, DCJ, WSW - 19 - 23 Feb 1995 Unstable cohomology operations This is a stronger structure than a stable module (when E*(E; o) is Hausdor* *ff, so that stable modules exist). Given a coaction aeM as above, Thm. 5.8(b) shows t* *hat the coaction aeM oeM M ---! AM ---! SM (5:10) makes M a stable module. One may think of AkM as the set of all candidates for the action of P E*(E_* *k) on a typical element of Mk, and aeM as the selection of a candidate for each x 2 * *Mk. The coaction axioms translate into the usual action axioms (sr)x = s(rx) and kx = x. As stably, it is sometimes useful to fix r: k ! m and express the first axiom a* *s the commutative square r Mk _______-Mm |ae |ae | M | M (5:11) |? !rM |? AkM ______Am-M where !rM denotes composition with P r*: P E*(E_m ) ! P E*(E_k). Theorem 5.12 Assume that E*(E_k) is a free E*-module for all k (as is true * *for E = H(F p), BP , MU, KU, or K(n) by Lemma 4.17(a)). Then: (a) aeX (defined in eq. (5.6)) factors through E*(X)^ as aeX : E*(X)^ ! A(E** *(X)^) to make E*(X)^ an additively unstable module for any space X; (b) If E*(E; o) is Hausdorff, we recover the stable coaction in [8, Thm. 10.* *16(a)] from aeX by diag. (5.10); (c) ae is universal: given an object N of FMod and an integer k, any add* *i- tive natural transformation of abelian groups X: Ek(X) ! FMod (N; E*(X)^) (or bX: Ek(X)^ ! FMod (N; E*(X)^)) that is defined on all spaces X is induced from* * aeX by a unique morphism f: N ! P E*(E_k) in FMod , as aeX k * * * X: Ek(X) ---! A (E (X)^) = FMod (P E (E_k); E (X)^) Hom(f;1) ------! FMod (N; E*(X)^) : Proof of Thms. 5.8 and 5.12 We prove parts (a) and (b) of both Theorems togeth* *er, in the same seven steps as the stable proof of Thms. 10.12 and 10.16 of [8]. As* * most steps are more or less repetitions of that proof, except for the insertion of i* *ndices everywhere, we indicate only the substantive changes for (a) and the additions * *needed to handle oe for the (b) parts. Instead of 2 E*(E; o), we have k 2 P E*(E_k). * *Instead of idA, we have the identity map idk: P E*(E_k) = P E*(E_k), considered as an e* *lement of AkP E*(E_k). We write aek for aeX when X = E_k. Step 1. We construct an E*-module structure on the graded group AM we de- fined in eq. (5.5). We start with the E*-module object n 7! P E*(E_n) in FMod * * op from Lemma 3.3(a), with v 2 E* acting by P (v)*. We apply the additive func- tor Mor (-; M): FMod op! Ab to obtain by [8, Lemma 7.6(a)] the E*-module object n 7! AnM in Ab, i. e. make AM an E*-module. JMB, DCJ, WSW - 20 - 23 Feb 1995 x5. What is an additively unstable module? Despite appearances, the square (3.4) does commute in the dual ca* *tegory FMod *op, to show that oeM: AM ! SM is an E*-module homomorphism. Step 2. We have defined aeX as a natural transformation of sets. For fixed * *X, the cohomology functor E*(-)^: Ho ! FMod op induces the natural transformation Ho (X; -) --! FMod (P E*(-)^; E*(X)^): Gp(Ho ) --! Set: We apply [8, Lemma 7.6(c)] to the E*-module object n 7! E_n to see that aeX is* * a morphism of E*-module objects, i. e. takes values in Mod . For Thm. 5.12(b), we note that given x 2 Ek(X), we have (oe(E*(X)^))x*U = x*UOoe*k= x*S, by [8, eq. (9.4)]. If X is a group object in Ho and x 2 P Ek(X), the associated map x: X ! E_k is a morphism of group objects (as remarked after Defn. 3.1) and so induces x*: P E*(E_k) ! P E*(X). If E*(X) (and hence P E*(X)) is Hausdorff, aeX restric* *ts to define P aeX : P E*(X) --! AP E*(X) : (5:13) Step 3. We filter AM exactly as we did SM in [8, x10], by the submodules F a(AM) = A(F aM), using naturality. The proof that AM is complete Hausdorff is formally the same as for SM. Our choice of filtrations and the naturality o* *f ae clearly make aeX and oeM continuous, so that aeX factors through E*(X)^ and oe * *takes values in FMod . Step 4. Whenever X is a group object in Ho and E*(X) is Hausdorff, we conve* *rt the object P E*(X) of FMod to the corepresented functor FPX = FMod (P E*(X); -): FMod --! Ab and the coaction P aeX in (5.13) to a natural transformation aePX : F* *PX ! FPX A: FMod ! Ab. Given M, aePX M: FPX M ! FPX AM is the homomorphism aePX M: FMod (P E*(X); M) --! FMod (P E*(X); AM) (5:14) that is defined on f: P E*(X) ! M as the composite PaeX * Af (aePX M)f: P E*(X) ----! AP E (X) ---! AM : Step 5. To construct = A: A ! AA, we take X = E_kin (5.14) and define ( M)k: FMod (P E*(E_k); M) --! FMod (P E*(E_k); AM) on the element f: P E*(E_k) ! M of AkM as the composite Paek * Af ( M)kf: P E*(E_k) ---! AP E (E_k) ---! AM : When we substitute the E*-module object n 7! E_nfor X in (5.14), [8, Lemma 7.6(* *c)] shows that ( M)k: AkM ! AkAM lies in Mod . As k varies, we obtain the natural transformation : A ! AA. Naturality in M also shows that M is filtered and so lies in FMod . Step 6. The other required natural transformation, ffl: A ! I, is defined o* *n M simply as the evaluation (fflM)k = (fflAM)k: AkM = FMod (P E*(E_k); M) --! M (5:15) JMB, DCJ, WSW - 21 - 23 Feb 1995 Unstable cohomology operations on k 2 P Ek(E_k). It is continuous by naturality. It is compatible with the s* *table version, fflA = fflS Ooe: A ! I, since given f 2 AkM, we have (fflSM)(oeM)f = ((oeM)f) = foe*k = fk = (fflAM)f : Step 7. To see that aeX is a coaction on E*(X), we use [8, Lemma 8.20] (ada* *pted to graded objects). We use R = P E*(E_n) (really, the graded object n 7! P E*(E* *_n)), 1R = n, and aeR = P aen. By [8, Lemma 8.22], A is a comonad in FMod . To see that oe: A ! S is a morphism of comonads, we apply [8, Lemma 8.24]. * *The first condition on u = oe*k: E*(E; o) ! P E*(E_k) is the commutative diagram oe*k Eh(E; o) _____________________________-P Ek+h(E_k) | | | | |Paek | | | |? ||aeE FMod (P E*(E_k+h); P E*(E_k)) | | | | |Hom(oe*k+h;1) | | |? Hom(1;oe* |? k) FMod h(E*(E; o); E*(E; o))__________-FMod k+h(E*(E; o); P E*(E_k)) A stable operation rS 2 Eh(E; o) restricts to an additive operation rU : k ! k * *+ h. On rS, the lower route gives by diag. [8, eq. (9.8)] oe*kOr*S= (-1)hk(rS Ooek)* = (-1)hk(oek+h OrU )* = (-1)hkr*UOoe*k+h: This agrees with the upper route, because oe*krS = (-1)hkrU by [8, eq. (9.9)]. * * The second condition needed is oe*k = k, which holds by the definition of oek. For Thm. 5.12(c), as in [8, Thm. 10.16(b)], it is enough to consider X. Bec* *ause E_k represents Ek(-), natural transformations are classified by the elements f* * = k: N ! E*(E_k), i. e. morphisms in FMod . The additivity (X)(x+y) = (X)(x) + (X)(y) of X on the universal example (2.6) yields *kOf = p*1Of + p*2Of: N --! E*(E_k x E_k): By Prop. 2.7(b), f factors through P E*(E_k). | 6 Unstable comodules Although the Fourth Answer of x5 is the cleanest and most general, the Seco* *nd Answer, in terms of unstable comodules, is usually the most practical and is av* *ailable in the cases of interest. The parallel with the stable theory of [8] is extreme* *ly close, in spite of the very different provenance of the two theories. Some of the mach* *inery was used in [6]; here we supply the missing definitions. We assume throughout this section that E*(E_k) and QE*(E_k) are free E*-mod* *ules for all k, so that we have available all the results of x5. The bigraded group Q(E)** As noted in x5, tensor products do not work correctly because the groups QE*(E_k) have the wrong degree; we therefore shift degrees. * *We JMB, DCJ, WSW - 22 - 23 Feb 1995 x6. Unstable comodules also adopt more efficient notation, that hides the details of construction and * *empha- sizes the algebraic aspects and the formal similarity to stable comodules. (We * *remind that homology Ei(X) has degree -i under our conventions.) Definition 6.1 We define the bigraded group Q(E)**as having the components Q(E)ki= QEi(E_k) (the component of QE*(E_k) in degree -i), except that we assign the degree k-i (instead of -i) to elements of Q(E)ki. (This is the degree that * *governs signs in formulae. We thus have the formal isomorphism k: QE*(E_k) ~=Q(E)k*of degree k.) We define the left action of v 2 Eh on kc 2 Q(E)k*, for c 2 QE*(E_k), by v(* *kc) = (-1)hkkvc, as in [8, eq. (6.7)], to make k: QE*(E_k) ~= Q(E)k*an isomorphism of E*-modules of degree k. We equip Q(E)k*with the projection k k qk: E*(E_k) --! QE*(E_k) ---! Q(E)* : (6:2) We define the stabilization -k Qoek* Q(oe): Q(E)k*---! QE*(E_k) ----! E*(E; o); (6:3) where Lemma 4.7(c) provides the factorization Qoek* of oek* . We thus have the factorization into E*-module homomorphisms oek* = Q(oe) Oqk: E*(E_k) --! Q(E)k*--! E*(E; o); (6:4) where we arranged for Q(oe) to have degree zero and qk to have degree k. Definition 6.5 Given an additive operation r: k ! m, i. e. an element rA 2 P Em (E_k), we define the associated E*-linear functional -k * : Q(E)k*---! QE*(E_k) -----! E (6:6) of degree m-k (with no sign). Now we can make the degree shift suggested by eq. (5.4). We have the strong duality P E*(E_k) ~=DQE*(E_k) from Lemma 4.16(a). Given an object M of FMod , we use [8, Lemma 6.16(b)] and the freeness of QE*(E_k) to define the natural is* *omor- phism of degree k Mk k FMod *(P E*(E_k); M) ~=M b QE*(E_k) -----! M b Q(E)* : (6:7) Lemma 6.8 Given an additive operation r: k ! m and an object M of FMod , the composite (formed using (6.7)) M FMod *(P E*(E_k); M) ~=M b Q(E)k*-------! M E* ~=M coincides with the evaluation homomorphism er: FMod *(P E*(E_k); M) ! M defined by erf = (-1)m deg(f)frA. JMB, DCJ, WSW - 23 - 23 Feb 1995 Unstable cohomology operations Proof We choose x 2 M, c 2 QE*(E_k), and evaluate. | With Defn. 6.5 in hand, we extend Prop. 2.7 and identify: (i)the additive operation r: Ek(-) ! Em (-); (ii)the cohomology class r = rA = rk 2 P Em (E_k); (iii)the morphism of group objects r: E_k! E_m in Ho ; (6.9) (iv)the E*-linear functional = : Q(E)k*! E*, of degree m-k, defined by eq. (6.6). (We drop the decorations A and Q on r except when we need to compare different versions.) As Q(E)**is smaller than P E*(E_*), (iv) is the preferred choice. * *We do have to be careful with degrees, as (ii) has a different degree from (i) and (i* *v), while (iii) has no degree at all. Scholium on signs We construct the duality diagram in FMod * rS rA rU oe*k E*(E; o)_______-P E*(E_k)_______-E*(E_k) | | | ||~= (-1)k ||~= (-1)k ||~= (6:10) | | | |? DQ(oe) |? Dq |? DE*(E; o) ______D(Q(E)k*)- ______DE*(E_k)-k whose center isomorphism is taken as d D(-k) k P E*(E_k) --! DQE*(E_k) -----! D(Q(E)*) : Because D is contravariant, each square commutes up to the sign (-1)k. On restriction to spaces, a stable operation r of degree h yields an additi* *ve unstable operation r: k ! k + h, and we obtain elements rS, rA, and rU lying in the indi* *cated groups. From these, we get the linear functionals , , and by eq* *. (6.6) also . We note that rS and rQ have degree h, while rA and rU have degre* *e k+h. The algebra forces us to work with the element rA and the functional ; * *we are not really interested in the functional , which appears only in the defi* *nition of , and the element rQ will occur nowhere. The complication is that these six elements do not all correspond in obvious ways under the morphisms of diag. (6.10). The first surprise was [8, eq. (9.9)* *], that oe*krS = (-1)khrA. Of course, rA and rU do correspond, because they are the sa* *me element regarded as being in different groups. The second surprise is that rA d* *oes not correspond to , because the definition [8, eq. (6.4)] of D(k) requires * *the sign (-1)k(h+k), which is absent from Defn. 6.5. In fact, matters are simpler if we* * work with elements and refrain from turning everything into E*-module homomorphisms. Proposition 6.11 In diag. (6.10): JMB, DCJ, WSW - 24 - 23 Feb 1995 x6. Unstable comodules (a) Given a stable operation r, the homomorphism DQ(oe) takes to , or in elements, = for c 2 Q(E)k*, (6:12) and also = for c 2 E*(E_k); (6:13) (b) Given an additive operation r: k ! m, the homomorphism Dqk takes to (-1)k(m-k), or equivalently, in elements, = for c 2 E*(E_k). (6:14) Proof We just proved (a), except for eq. (6.13), which combines eqs. * * (6.12) and (6.14). In (b), is simply the restriction of , so that = = : But the definition of Dqk adds the unwanted sign (-1)k(m-k). | Q(E)**as an algebra There is much structure on Q(E)**. First, it is by constru* *ction a left E*-module. Proposition 6.15 For any ring spectrum E, Q(E)**has the properties: (a) Q(E)**is a bigraded E*-algebra, with multiplication Q(OE) defined by the* * com- mutative diagram (6.16) x OEU* E*(E_k) E*(E_m ) ______E*(E_kxE_m-) _________-E*(E_k+m ) | | | | |qkqm |qk+m | | |? Q(OE) |? Q(E)k* Q(E)m* ______________________________-Q(E)k+m* (6:16) | | | | |Q(oe)Q(oe) |Q(oe) | | |? x OES |? E*(E; o) E*(E; o) ______E*(E^E;-o) ___________E*(E;-o)* and unit Q(j) defined by the commutative diagram E*(T )________-E*= _______E*(T-+;=o) | | | |jU* |Q(j) |jS* (6:17) | | | |? q |? |? 0 Q(oe) E*(E_0) ______Q(E)0*-______-E*(E; o) (b) The stabilization Q(oe): Q(E)**! E*(E; o) is a homomorphism of E*-algebr* *as. Proof Q(OE) is inherited, with a shift, from the multiplication on QE*(E_*) co* *n- structed by Lemma 4.9. It thus fills in diag. (6.16), which is derived from [8,* * eq. (9.15)] by applying E-homology and the factorization (6.4). We simply define Q(j) = q0O* * jU*, JMB, DCJ, WSW - 25 - 23 Feb 1995 Unstable cohomology operations to fill in diag. (6.17). This comes from diag. [8, eq. (9.4)] by taking x = 1T * *2 E*(T ). The algebraic properties of Q(OE) and Q(j) are inherited from the E*-algebra ob* *ject n 7! E_nin Ho . Part (b) is clear from the diagrams. | Q(E)**as a bimodule We also need the right E*-action. By Lemma 4.5, the functor QE*(-): Gp(Ho ) ! Mod preserves finite products. We apply [8, Lemma 7.6(a)] to the E*-module object n 7! E_nin Gp(Ho ), to obtain, for each v 2 Eh, homomorphi* *sms Q(v) that fill in the commutative diagram qk Q(oe) E*(E_k) ________Q(E)k*-_______-E*(E; o) | | | | | | |(Uv)* |Q(v) |(Sv)* (6:18) | | | |? q |? |? k+h Q(oe) E*(E_k+h) ______Q(E)k+h*-______-E*(E; o) and make Q(E)**a module object in Mod *, i. e. an E*-bimodule. This diagram came from diag. [8, eq. (9.8)] by taking r = v. We have the additive analogue of the stable right unit. Definition 6.19 We define the right unit function jR: E* ! Q(E)**on v 2 Eh = Eh(T ) by jRv = qhv*1 2 Q(E)h0, using the homology homomorphism v*: E* ~=E*(T ) ! E*(E_h) induced by the map v: T ! E_h. It is clear from [8, eq. (9.4)] and the factorization (6.4) that compositio* *n with Q(oe) yields the stable right unit jR: E* ! E*(E; o) of [8, Defn. 11.2]. Proposition 6.20 For any ring spectrum E, the algebra Q(E)**has the properti* *es: (a) It is a bigraded E*-bimodule, with components Q(E)ki= QEi(E_k) which are assigned the degree k-i; (b) It has the well-defined unit element 1 = Q(j)1 = jR1 2 Q(E)00; (c) The left action of v 2 Eh is left multiplication by v1 2 Q(E)0-h; (d) The right action of v 2 Eh is right multiplication by jRv 2 Q(E)h0; (e) The stabilization Q(oe): Q(E)** ! E*(E; o) is a homomorphism of * *E*- bimodules. Remark Props. 6.15 and 6.20 are similar to [8, Prop. 11.3], except that Q(E)** **is bigraded and the conjugation O is conspicuous by its absence. The examples of x* *16 show that O does not exist, at least, not in any obvious sense. (This is why we eschewed O in [8].) Proof Most of the proof is formally identical to the stable case [8, Prop. 11.* *3]. For (d), we apply E-homology to the factorization [8, eq. (3.27)] of v. Part (e) is* * clear from diag. (6.18). | We write the left and right E*-actions as L: Eh Q(E)ki ! Q(E)ki-hand R: Q(E)ki Eh ! Q(E)k+hi. Explicitly, the signs for R are R(c v) = c . v = c(jRv) = (-1)h deg(c)(jRv)c = (-1)h deg(c)Q(v)c;(6:21) JMB, DCJ, WSW - 26 - 23 Feb 1995 x6. Unstable comodules where v 2 Eh and c . v denotes the right action. For future use, we rewrite (d)* * as the commutative square Q(OE) Q(E)k* Q(E)m* ________-Q(E)k+m* | | | | |Q(v)1 |Q(v) (6:22) | | |? Q(OE) |? Q(E)k+h* Q(E)m* ______-Q(E)k+m+h* The functor A0 Given an E*-module M, we define (as promised in x5) the graded group A0M as having the components (A0M)k = Mi biQ(E)ki= (M b Q(E)k*)k (6:23) (where the tensor product b iis formed using the two E*-actions indexed by i. * *We have no use for the rest of M b Q(E)k*!) We use the isomorphism (6.7) to define* * the isomorphism AM ~=A0M as having the components (AM)k = AkM = FMod (P E*(E_k); M) ~=Mi biQ(E)ki= (A0M)k: (6:24) We use this isomorphism to transfer all the structure of x5 from A to A0and mak* *e A0 a comonad, just as we did stably in [8]. (We generally drop the decorations 0ex* *cept when comparing different versions.) In particular, we use (6.24) to convert modules to comodules. If M is an a* *ddi- tively unstable module with coaction aeM : M ! AM (as in Defn. 5.9), we deduce * *the equivalent coaction ae0M: M ! A0M with components ae0M: Mk --! (A0M)k = Mi biQ(E)ki M b Q(E)k*: (6:25) In particular, for a space X, we convert the action aeX in (5.6) to ae0X: Ek(X) --! Ei(X) biQ(E)ki E*(X) b Q(E)k*: (6:26) Q(E)**as a coalgebra The stable discussion carries over, except that Q(E)**is bigraded. The comonad structure ( ; ffl) on A translates into a comonad struct* *ure ( 0; ffl0) on A0. By naturality and the case M = iE*, 0M: (A0M)k ! (A0A0M)k must take the form M b for a certain comultiplication = Q( ): Q(E)ki--! Q(E)jij Q(E)kj (6:27) (where we sum over j as in eq. (6.23)), and ffl0M: (A0M)k ! Mk must take the fo* *rm M b ffl for a certain counit ffl = Q(ffl): Q(E)ki--! Ek-i: (6:28) By construction, these are both E*-bimodule homomorphisms of degree zero. Proposition 6.29 Assume that E*(E_k) and QE*(E_k) are free E*-modules for all k. Then: (a) The homomorphisms = Q( ) and ffl = Q(ffl) in diags. (6.27) and (6.28) * *make Q(E)**a coalgebra over E*; (b) If E*(E; o) is also free, the stabilization Q(oe): Q(E)**! E*(E; o) is a* * morphism of coalgebras (cf. [8, Lemma 11.8]). JMB, DCJ, WSW - 27 - 23 Feb 1995 Unstable cohomology operations Proof By taking M = iE*, the comonad axioms [8, eq. (8.6)] for A0 yield the coassociativity Q( ) Q(E)kh _________________-Q(E)jhj Q(E)kj | | |Q( ) |1Q( ) (6:30) | | |? Q( )1 |? Q(E)ihiQ(E)ki ________-Q(E)ihiQ(E)jij Q(E)kj of Q( ) and the two counit axioms Q( ) k Q( ) j k Q(E)ki ______Q(E)jij-Q(E)kj Q(E)i ______Q(E)i-j Q(E)j | | | | | | | |= |Q(ffl)1 |= | | | | |1Q(ffl) (6:31) | | | | |? |? |? |? L k R j k-j Q(E)ki _______Ej-ijoQ(E)kje Q(E)i ______oQ(E)iej E Part (b) is the translation of Thm. 5.8(b). | Comodules Now that we have the coalgebra Q(E)**, we can convert Defn. 5.9 and Thm. 5.12. Definition 6.32 An unstable (E-cohomology) comodule is an A0-coalgebra* * in FMod . In detail, given a complete Hausdorff filtered E*-module M (i. e. object of* * FMod ), an unstable comodule structure on M consists of a coaction aeM : M ! A0M, with * *com- ponents Mk ! Mi biQ(E)kias in diag. (6.25), that is a continuous homomorphism of E*-modules (i. e. morphism in FMod ) and satisfies the axioms aeM aeM M ______M-b Q(E)** M ______________-M b Q(E)** Q Q ~= | |aeM |MQ( ) Q |MQ(ffl) |? ae |? Q | M 1 Qs |? (6:33) M b Q(E)** ______M-b Q(E)**bQ(E)** M E* (i) (ii) This is a stronger structure than a stable comodule (assuming that E*(E; o)* * is free, so that stable comodules can be defined). Given a coaction aeM as above, Prop. * *6.29(b) shows that the coaction aeM * MQ(oe) M ---! M b Q(E)* ------! M b E*(E; o) (6:34) makes M a stable comodule. Remark We regard comodules as essentially additive constructs, as we find no a* *na- logue in the fully unstable context. We therefore omit the adjective "additive"* * from comodules. JMB, DCJ, WSW - 28 - 23 Feb 1995 x6. Unstable comodules Theorem 6.35 Assume that E*(E_k) and QE*(E_k) are free E*-modules for all k (which is true for E = H(F p), BP , MU, KU, or K(n) by Lemma 4.17(a).) Then given a complete Hausdorff filtered E*-module M (i. e. object of FMod ), an add* *itively unstable module structure on M in the sense of Defn. 5.9 is equivalent to an un* *stable comodule structure on M in the sense of Defn. 6.32. Proof We have the isomorphism AM ~= A0M in eq. (6.24). The axioms (6.33) are just the general coaction axioms [8, eq. (8.7)] interpreted for A0. | Theorem 6.36 Assume that E*(E_k) and QE*(E_k) are free E*-modules for all k (which is true for E = H(F p), BP , MU, KU, or K(n) by Lemma 4.17(a).) Then: (a) For any space X, there is a natural coaction aeX : E*(X) --! E*(X) b Q(E)** that makes E*(X)^ an unstable comodule, which corresponds by Thm. 6.35 to the additive module structure given by Thm. 5.12; (b) If also E*(E; o) is free, we recover the stable coaction [8, eq. (11.15)* *] on E*(X) from aeX as in diag. (6.34); (c) ae is universal: given a discrete E*-module N and an integer k, any addi* *tive natural transformation X: Ek(X) ! E*(X) b N (or bX: Ek(X)^ ! E*(X)^ bN) that is defined for all spaces X is induced from aeX by a unique homomorphism f: Q(E)k*! N of E*-modules as aeX * k 1f * X: Ek(X) ---! E (X) b Q(E)* ---! E (X) b N : Proof We deduce (a) from Thm. 5.12(a) and Thm. 6.35, just as we did stably in * *[8, Thm. 11.14]. In eq. (6.26), we defined the coaction ae0Xas corresponding to aeX* * . In (b), the stabilization Q(oe) clearly dualizes to oe*k: E*(E; o) ! P E*(E_k), which w* *e used in eq. (5.7) to define the stabilization oe: A ! S of comonads. In (c), the natural transformation is classified by the element u = k 2 E*(E_k) b N. Additivity of for the universal example (2.6) states that (*kN)u = (p*1N)u + (p*2N)u in E*(E_kxE_k) b N. By [8, Lemma 6.16(a)], u corresponds to a homomorphism f: E*(E_k) ! N of E*- modules. The above property dualizes to f Ok * = f Op1*+ f Op2*: E*(E_kxE_k) --! N; which shows that f factors through Q(E)k*as required. | Remark Just as stably, (c) allows us to use diags. (6.33) to define Q( ) and Q* *(ffl) in terms of ae. Three applications of the uniqueness in (c) show that Q( ) is coas* *sociative and has Q(ffl) as a two-sided counit. Linear functionals Theorem 6.35 establishes the equivalence between unstable modules and comodules. For applications, we need the details. All our formulae * *sta- bilize to the corresponding formulae of [8, x11] by applying Q(oe), which conve* *niently has degree zero. JMB, DCJ, WSW - 29 - 23 Feb 1995 Unstable cohomology operations Given an unstable comodule M, we recover the action of the additive operati* *on r: k ! m on M from Lemma 6.8 as aeM k M * r: Mk ---! M b Q(E)* ------! M E ~=M: (6:37) Because has degree m-k, r takes values in Mm . To make this action expli* *cit, let us choose x 2 Mk and write X aeM x = (-1)deg(xff) deg(cff)xff cff in M b Q(E)k*, (6:38) ff where the sum may be infinite, and of course deg(xff) = k - deg(cff). (As in [8* *], we insert signs here to keep the next formula simple.) Then X rx = xff in M, for all r: k ! m, (6:39) ff where the cffand xffdepend only on x, not on r. Because M is assumed complete, * *this sum converges if it is infinite. (Recall that Q(E)k*always has the discrete top* *ology.) Remark It is important for our applications not to require the cffto form a ba* *sis of Q(E)k*, or even be linearly independent; but if they do form a basis, the xf* *fare uniquely determined by eq. (6.39) as xff= c*ffx, where c*ffdenotes the operatio* *n dual to cff. The fact that aeM is an E*-module homomorphism is expressed by X X r(vx) = xff= (-1)h deg(cff) xff in M,(6:40) ff ff for any v 2 Eh and all operations r: k + h ! m. Because Q(ffl): Q(E)k*! E* corresponds to ffl in eq. (5.15), which is evalu* *ation on k, we have immediately = Q(ffl): Q(E)k*--! E*; (6:41) as is obvious by comparing axiom (6.33)(ii) with eq. (6.37). In other words, i* *n the list (6.9), the identity operation k corresponds to the functional Q(ffl). The cohomology of a point Our first test space is the one-point space T . Proposition 6.42 In the unstable comodule E*(T ) = E*: (a) The action of the additive operation r: k ! m on v 2 Ek is given by rv = in E*(T ) = E*; (6:43) (b) The coaction aeT: E* ! E* Q(E)**~= Q(E)**coincides with the right unit jR: E* ! Q(E)*0(see Defn. 6.19). Proof We imitate [8, Prop. 11.22]. The map v: T ! E_kyields rv = = = = = ; by eq. (6.14) and Defn. 6.19 of jR. We compare eqs. (6.38) and (6.39) and rewr* *ite this as aeTv = 1 jRv, to give (b). | Homology homomorphisms A class x 2 Ek(X) may be regarded as a map x: X ! E_k. We need information about the induced homology homomorphism x*: E*(X) ! E*(E_k). JMB, DCJ, WSW - 30 - 23 Feb 1995 x6. Unstable comodules Proposition 6.44 Assume that E*(E_k) and QE*(E_k) are free E*-modules for all k. Given x 2 Ek(X), suppose that rx is given by eq. (6.39). Then the homomorphi* *sm qk Ox*: E*(X) ! Q(E)k*induced by the map x: X ! E_kis given on z 2 Eh(X) by X X qkx*z = (-1)deg(cff)(deg(xff)+h) cff= cff in Q(E)k*.* *(6:45) ff ff Proof For any additive r: k ! m, we have = by eq. (6.1* *4). The rest of the proof is formally identical to the stable analogue [8, Prop. 11.26]* *. | Conversely, we can recover aeX x from x* when X is well behaved, just as we* * did stably. If E*(X) is free, we have strong duality E*(X) ~=DE*(X) by Thm. 1.18(a), and [8, Lemma 6.16(a)] supplies the isomorphism E*(X) b Q(E)k*~=Mod *(E*(X); Q(E)k*) : (6:46) Proposition 6.47 Assume that E*(X), E*(E_k), and QE*(E_k) are free E*-modules for all k. Take x 2 Ek(X). Then under the isomorphism (6.46), the element aeX* * x corresponds to the homomorphism qk Ox*: E*(X) ! E*(E_k) ! Q(E)k*. Proof We apply the isomorphism to eq. (6.38) and compare with eq. (6.45). | In particular, it is important to know the homomorphism of E*-modules Qr* m Q(r): Q(E)k*~=QE*(E_k) ---! QE*(E_m ) ~=Q(E)* (6:48) induced by an additive operation r: k ! m (which by Prop. 2.7(c) is a morphism of group objects in Ho ). It has degree m-k. The Q(r) provide a convenient fait* *h- ful representation of the additive operations. The translation of diag. (5.11)* * is the commutative square r Mk ________________-Mm |aeM |aeM (6:49) |? MQ(r) |? Mi iQ(E)ki ________Mi-iQ(E)mi which stabilizes to diag. [8, eq. (11.29)]. Just as stably, we easily recover the functional from Q(r) as Q(r) Q(ffl) : Q(E)k*----! Q(E)m*---! E*: (6:50) Conversely, we have the additive analogue of [8, Lemma 11.31]. Lemma 6.51 Assume that E*(E_k) and QE*(E_k) are free E*-modules for all k. * *If r: k ! m is an additive operation, then the homology homomorphism Q(r): Q(E)k*! Q(E)m*in diag. (6.48) has the properties: (a) The diagram Q(r) Q(E)k* ________________-Q(E)m* | | |Q( ) |Q( ) (6:52) |? 1Q(r) |? Q(E)** Q(E)k* ________Q(E)**-Q(E)m* JMB, DCJ, WSW - 31 - 23 Feb 1995 Unstable cohomology operations commutes; in other words, Q(r) is a morphism of left Q(E)**-comodules; (b) Q(r): Q(E)k*! Q(E)m*is the unique homomorphism of left E*-modules that satisfies eq. (6.50) and is a morphism of left Q(E)**-comodules in the sense of* * (a); (c) Q(r) is given in terms of the functional as Q( ) j 1 j Q(r): Q(E)ki----! Q(E)i j Q(E)kj -----! Q(E)i j Em-j R m ---! Q(E)i : | We deduce from (c) that the composite sr: k ! n of the operations r: k ! m * *and s: m ! n corresponds to the functional Q( ) j 1 j : Q(E)ki----! Q(E)i j Q(E)kj-----! Q(E)i j Em-j R m n-i (6:53) ---! Q(E)i ----! E : Remark From diags. (6.30) and (6.31)(ii) we observe that for fixed h, Q( ) mak* *es the graded group n 7! Qnhan additively unstable comodule, if we use the right E*-mo* *dule action (6.21). Then by (c), the action of r: k ! m is just Q(r), and diag. (6.* *52) becomes a special case of diag. (6.49). 7 What is an additively unstable algebra? In this section, we define an additively unstable algebra by enriching each* * of the four Answers in x5 with multiplicative structure. The treatment is closely par* *allel to the stable case [8, x12] and we give only the significant additions. The lo* *gical sequence is made slightly complicated by the fact that the monoidal structure i* *s most easily described in the context of the Second (or Third) Answer, while the como* *nad structure prefers the Fourth Answer. In Defn. 7.13 we introduce the collapse operation, which detects the connec* *tedness of a space. We assume throughout this section that E*(E_k) and QE*(E_k) are free E*-mod* *ules for all k, which is true for our five examples by Lemma 4.17(a). Then by Cor. 2* *.9, P E*(E_k) is an object of FMod . First Answer We have, for any space X, the additively unstable action (5.1) O: P Em (E_k) x Ek(X) --! Em (X) : Given x 2 Ek(X), y 2 Em (X), and r 2 P E*(E_k+m ), we would like to have a Cart* *an formula X r(xy) = (r0ffx)(r00ffy) in E*(X), (7:1) ff for suitably chosen operations r0ffand r00ff(depending on k and m as well as r)* *. For the universal example X = E_kx E_m; with x = k x 1, y = 1 x m , xy = OE = k x m , (7:2) JMB, DCJ, WSW - 32 - 23 Feb 1995 x7. What is an additively unstable algebra? where OE: E_kxE_m ! E_k+m denotes the multiplication map of [8, Thm. 3.25], eq.* * (7.1) reduces to X OE*r = r0ffx r00ff in E*(E_kxE_m ) . ff To ensure that OE*r is expressible in this form, we need to allow infinite sums* * and use the K"unneth homeomorphism E*(E_kxE_m ) ~=E*(E_k) b E*(E_m ) from Thm. 1.18(c). We need to know more, that r0ff; r00ff2 P E*(E_*). We have enough duality i* *somor- phisms to dualize the multiplication in Lemma 4.9 and define a comultiplication* * P by the commutative diagram P P E*(E_k+m )__________________________-P E*(E_k) b P E*(E_m ) | | | | | | (7:3) | | |? OE* ~ |? E*(E_k+m )_________-E*(E_kxE_m ) ______E*(E_k)obeE*(E_m=) P 0 00 Then we write P r = ffrff rff, as required. We must not forget the unit element 1X 2 E*(X). We define the counit fflP : P E*(E_0) ! E* as the restriction of j*: E*(E_0) ! E*(T ) = E*, s* *o that r1X = (fflP r)1X in E*(X). It is now clear what an additively unstable algebra should be. Given an E*-* *algebra M, we need actions P Em (E_k) x Mk ! Mm that compose correctly, are biadditive and E*-bilinear in the sense of diag. (5.2), satisfy the Cartan formula (7.1), * *and respect the unit in the sense that r1M = (fflP r)1M . In the classical case E = H(F p)* *, there is a good Cartan formula and this approach is useful. For more general E, such as * *MU and BP , this structure seems even more impractical than it was stably. Second Answer We have the coaction (6.26), aeX : Ek(X) --! Ei(X) biQ(E)ki: In contrast to the Cartan formula of the First Answer, and just as stably in [8* *], all we have to do is observe that as k varies, aeX is a homomorphism of E*-algebras, w* *here we use the bigraded algebra structure on Q**= Q(E)**from Prop. 6.15. Explicitly, if for particular x; y 2 E*(X) we have, as in eq. (6.39), X X rx = xff; ry = yfi; for all r, ff fi the Cartan formula (7.1) becomes (cf. the stable analogue [8, eq. (12.5)]) X X r(xy) = (-1)deg(dfi) deg(xff) xffyfi in E*(X)^, for(all7* *r.:4) ff fi Lemma 7.5 Assume that E*(E_k) and QE*(E_k) are free E*-modules for all k. Th* *en the homomorphisms Q( ) and Q(ffl) in (6.27) and (6.28) are multiplicative and r* *espect the unit element. We defer the proofs until after Thm. 7.9, as the coalgebra structure on Q(E* *)**is not easily handled directly. The Lemma makes the following definition reasonabl* *e. JMB, DCJ, WSW - 33 - 23 Feb 1995 Unstable cohomology operations Definition 7.6 We call an unstable comodule M in the sense of Defn. 6.32 an unstable (E-cohomology) comodule algebra if M is a filtered algebra (i. e. obje* *ct of FAlg) and its coaction aeM : M ! M b Q(E)**is a homomorphism of E*-algebras. In detail, M is a complete Hausdorff commutative filtered E*-algebra, equip* *ped with a structure map aeM that is a continuous homomorphism of E*-algebras and makes diags. (6.33) commute. Theorem 7.7 Assume that E*(E_k) and QE*(E_k) are free E*-modules for all k (which is true for E = H(F p), BP , MU, KU, or K(n) by Lemma 4.17(a)). Then: (a) For any space X, aeX makes E*(X)^ an unstable comodule algebra in the se* *nse of Defn. 7.6; (b) ae is universal: given a (possibly bigraded) discrete E*-algebra B, any * *natural transformation of rings X: E*(X) ! E*(X) b B (or bX: E*(X)^ ! E*(X)^ bB) that is defined for all spaces X is induced from aeX by a unique homomorphism f: Q(E)**! B of left E*-algebras as aeX * * 1f * X: E*(X) ---! E (X) b Q(E)* ---! E (X) b B : Proof This will follow from Thm. 7.9 in the same way that the stable result Th* *m. 12.8 followed from Thm. 12.10 in [8]. | Third Answer We use the multiplication Q(OE): Qk* Qm*! Qk+m*from Prop. 6.15 to make A0a symmetric monoidal functor (A0; iA0; zA0) in FMod , with iA0(M; N): (A0M)k b(A0N)m --! (A0(M b N))k+m given by iA0(M; N): M b Qk*b N b Qm*~= M b N b(Qk* Qm*) (7:8) --! M b N b Qk+m* and zA0 = jR: Eh ! E* Qh*~= Qh*. Thus when M is an E*-algebra, so is A0M. We see that A0, equipped with natural transformations 0: A0! A0A0and ffl0: A0!* * I constructed from Q( ) and Q(ffl), becomes a symmetric monoidal comonad in FMod and therefore a comonad in FAlg. Fourth Answer For suitable E, we can make A a comonad in FAlg. Theorem 7.9 Assume that E*(E_k) and QE*(E_k) are free E*-modules for all k (which is true for E = H(F p), BP , MU, KU, or K(n) by Lemma 4.17(a)). Then: (a) We can enrich A to make it a symmetric monoidal comonad in FMod and therefore a comonad in FAlg; (b) If also E*(E; o) is free, the stabilization oe: A ! S is a monoidal natu* *ral trans- formation in FMod . The relevant definition is now clear. Definition 7.10 An additively unstable (E-cohomology) algebra is an A-coalgeb* *ra in FAlg, i. e. a complete Hausdorff commutative filtered E*-algebra M equipped * *with a morphism aeM : M ! AM in FAlg that satisfies the coaction axioms [8, eq. (8.7* *)]. JMB, DCJ, WSW - 34 - 23 Feb 1995 x7. What is an additively unstable algebra? If the closed ideal L M is invariant, the quotient algebra M=L inherits a * *well- defined A-coalgebra structure. Theorem 7.11 Assume that E*(E_k) and QE*(E_k) are free E*-modules for all k (which is true for E = H(F p), BP , MU, KU, or K(n) by Lemma 4.17(a)). Then given a complete Hausdorff commutative filtered E*-algebra M (i. e. object of F* *Alg), an unstable comodule algebra structure on M in the sense of Defn. 7.6 is equiva* *lent to an additively unstable algebra structure on M in the sense of Defn. 7.10. Theorem 7.12 Assume that E*(E_k) and QE*(E_k) are free E*-modules for all k (which is true for E = H(F p), BP , MU, KU, or K(n) by Lemma 4.17(a)). Then: (a) For any space X, the coaction aeX : E*(X) ! A(E*(X)^) in diag. (5.6) is a homomorphism of E*-algebras and makes E*(X)^ an additively unstable algebra; (b) ae is universal: given a graded monoid object n 7! Cn in FMod op, so th* *at (by [8, Lemma 7.9]) n 7! Gn(X) = FMod (Cn; E*(X)^) is a graded ring, any natural transformation of graded rings X: E*(X) ! G*(X) (or bX: E*(X)^ ! G*(X)), that is defined for all spaces X, is induced from aeX by a unique morphism in FMod * *opof graded monoid objects with components fn: Cn ! P E*(E_n) in FMod , as aeX * * Hom(fn;1) n * X: En(X) ---! FMod (P E (E_n); E (X)^) -------! FMod (C ; E (X)^) : Proof of Thms. 7.9 and 7.12 The main proof proceeds by the same five steps as * *stably for [8, Thms. 12.10, 12.13], except based on Thm. 5.8 instead of [8, Thm. 10.12* *]. We give only the major changes. We recall the universal class k 2 Ek(E_k), element idk 2 AkP E*(E_k), and aek from the proof of Thm. 5.8. Step 1. We construct the symmetric monoidal functor (A; iA; zA): (FMod ; b; E*) --! (Mod ; ; E*) : Then A will take monoid objects in FMod (i. e. objects of FAlg) to monoid obje* *cts in Mod (i. e. E*-algebras). By Lemma 4.16(a), we can construct the diagram (7.3) that defines P and ve* *rify its properties, which are dual to those of Q(OE) in Props. 6.15 and 6.20. The c* *ounit fflP : P E*(E_0) ! E* is the restriction of j*: E*(E_0) ! E*(T ) = E*. These m* *ake n 7! P E*(E_n) an E*-algebra object in FMod *op, to which we apply [8, Lemma 7.* *14]. The necessary compatibility axiom [8, eq. (7.13)] is the dual of diag. (6.22). * *As stably, we use [8, eq. (7.15)] to identify zA with aeT: E*(T ) ! AE*(T ). If E*(E; o) is also free, we can dualize Prop. 6.15(b) to see that the dest* *abi- lizations oe*n: E*(E; o) ! P E*(E_n) form a morphism of graded monoid objects in (FMod *op; b; E*). Then [8, Lemma 7.9(b)] shows that oe: A ! S is monoidal. Step 2. The proof that ae is monoidal is similar to the stable case. Here,* * the universal example is X = E_k and Y = E_m , with the element k m . The two elements of Ak+m E*(E_kxE_m ) to be compared are P * * * * P E*(E_k+m ) ---! P E (E_k) b P E (E_m )E (E_k) b E (E_m ) x * --! E (E_kxE_m ) JMB, DCJ, WSW - 35 - 23 Feb 1995 Unstable cohomology operations and OE* * P E*(E_k+m ) E*(E_k+m ) --! E (E_k x E_m): These agree by diag. (7.3). The second condition needed is just zA = aeT. Step 3. The analogue of diag. [8, eq. (12.17)] for this situation is fig. 1* *. To Figure 1: Additive operations and comultiplication P P E*(E_k+m )________-P E*(E_k) b P E*(E_m ) | | |Pae Paem | | k | |? ||Paek+m AP E*(E_k) b AP E*(E_m ) | | | | |iA |? A |? AP E*(E_k+m ) ______A(P-E*(E_k)Pb P E*(E_m )) establish this, we proceed as in [8, Thm. 12.10]. Because ae is monoidal and na* *tural, we have the commutative diagram fig. 2 (cf. diag. [8, eq. (12.16)]) which incl* *udes an Figure 2: The monoidality of ae aekaem E*(E_k) b E*(E_m )______AE*(E_k)-b AE*(E_m ) | | |iA | | | |? ~=||x A(E*(E_k) b E*(E_m )) | | ~ | | = |Ax |? ae |? E*(E_kxE_m ) ___________-AE*(E_kxE_m ) |6 * |6 * |OE |AOE | aek+m | E*(E_k+m )______________AE*(E_k+m-) isomorphism from Thm. 1.18(c). Figure 1 is obtained from this by restriction, u* *sing the coaction (5.13) and diag. (7.3). Step 4. The monoidality of follows formally from that of ae, just as stab* *ly (cf. diags. [8, eq. (12.18)]). The universal example is M = P E*(E_m ) and N = P E*(* *E_n), with element idm idn. We use fig. 1 instead of diag. [8, eq. (12.17)]. Step 5. The proof that ffl is monoidal is formally the same as stably, exce* *pt for the insertion of indices. In Thm. 7.12(b), C has comultiplications C : Ck+m ! Ck b Cm and a counit fflC : C0 ! E* which make n 7! FMod (Cn; E*(X)^) a graded ring. For each n, Thm. 5.12(c) provides a morphism fn: Cn ! P E*(E_n) in FMod . For the universal JMB, DCJ, WSW - 36 - 23 Feb 1995 x7. What is an additively unstable algebra? example (7.2), the multiplicativity (X)(xy) = ((X)x)((X)y) reduces to the com- mutativity of the outside of the diagram in fig. 3. The lower rectangle is dia* *g. (7.3). Figure 3: Comparison of comultiplications C Ck+m _______________Ck-bCm | |fk+m |fkfm |? |? P E*(E_k+m )______P-E*(E_k)PbP E*(E_m ) | | | | |? | || E*(E_k) bE*(E_m ) | | |~= |? OE* |? E*(E_k+m )__________-E*(E_kxE_m ) It follows that the upper square commutes, so that f preserves the comultiplica* *tion. Similarly, (T )1 = 1 yields fig. 4, which shows that f preserves the counit. | Figure 4: Comparison of counits ________-f0* ______-* C0 P E (E_0) E (E_0) Q Q | j QQfflC ||fflPj j Q Qs ||?jjj+j* E* Proof of Thm. 7.11 We use the isomorphism (6.7) to translate the monoidal stru* *cture of A to A0. From iA, which is given by [8, eq. (7.11)], we obtain eq. (7.8). We* * have identified both zA and zA0 with the coaction aeT. | Proof of Lemma 7.5 Theorem 7.9(a) shows in particular that : A ! AA and ffl: * *A ! I are monoidal natural transformations. By the isomorphism (6.24), so are 0: A* *0! A0A0and ffl0: A0! I. Evaluation of the relevant diagrams involving i for M = N * *= E* show precisely that Q( ) and Q(ffl) are multiplicative. Since zA0 = jR: Eh ! E* Q(E)h*~= Q(E)h*, the two diagrams involving z show that 1 = 1 1 and ffl1 = 1, simply because jR1 is the unit element of Q(E)**. | Proof of Thm. 7.7 Part (a) follows from Thm. 7.12(a). In (b), Thm. 6.36(c) pr* *ovides for each n the E*-module homomorphism fn: Q(E)n*! B that induces X: En(X) ! E*(X) b B. As in the proof of [8, Thm. 12.8(b)], the resulting f: Q(E)**! B is * *an E*-algebra homomorphism. | Connectedness There is a particular operation that is useful for expressing the concept of connectedness in a cohomology algebra. It sees only the path compone* *nts of a space. JMB, DCJ, WSW - 37 - 23 Feb 1995 Unstable cohomology operations Definition 7.13 For each n, we define the collapse operation n: n ! n as the * *map n: E_n! E_n(well defined up to homotopy) that sends each path component of E_n to one point in that path component. It is clearly additive, multiplicative ((xy) = (x)(y)), unital (01X = 1X ),* * and idempotent. It commutes with all operations in the sense that m Or = r Ok: k ! m for all r: k ! m; in particular, is E*-linear. It is zero in any degree n for* * which En = 0. In spite of being defined in all degrees, it is not at all stable, as n* * = 0. All these properties carry over to any additively unstable algebra M; in particular* *, we always have the E*-module decomposition M = Im Ker, with (E*)1M Im . For a connected space X with basepoint o, it is clear that the augmentation* * ideal E*(X; o) E*(X) is precisely Ker . In general, Ker = F 1E*(X) for any space X, the first stage of the skeleton filtration. This suggests the following definit* *ion. Definition 7.14 We call the additively unstable algebra M connected if Im = (E*)1M . We call M spacelike if it is a product (in FAlg) of connected algebras. In particular, for a space X, E*(X)^ is always spacelike, and is connected * *if and only if X is connected. 8 What is an unstable object? In this section, we interpret what it means to have an algebra over all the* * un- stable operations on E-cohomology. Tensor products rapidly become unworkable for nonadditive operations, with the effect that only the First and Fourth Answers * *from x5 survive intact. We generally assume that E*(E_k) is a free E*-module for all k. Then Thm. 1* *.18 provides all the K"unneth and duality isomorphisms and homeomorphisms we need. Of course, when we compare with the additive or stable theory, we impose the ap- propriate extra conditions. As in (2.1), we identify: (i)The cohomology operation r: Ek(-) ! Em (-); (ii)The class r = r(k) 2 Em (E_k); (iii)The representing map r: E_k! E_m; and write any of these as r: k ! m. (We shall retain the parentheses in r(x) wh* *enever r is nonadditive.) We first deal with the constant operations r: k ! m, those of the form r(x)* * = v1X 2 Em (X) for all x 2 Ek(X) and all spaces X, where v 2 Em . Lemma 8.1 Any operation r: k ! m decomposes uniquely as the sum of a based operation s: k ! m and a constant operation. Proof We set v = r(0) 2 Em (T ) = Em and define the operation s by s(x) = r(x) - v1X in E*(X), to make s(0) = 0. | JMB, DCJ, WSW - 38 - 23 Feb 1995 x8. What is an unstable object? First Answer Since Ek(-) is represented in Ho by E_k, we have as in (5.1) the actions O: Em (E_k) x Ek(X) --! Em (X); (8:2) except that we cannot write them using tensor products. Instead, we need a Cart* *an formula for r(x+y) as well as for r(xy). To find r(x+y), we consider the abelian group object E_k of Ho provided by * *[8, Cor. 7.8], which is equipped with the addition map k: E_kx E_k! E_kand zero map !k: T ! E_k. By Lemma 8.1, we may restrict attention to based operations r. The group axioms on E_klead (as in any Hopf algebra) to a formula of the form X *kr = rx1 + r0ffxr00ff+ 1xr in E*(E_kxE_k) ~=E*(E_k) b E*(E_k), ff where the r0ffand r00ffare also based. The only novelty is that the sum may be * *infinite. This translates into the desired Cartan formula X r(x + y) = r(x) + r0ff(x) r00ff(y) + r(y) in E*(X) (8:3) ff for any x; y 2 Ek(X). There is a similar Cartan formula for multiplication, given x 2 Ek(X) and y* * 2 Em (X), of the form X r(xy) = r0ff(x) r00ff(y) in E*(X), (8:4) ff for certain (other) based operations r0ffand r00ff(which depend on k and m). This suggests that an unstable algebra should consist of an E*-algebra M eq* *uipped with operations r that compose correctly and satisfy both Cartan formulae. This requires knowing the operations r0ffand r00ffin eqs. (8.3) and (8.4) for all r.* * In x10, we shall in effect expand both Cartan formulae explicitly. Second Answer We convert the First Answer to adjoint form, corresponding to the Fourth Answer in x5. (We skip the Second and Third Answers.) Everything becomes far cleaner, more evidence that this is the natural answer. Any element x 2 Ek(X), regarded as a map x: X ! E_k, induces the continuous homomorphism x*: E*(E_k) ! E*(X) of E*-algebras. By Thm. 1.18(a), E*(E_k) is Hausdorff and so in FAlg; we may therefore define, for any object M of FAlg, UkM = FAlg(E*(E_k); M); (8:5) the set of all continuous E*-algebra homomorphisms E*(E_k) ! M. This encodes the set of all possible actions on a typical element of degree k. We convert the a* *ction (8.2) to what we continue to call a coaction, aeX : Ek(X) --! Uk(E*(X)^) = FAlg(E*(E_k); E*(X)^); (8:6) by defining aeX x = x*, completing E*(X) if necessary to get it into FAlg. We a* *ssemble the sets UkM to form the graded set UM, which has the component (UM)k = UkM in degree k, and obtain aeX : E*(X) ! U(E*(X)^). JMB, DCJ, WSW - 39 - 23 Feb 1995 Unstable cohomology operations We compare UM with the stable and additive versions. Restriction to P E*(E_* *k) induces the natural transformation (oM)k: UkM = FAlg(E*(E_k); M) --! FMod (P E*(E_k); M) = AkM : (8:7) These form oM: UM ! AM. Composition with oeM: AM ! SM (see eq. (5.7)) yields UkM = FAlg(E*(E_k); M) --! FMod k(E*(E; o); M) = (SM)k; which is induced by the destabilization oe*k: E*(E; o) ! P E*(E_k) E*(E_k). Apparently only a morphism of graded sets, aeX has far more structure, than* *ks to the rich structure on the spaces E_k. Theorem 8.8 Assume that E*(E_k) is a free E*-module for all k (which is true* * for E = H(F p), BP , MU, KU, or K(n) by Lemma 4.17(a)). Then: (a) We can make the functor U, defined in eq. (8.5), a comonad in the catego* *ry FAlg of filtered E*-algebras; (b) If QE*(E_k) is a free E*-module for all k, o: U ! A (see (8.7)) is a mor* *phism of comonads in FAlg; (c) If E*(E; o) is a free E*-module, oe Oo: U ! S (see (8.7) and (5.7)) is a* * morphism of comonads in FAlg. Our main definition is now clear. Definition 8.9 An unstable (E-cohomology) algebra is just a U-coalgebra in FA* *lg, i. e. a complete Hausdorff filtered E*-algebra M equipped with a continuous mor* *phism aeM : M ! UM of E*-algebras that satisfies the coaction axioms [8, eq. (8.7)]. * *We then define the action of r 2 E*(E_k) on x 2 Mk by r(x) = aeM (x)r 2 M. A closed ideal J M is called (unstably) invariant if the quotient algebra * *M=J inherits a well-defined unstable algebra structure from M. It follows that the Cartan formulae (8.3) and (8.4) hold in M. The constant* * op- erations behave correctly because aeM (x) is required to be a morphism of E*-al* *gebras. We need to be able to recognize invariant ideals. Lemma 8.10 Given an unstable algebra M, a closed ideal J M is unstably in- variant if and only if r(y) 2 J for all y 2 J and all based operations r. Proof To make aeM=J well defined, we need r(x+y) r(x) mod J, for all x 2 M and y 2 J. This is trivial for constant operations r, and so by Lemma 8.1, we need * *only check for based r. The stated condition is obviously necessary, by taking x = 0* *. It is also sufficient, by eq. (8.3). | Theorem 8.11 Assume that E*(E_k) is a free E*-module for all k (which is tru* *e for E = H(F p), BP , MU, KU, or K(n) by Lemma 4.17(a)). Then: (a) For any space X, the coaction (8.6) factors through E*(X)^ to make E*(X)^ an unstable E-cohomology algebra; JMB, DCJ, WSW - 40 - 23 Feb 1995 x8. What is an unstable object? (b) We recover the additively unstable coaction (5.6) from aeX as aeX * o * E*(X) ---! U(E (X)^) --! A(E (X)^); (c) If E*(E; o) is Hausdorff, we recover the stable coaction [8, eq. (10.10)* *] from aeX as aeX o oe E*(X) ---! U(E*(X)^) --! A(E*(X)^) --! S(E*(X)^); (d) ae is universal: given an object B of FAlgand an integer k, any natural * *transfor- mation of sets X: Ek(X) ! FAlg(B; E*(X)^) (or bX: Ek(X)^ ! FAlg(B; E*(X)^), that is defined for all spaces X, is induced from aeX by a unique morphism f: B* * ! E*(E_k) in FAlg as aeX * * * X: Ek(X) ---! U(E (X)^) = FAlg(E (E_k); E (X)^) Mor(f;1) ------! FAlg(B; E*(X)^) Proof of Thms. 8.8 and 8.11 The proof breaks up into the same seven steps as a* *ddi- tively (and stably), in Thms. 5.8 and 5.12. However, it is far simpler than Thm* *s. 7.9 and 7.12 on algebras, because we are able to treat the multiplicative and module structures together. At each step, we also discuss o and o Ooe, assuming the e* *xtra conditions hold. Corollary 7.8 of [8] provides the E*-algebra object n 7! E_nin Ho. We again* * write aek for aeX when X = E_k. Step 1. We endow the functor U with an E*-algebra structure. For each obje* *ct M of FAlg, we observe that according to [8, Lemma 6.9], the functor E*(-)^ op Mor(-;M) FAlg(E*(-)^; M): Ho -----! FAlg -------! Set preserves enough products that by [8, Lemmas 7.6(a), 7.7(a)] it takes the E*-al* *gebra object n 7! E_n to the E*-algebra object UM in Set; i. e. UM is an E*-algebra. * *It is clear that UM is functorial in M. We shall filter it in Step 3. To see that oM is a homomorphism of E*-modules, we apply [8, Lemma 7.6(c)] to the E*-module object n 7! E_nin Gp(Ho ), using the natural transformation FAlg(E*(-)^; M) --! FMod (P E*(-)^; M) defined by restriction. To see that o is monoidal, we apply [8, Lemma 7.9(b)]. * *The monoidal structure of U is simply the multiplicative part of the algebra struct* *ure, and diag. (7.3) shows that the inclusions P E*(E_n) E*(E_n) form a morphism of graded monoid objects in FMod op. The units are correct by definition. For o Oo* *e, we bypass P E*(E_n) and use the duals of diags. (6.16) and (6.17) instead. Step 2. In order to define aeX (in (8.6)) as a morphism of E*-algebras, we * *consider the Set-valued natural transformation Ho (X; -) --! FAlg(E*(-)^; E*(X)^) induced by E*(-)^: Hoop ! FAlg. We apply [8, Lemma 7.6(c)] to the E*-algebra object n 7! E_n, to obtain Thm. 8.11(a). Then Thm. 8.11(b) is clear by comparing with the additive coaction (5.6), and for Thm. 8.11(c), we combine with Thm. 5.* *12(b). JMB, DCJ, WSW - 41 - 23 Feb 1995 Unstable cohomology operations Step 3. For U to take values in FAlg, we must filter UM. If M is filtered b* *y the ideals F aM, we filter UM by the ideals " ! # M F a(UM) = Ker UM --! U _____ F aM Just as stably, this filtration is complete Hausdorff and makes aeX continuous* * by naturality. This allows us to factor aeX through E*(X)^. Similarly, oM and oeM * *OoM are also filtered and therefore continuous. Step 4. We convert the object E*(X)^ of FAlg to the corepresented functor FX = FAlg(E*(X)^; -): FAlg ! Set. For example, when X = E_k, FX = Uk. As suggested by [8, eq. (8.16)], we also convert the coaction aeX to the natural t* *ransfor- mation aeX : FX ! FX U: FAlg! Set. Given M in FAlg, aeX M: FX M ! FX UM is thus defined on f 2 FX M = FAlg(E*(X)^; M) as (aeX M)f = Uf OaeX : E*(X)^ --! U(E*(X)^) --! UM; (8:12) an element of FX UM. Step 5. We define the natural transformation M: UkM = FAlg(E*(E_k); M) --! FAlg(E*(E_k); UM) = UkUM (8:13) by taking X = E_kin eq. (8.12). On the element f: E*(E_k) ! M of UkM, it is aek * Uf ( M)f: E*(E_k) --! UE (E_k) ---! UM : (In terms of elements, this is r 7! [s 7! f(r*s) = f(sr)].) If we substitute t* *he E*- algebra object n 7! E_n for X in eq. (8.12), [8, Lemma 7.6(c)] shows that M ta* *kes values in Alg. Naturality in M shows that M is filtered and so takes values in* * FAlg as required. Step 6. The other required natural transformation, fflM: UkM = FAlg(E*(E_k); M) --! M; is defined simply as evaluation on k 2 E*(E_k). As before, naturality in M sho* *ws that fflM is filtered, but we have to calculate that ffl is an E*-algebra homom* *orphism. Take any binary operation s(-; -) in E*-algebras (addition, multiplication,* * or any other), represented in Ho by the map s: E_kx E_m! E_q, which therefore indu* *ces s*q = s(p*1k; p*2m ). We need to show that the square s UkM x Um M ______UqM- |fflxffl ||ffl |? |? s Mk x Mm ________Mq- commutes. We evaluate on f 2 UkM and g 2 Um M. Because E*(E_kxE_m ) is by [8, Lemma 6.9] the coproduct in FAlg, there is a unique h: E*(E_kxE_m ) ! M in FAlg such that h Op*1= f and h Op*2= g. Then by definition of the algebra structure* * of UM, s(f; g) = h Os*: E*(E_q) ! M. Since h is an algebra homomorphism, ffls(f; g) = hs*n = hs(p*1k; p*2m ) = s(hp*1k; hp*2m ) = s(fk; gm ) = s(ffl* *f; fflg): JMB, DCJ, WSW - 42 - 23 Feb 1995 x9. Unstable, additive, and stable objects For unary and 0-ary operations, we may adapt the above proof, or simply thr* *ow away any unwanted arguments. (For example, given v 2 E*, we could define the constant binary operation s(x; y) = v1 in any E*-algebra, to deduce that fflv =* * v.) Step 7. The proof that E*(X)^ is a U-coalgebra and that U is a comonad is formally identical to the stable case, except that we need versions of [8, Lemm* *as 8.20, 8.22] for graded objects. We use [8, Lemma 8.24] to show that o: U ! A is a natural transformation of comonads. We take R as n 7! E*(E_n), R0 as n 7! P E*(E_n), 1R = 10Ras n 7! n, and u: P E*(E_k) E*(E_k) as the inclusion. The first hypothesis on u is the commutativity of the diagram P E*(E_k) ______________________-E*(E_k) | | | |aek | |? | |Paek FAlg(E*(E_k); E*(E_k)) | | | | | |? |? FMod (P E*(E_k); P E*(E_k))_____FMod- (P E*(E_k); E*(E_k)) which is obvious by construction, as r 2 P E*(E_k) yields r*|P E*(E_k). The proof of Thm. 8.11(d) is formally the same as stably. Since Ek(-) is re* *pre- sented by k 2 Ek(E_k), is classified by f = (E_k)k 2 FAlg(B; E*(E_k)). | 9 Unstable, additive, and stable objects In previous sections and [8], we constructed five different kinds of object* *: stable modules and algebras, additively unstable modules and algebras, and unstable al* *ge- bras. In this section we compare them all. Unstable modules are conspicuous by * *their absence; Thm. 9.4 will show that they cannot be defined compatibly with our oth* *er objects. Each kind of object is defined by a comonad. Theorems 8.8(b) and 7.9(b) pro* *vide natural transformations o oe U --! A --! S in FAlg (9:1) between the comonads that define unstable, additively unstable, and stable alge* *bras. Theorem 5.8(b) provides the natural transformation __ _oe__ A --! S in FMod (9:2) between the comonads that define additively unstable and_stable_modules (where * *we temporarily rename the module versions of A and S toA andS ). They are related* * __ to the algebra_versions by the forgetful functor V : FAlg! FMod , so that V A =* *A V and V S =S V . We have the category, e. g. U-coalgebras, of each kind of object. We consi* *der the diagram of categories and functors in fig. 5. For example, a stable alge* *bra B with coaction aeB :_B_! SB in FAlg yields the stable module V B with coaction V aeB : V B ! V SB =S V B in FMod . JMB, DCJ, WSW - 43 - 23 Feb 1995 Unstable cohomology operations Figure 5: Five kinds of object unstable ______-additivelyunstableo_stable-oe algebras algebras algebras | | |V |V |? ||? additively _oe unstable ______- stable modules modules Theorem 9.3 Assume that E*(E_k), QE*(E_k), and E*(E; o) are free E*-modules for all k (which is true for E = H(F p), BP , MU, KU, or K(n) by Lemma 4.17(a) and [8, Lemma 9.21]). Then we have the diagram fig. 5 of categories and functor* *s. For any space X, E*(X)^ is an object in each of the five categories, relate* *d by these functors. Proof The last assertion combines Thms. 5.12, 7.12, and 8.11 with Thms. 10.16 and 12.13 of [8]. | There is a glaring gap: we have not defined unstable modules. We now show that this gap cannot be filled, for rather silly reasons. In fact, the three mo* *st natural definitions are for stable modules, additively unstable modules, and unstable a* *lgebras. We can enrich the two kinds of module with multiplicative structure, but it is * *not possible to remove the multiplicative structure from the definition of unstable* * algebra. This is already strongly suggested by the appearance of multiplication in the C* *artan formula (8.3) for r(x+y). We ignore most of the structure and the topology, fix k, and restrict atten* *tion to the two functors Uk: FAlg! Ab and Ak: FMod ! Ab and the natural transformation ok: UkV ! Ak. Theorem 9.4 Even in the classical case E = H(F p), unstable_modules do not e* *xist in the sense that we cannot insert a suitable comonadU into diag. (9.2). Speci* *fically, for fixed k > 0 there do not exist: __k (i)a functorU : FMod ! Ab; __k (ii)a natural isomorphismU V ~= Uk: FAlg! Ab; __k (iii)a natural transformation _ok:U ! Ak of functors FMod ! Ab; __k such that on FAlg, _okV :U V ! Ak agrees with ok: Uk ! Ak. __k _ Proof We assume thatU and ok exist as stated and derive a contradiction. Giv* *en any (filtered) graded Fp-module M, we construct the Fp-algebra M+ = Fp M with the unit element 1 2 Fp and xy = 0 for all x; y 2 M. Then M is a retract in FMod JMB, DCJ, WSW - 44 - 23 Feb 1995 x9. Unstable, additive, and stable objects of V M+ and we can compute _okM from the commutative diagram FAlg(Ak; M+ ) |= ~ |? __k ________-__k + _______-=k + U M U V M U M |_okM |_okV M+ |okM+ |? |? |? __k ________o__ke + _______oek~+= A M A V M A M |= |= |? |? FMod (P Ak; M) FMod (P Ak; M+ ) where Ak = H*(H_k). Because M+ has no decomposables, every homomorphism P Ak ! M in the image of _okM kills the decomposable elements of P Ak (of which there are many). But for a general algebra B, okB: UkB ! AkB does not have this property, e.* * g. (okAk)idk 2 AkAk is the inclusion P Ak Ak. Taking M = V B shows that _okV B does not agree with okB. | Objects in ordinary cohomology Theorem 9.4 demands an immediate explana- tion of our terminology even in the case of ordinary cohomology. We give detail* *s for E = H(F 2); the case E = H(F p) for odd p is similar, with the usual changes. The Steenrod algebra A = E*(E; o) is exactly as expected: it is the F2-alg* *ebra generated by the Steenrod squares Sq ifor i > 0, subject to the standard Adem relations. It is useful to write Sq0 = . We note that for E = H(F 2): (i)oe*kmakes P E*(E_k) a quotient of E*(E; o); (ii)E*(E_k) is a primitively generated Hopf algebra. Below, M is to be an object of FMod (or FAlg), i. e. a complete Hausdorff * *filtered graded F2-module (or commutative F2-algebra). Topological conditions apply (whi* *ch we ignore for now). We list the five kinds of object we have defined, under our* * names for them: (i)A stable module M is an A-module. (ii)A stable algebra M is both an F2-algebra and an A-module that satisfies the Cartan formula Xk Sq k(xy) = (Sq ix)(Sq k-iy) for k > 0. i=0 It follows by induction that Sqk1M = 0 for all k > 0. (iii)An additively unstable module M is an A-module that satisfies the ext* *ra condition Sqix = 0 for all x 2 M and all i > deg(x). (9:5) Since Sq0x = x, it follows that Mn = 0 for all n < 0. (iv)An additively unstable algebra is a stable algebra that satisfies (9.5* *). JMB, DCJ, WSW - 45 - 23 Feb 1995 Unstable cohomology operations (v) An unstable algebra M is a stable algebra that satisfies (9.5) as well* * as the extra condition Sqk x = x2 for x 2 M and k = deg(x). The objects normally known as unstable modules appear here as additively unstab* *le modules (although the word "additively" could well be omitted, there being no d* *anger of confusion with something that does not exist). However, we do have two kinds of unstable algebra. We emphasize that in (iv* *), the squaring operation Mk ! M2k given by x 7! x2 (which looks additive but from our point of view is not, because it is defined only when M is an algebra) is u* *nrelated to Sqk. We have equivalent comodule descriptions in terms of E*(E; o) = F2[1; 2; 3;* * : :]: and the corresponding bigraded algebra Q(E)**= F2[0; 1; 2; : :]:, which has pol* *yno- mial generators i2 Q(E)12i(as we shall see in Thm. 16.2): (i)A stable comodule M has a coaction aeM : M --! M b E*(E; o) = M b F2[1; 2; 3; : :]: that satisfies the usual axioms [8, eq. (8.7)]. Then Sqk is dual to k1. (ii)A stable comodule algebra M is both a stable comodule and a commutative F2-algebra, in such a way that aeM is an algebra homomorphism. (iii)An unstable comodule M has coactions aeM : Mk --! Mi biQ(E)ki M b F2[0; 1; 2; : :]: that satisfy the coaction axioms (6.33). The unstable operation Sqi: * *k ! k + i is now dual to k-i0i1for i k, or is zero if i > k. (iv)An unstable comodule algebra M is an unstable comodule that is also a * *com- mutative F2-algebra, in such a way that aeM is an algebra homomorphis* *m. The special features of H(F 2) allow us to handle unstable algebras too: (v) For any x 2 Mk, aeM x contains the term x2 k1. Remark There is one candidate for an unstable module, but it does not work. One could try defining GkM = FMod (E*(E_k); M) for any object M of FMod , with aeX : Ek(X) ! GkE*(X) defined as usual, by aeX x = x*. We would like aeX to be at least additive, but the standard additive structure on FMod does not give t* *his. Indeed, it is easy to see that in general no abelian group structure on GkM* * makes aeX additive (not even for E = H(F p)). By [8, Lemma 7.7(d)], such a structure * *would have to be induced by some morphism : E*(E_k) ! E*(E_k) E*(E_k) in FMod . Take any r 2 E*(E_k) and write r = (r0; r00). Then additivity of aeX translate* *s into r(x+y) = r0x + r00y for all x; y 2 Ek(X), which is absurd unless r happens to be additive. In fact, these objects appear to be particularly devoid of interest. In th* *e case E = H(F 2), for example, they are modules equipped not only with Steenrod squar* *es Sq ithat behave as expected, but also operations such as x 7! (Sq 2x)(Sq 3x), w* *ithout having cup products. JMB, DCJ, WSW - 46 - 23 Feb 1995 x10. Enriched Hopf rings 10 Enriched Hopf rings In Defn. 8.9 we condensed all the structure of an unstable algebra down to * *the single word U-coalgebra. In this section, we unpack the information again to gi* *ve a complete description of an unstable algebra in the language of Hopf rings, enri* *ched with certain additional structure. This description is summarized in Thm. 10.4* *7, which may be regarded as the unstable analogue of Thm. 11.14 of [8] and Thm. 6.* *36. Indeed, we find a whole new paradigm for handling unstable operations, making computations with them reasonably practical and efficient. It serves as the tr* *ue successor to the Second Answer of x5 and [8, x10]. We assume in this section that E*(E_k) is a free E*-module for all k, which* * is true for our five examples by Lemma 4.17(a). Thus all the results of x8 are ava* *ilable, and by [8, Lemma 6.16(c)], the topological dual FMod *(E*(E_k); E*) of E*(E_k)* * is E*(E_k). We shall consistently identify (with some abuse of notation): (i)the cohomology operation r: Ek(-) ! Em (-); (ii)the cohomology class r(k) 2 Em (E_k), which we often write simply as r 2 Em (E_k); (10.1) (iii)the representing map of spaces r: E_k! E_m; (iv)the E*-linear functional : E*(E_k) ! E* of degree m. Remark In some situations, these identifications can obscure the correct signs* * in formulae. Considered as a cohomology class or functional, r has degree m, while* * its degree as an operation is m-k, and as a map of spaces, r has no degree at all. In any unstable algebra M, including E*(X)^ for any space X, Defn. 8.9 give* *s, for each x 2 Mk, the homomorphism aeM (x): E*(E_k) ! M. Then we defined r(x) = aeM (x)r 2 M for any operation (i. e. class) r 2 E*(E_k). In practice, we find * *it more convenient to revert to the First Answer r(x) of x8, although the Second Answer* *, in terms of aeM , will continue to inform us as to what to do, even when only impl* *icit. Classically, one investigates cohomology operations by studying what happens to* * r(x) when r is fixed and x varies; but it is clear from x8 that what we should do is* * fix x and allow r to vary. Linear functionals We need to develop a computational description of aeM in an unstable algebra M. We start from the fact that aeM (x) is E*-linear, i. e. r(* *x) is E*-linear in r. Definition 10.2 Let M be an unstable algebra, and fix an element x 2 Mk. We say r(x) is written in standard form if X r(x) = xff in M (for all r); (10:3) ff for suitable choices cff2 E*(E_k) and xff2 M, where deg(xff) = - deg(cff). If * *the sum is infinite, we require each ideal F aM in the filtration of M to contain a* *ll except finitely many of the xff. JMB, DCJ, WSW - 47 - 23 Feb 1995 Unstable cohomology operations This is the closest we will come to an unstable replacement for the tensor * *products and homomorphisms of x6 and [8, x11]. Our convention here and in all similar formulae is that r runs through all unstable cohomology operations having the c* *orrect domain degree (different in nearly every formula, and rarely specified) but arb* *itrary target degree. The indexing set for ff is often left implicit. It is easy to achieve eq. (10.3) in the universal form X r(x) = rff(x) in M (for all r); (10:4) ff by allowing cffto run through some basis of E*(E_k), which forces us to take xf* *f= rff(x), where rffdenotes the operation (linear functional) dual to cff. Contin* *uity of aeM (x): E*(E_k) ! M assures the finiteness condition in Defn. 10.2. We may the* *refore always assume that r(x) is written in standard form. Where we depart from tradition is in not picking a definite basis of E*(E_k* *) in advance. We do not even insist on the cffbeing linearly independent. Nor do we require the cffto span; we may obviously omit zero terms. This does not affect the linearity of eq. (10.3) and allows the flexibility that our formulae requir* *e. One consequence is that most cohomology operations will never acquire names. We have the analogue of Prop. 6.44. Proposition 10.5 Given x 2 Ek(X), regarded as a map of spaces x: X ! E_k, assume that r(x) is given by eq. (10.3). Then x*: E*(X) ! E*(E_k) is given by X X x*z = (-1)deg(cff)(deg(xff)+deg(z)) cff= cff :* * | ff ff The nonuniqueness in eq. (10.3) is really not a problem because we are usin* *g it to describe, not define the structure on M. The real definitions are all in x8; he* *re, we are only reinterpreting them. Nevertheless, it is easy to convert one standard* * form to another. Lemma 10.6 Any standard form (10.3) can be transformed into the universal fo* *rm (10.4), and hence into any other standard form, by iterating three kinds of rep* *lacement (in either direction): (i) =x0 x0+ x0; (ii) x0= (-1)deg(c) deg(v) vx0; (iii) x0+ =x00 (x0+ x00): (Infinitely many replacements may be needed; however, each F aM contains x0for * *all except finitely many of them.) | Stabilization We need to record how eq. (10.3) behaves when we restrict the op- eration r to be additive or stable. We recall from [8, Defn. 9.3] the stabiliz* *ation homomorphism oek*: E*(E_k) ! E*(E; o) and from eq. (6.2) the algebraic homomor- phism qk: E*(E_k) ! Q(E)k*, both of which have degree k under our conventions. JMB, DCJ, WSW - 48 - 23 Feb 1995 x10. Enriched Hopf rings Lemma 10.7 Let M be an unstable algebra, and assume that r(x) is expressed in the standard form (10.3), where x 2 Mk. Then: (a) The unstable comodule coaction aeM : Mk ! M b Q(E)k*is given by X aeM x = (-1)deg(xff)(k-deg(xff))xff qkcff in M b Q(E)k*, ff provided QE*(E_k) is a free E*-module; (b) The stable comodule coaction aeM : M ! M b E*(E; o) is given by X aeM x = (-1)deg(xff)(k-deg(xff))xff oek*cff in M b E*(E; o), ff provided E*(E; o) is a free E*-module. The signs are as expected, once we remember that if deg(xff) = i, then deg(* *cff) = -i and deg(qkcff) = deg(oek*cff) = k - i. P Proof For additive r, Prop. 6.11 converts eq. (10.3) to rAx = ff* *xff. We deduce aeM x in (a) by comparing eqs. (6.38) and (6.39). Part (b) is similar, u* *sing [8, eq. (11.18), eq. (11.19)] instead. | Unstable algebra structure Our task is to convert all the algebraic structure * *of an unstable algebra M in Defn. 8.9 into the current context. There are in effec* *t four pairs of axioms: (a) Two axioms to make aeM (x): E*(E_k) ! M an E*-algebra homomorphism, rather than merely E*-linear: (r ^ s)(x) = r(x)s(x) and 1(x) = 1M , wh* *ich will become eqs. (10.14) and (10.15); (b) Two axioms to make aeM : M ! UM E*-linear: aeM (x+y) = aeM (x) + aeM (* *y) and aeM (vx) = vaeM (x), which will become eqs. (10.20) and (10.16); (c) Two axioms to make aeM : M ! UM multiplicative: aeM (1M ) = 1UM and aeM (xy) = aeM (x)aeM (y), which will become eqs. (10.41) and (10.34); (d) Two axioms to make M a U-coalgebra: (sr)(x) = s(r(x)) and kx = x, which will become eqs. (10.45) and (10.43). The natural language for expressing the first three pairs is that of Hopf rings* *, while the last requires some additional structure. Hopf rings We recall from [8, Lemma 6.12] that in Coalg, tensor products of co* *al- gebras serve as products and E* is the terminal object. A commutative (graded) * *ring object in Coalg is called a Hopf ring over E*. (The terminology and some of the notation were suggested by Milgram [17]; see [23, x1] for a detailed exposition* *.) We start from the E*-algebra object n 7! E_nin Ho provided by [8, Cor. 7.8]* *. We apply [8, Lemma 7.6(a)], using the homology functor E*(-), which takes values in Coalg on the spaces we need and preserves enough products to make n 7! E*(E_n) an E*-algebra object in Coalg. In particular, this is an E*-module object, and * *each E*(E_k) is an abelian group object in Coalg and thus a Hopf algebra. There are seven parts to the Hopf ring structure of n 7! E*(E_n): two from * *the coalgebra, three from the abelian group object E_k, and two from the multiplica* *tive JMB, DCJ, WSW - 49 - 23 Feb 1995 Unstable cohomology operations monoid object, in addition to the underlying E*-module structure on E-homology. They are as follows (for each k and m, where relevant): (i) : E*(E_k) ! E*(E_k) E*(E_k), the comultiplication induced by the dia* *go- nal map : E_k! E_kx E_k; (ii)ffl: E*(E_k) ! E*, the counit for , induced by the map q: E_k! T ; (iii)*: E*(E_k) E*(E_k) ! E*(E_k), a multiplication, induced by the addit* *ion map k: E_kx E_k! E_k; (iv)1k = !k*1 2 E0(E_k), the *-unit element, induced by the zero map !k: T* * ! E_k; (v) O: E*(E_k) ! E*(E_k), the canonical (anti)automorphism of the Hopf alg* *ebra E*(E_k), induced by the inversion map k: E_k! E_k; (vi)O: E*(E_k) E*(E_m ) ! E*(E_k+m ), another multiplication, induced by * *the multiplication map OE: E_kx E_m ! E_k+m; (vii)[1] = j*1 2 E0(E_0), the O-unit element, induced by the algebra unit m* *ap j: T ! E_0. Because n 7! E*(E_n) is an E*-algebra object rather than merely a ring obje* *ct, we have, for each v 2 Eh, the actions (v)*: E*(E_k) ! E*(E_k+h). As in x6, this re* *duces to a simpler structure. Definition 10.8 We define the right unit function jR: E* ! E*(E_*). We regard v 2 Eh = Eh(T ) as a map v: T ! E_h, and use the induced homomorphism v*: E* ~= E*(T ) ! E*(E_h) to define [v] = v*1 2 E0(E_h) and jR(v) = [v]. In particular, this includes [1] = j*1 as in (vii), and [0k] = !k*1 = 1k as* * in (iv). It is clear from Defn. 6.19 and [8, Defn. 11.2] that qh[v] and oeh*[v] are the add* *itive and stable versions of jRv. The elements [v] determine the E*-module object struct* *ure completely, because when we apply E-homology to [8, eq. (7.5)], we obtain (v)*c = [v] Oc for all c 2 E*(E_*). (10:9) For the sake of completeness, we list all 33 laws that a Hopf ring satisfie* *s, beyond the usual axioms for an E*-module. (Your count may vary.)PMost need no comment. They are as follows, where in several we write c = ic0i c00i: (i)The five operations are (bi)additive: (b+c) = b + c, ffl(b+c) = ffl* *b + fflc, (a+b) * c = a*c + b*c, O(b+c) = Ob + Oc, and (a+b) Oc = aOc + bOc; P 0 00 (ii)The five operations are E*-linear: (vc) = ivcici, ffl(vc) = vfflc, * *(vb)*c = v(b * c), O(vc) = vOc, and (vb) Oc = v(b Oc), for all v 2 E*; (iii)Three coalgebra axioms: is coassociativePand cocommutative (with t* *he standard sign), and ffl is a counit: i(fflc0i)c00i= c; (iv)The five parts of the ring object structure respect : (b * c) = ( b)* * * ( c) (where we give E*(E_k) E*(E_k) the obvious *-multiplication,Pwith sig* *ns), (b Oc) = ( b) O( c) (similarly), 1k = 1k 1k, Oc = iOc0i Oc00i, a* *nd [1] = [1] [1]; JMB, DCJ, WSW - 50 - 23 Feb 1995 x10. Enriched Hopf rings (v) The five parts of the ring object structure respect ffl: ffl(b * c) =* * (fflb)(fflc), ffl1k = 1, fflOc = fflc, ffl(b Oc) = (fflb)(fflc), and ffl[1] = 1; (vi)Four abelian group object axioms: associativity (a * b) * c = a * (b * ** c), commutativity b * c = (-1)ijcP* b (where i = deg(b), j = deg(c)), unit 1k * c = c, and inverse iOc0i* c00i= (fflc)1k; (vii)Three axioms for a commutative monoid: associativity (a Ob) Oc = a O(b* * Oc), commutativity, which takes the somewhat complicated form (see * *[23, Lemma 1.12(c)(v)]) b Oc = (-1)ijOkmc Ob (10:10) for b 2 Ei(E_k) and c 2 Ej(E_m ) (where Okm = O if k and m are odd, an* *d is the identity otherwise, as in Prop. 10.12(b) below), and [1] Oc = c; (viii)Three ring object axioms to state that - Oc respects the abelian group* * object structure: for addition, which yields the distributive law, in the com* *plicated form [ibid. (vi)] X 0 (a * b) Oc = (-1)deg(ci) deg(b)aOc0i* bOc00i;(10:11) i for the zero, 1m Oc = (fflc)1m+k [ibid. (ii)]; and for the inverse, O* *(b Oc) = (Ob) Oc. Many standard laws follow from these axioms. In order to simplify notation in eq. (10.11) and elsewhere, we give O-multiplication greater binding strength th* *an *- multiplication, so that a * b Oc always means a * (b Oc), never (a * b) Oc. In * *all our Hopf rings, Prop. 11.2 will provide the laws relating the added elements [v] and ide* *ntify the useful element O[1] with [-1]. Proposition 10.12 In any Hopf ring, the operation O has the following proper* *ties: (a) Oc = O[1] Oc, so that O[1] determines O; (b) OOc = c; (c) O(a * b) = Oa * Ob; (d) O[1] OO[1] = [1]. Proof For (a), Oc = O([1] Oc) = O[1] Oc. Since [1] = [1] [1] and hence O[1* *] = O[1] O[1], the distributive law gives (c), by O(a * b) = O[1] O(a * b) = O[1]Oa * O[1]Ob = Oa * Ob : Also, we have O[1] * [1] = 10 and similarly OO[1] * O[1] = 10, which yield OO[1] = OO[1] * 10 = OO[1] * O[1] * [1] = 10 * [1] = [1]: But (a) gives OO[1] = O[1] OO[1], and hence (d) and the general case of (b). | Generators We wish to use the laws to reduce any element of a Hopf ring to some standard form. The distributive law (10.11) plays a key role. We shall describe* * our Hopf rings H by specifying two sets of elements: (i)the O-generators of H; JMB, DCJ, WSW - 51 - 23 Feb 1995 Unstable cohomology operations (ii)the *-generators of H, each of which is a O-product of O-generators and possibly O[1], where we allow the empty O-product [1]. We require every element of H to be an E*-linear combination of *-products of t* *he *-generators of H; in other words, the *-generators generate H as an E*-algebra. For each O-generator g, we need formulae for g (so we can expand eq. (10.11)),* * fflg, and Og. Although Hopf rings tend to be huge, each of our examples (see x17) has* * a conveniently small set of O-generators. Hopf rings over Fp One can define the Frobenius operator F c = c*pin any algeb* *ra with multiplication *, and it is multiplicative if * is commutative. It is addi* *tive if also the ground ring has characteristic p. It is most useful when the ground ring i* *s Fp, because it is then automatically Fp-linear. Commutativity of *-multiplication i* *mplies that F c = 0 whenever c has odd degree (unless p = 2). Moreover, in a Hopf ring (or cocommutative Hopf algebra) H over Fp, one has dually the Verschiebung operator V : H ! H, defined so that DV = F : DH ! DH in the dual Hopf algebra. It divides degrees by p. Then V c = 0 unless deg(c)* * is divisible by 2p (if p 6= 2). Both F and V preserve all the Hopf algebra struc* *ture: F (a * c) = F a * F c, F 1k = 1k, F c = (F F ) c, fflF c = fflc, and dually V* * (a * c) = V a * V c, V 1k = 1k, V c = (V V ) c, and fflV c = fflc. For O-products, we c* *an iterate eq. (10.11) and obtain the identity a O(F c) = F (V a Oc); (10:13) which is useful for reducing elements of the Hopf ring to standard form. (Norma* *lly, a and c both have even degree.) Multiplication of operations The first pair of axioms on M we listed earlier, * *that for fixed x 2 M, aeM (x) is a homomorphism of E*-algebras, is easily translated* * into Hopf rings. Because the diagonal map in E_kinduces both the cup product r ^ s a* *nd the comultiplication on E*(E_k), we can write down the cup product from eq. (* *10.3) as X X (r ^ s)(x) = xfl= xfl in M : fl fl The product r(x)s(x) becomes, after some shuffling, X X r(x)s(x) = (-1)deg(xff) deg(xfi) xffxfi: ff fi Since (r ^ s)(x) = r(x)s(x) has to hold for all r and s, we deduce the identity X X X cfl xfl= (-1)deg(xff) deg(xfi)cff cfi xffxfi (10:14) fl ff fi in (E*(E_*) E*(E_*)) b M, where the tensor products are formed using only the usual left E*-actions. The identity element 1k 2 E0(E_k) is the constant operation Ek(X) ! E0(X) that sends everything to 1X ; regarded as a linear functional, it is simply ffl* *. In terms of eq. (10.3), the axiom 1k(x) = 1M becomes X (fflcff) xff= 1M in M : (10:15) ff JMB, DCJ, WSW - 52 - 23 Feb 1995 x10. Enriched Hopf rings Linear structure We next decode the statement that aeM : M ! UM is linear, namely that aeM (x+y) = aeM (x) + aeM (y) and aeM (vx) = vaeM (x). Related to t* *he first is the formula for r*(b * c), which can be shown to be the translation of the s* *tatement that M: UM ! UUM is additive. We assume that r(x) is given by eq. (10.3), where x 2 Mk. The v-action UkM ! Uk+hM was given by composing with (v)*: E*(E_k+h) ! E*(E_k); dually, we use eq. (10.9) to translate aeM (vx) = vaeM (x) into X r(vx) = xff in M (for all r): (10:16) ff For addition, the idea is that k: E_kx E_k ! E_k induces both the additive structure in UM and the *-multiplication in E*(E_k). Of course, r*c is not addi* *tive in r. Given two operations r; s: k ! m, their sum may be constructed as rxs m r + s: E_k--! E_kx E_k---! E_m x E_m ---! E_m ; as we can check by composing with x: X ! E_k. When we apply E-homology, we find X (r + s)*c = r*c0i* s*c00i in E*(E_m ), (10:17) i P 0 00 if we write c = ici ci for c 2 E*(E_k). (In other words, we add r* and s* according to the group structure on Mod (E*(E_k); E*(E_m )) described by Milnor* * and Moore in [19, Defn. 8.1], which makes use of the coalgebra structure of E*(E_k)* * and the algebra structure of E*(E_m ).) To add more than two operations, we need it* *erated coproducts: given any finite indexing set , we write the iterated comultiplica* *tion : E*(E_k) ! ff2E*(E_k) in the form X c = ci;ff in ff2E*(E_k) (10:18) i ff for suitable elements ci;ff2 E*(E_k). We can of course replace E_k by any space* * for which we have the necessary K"unneth formulae. Theorem 10.19 Let M be an unstable algebra and assume that E*(E_k) is a free E*-module for all k. Take x; y 2 Mk and assume that r(x) is in the standard form (10.3). Then: (a) We have the Cartan formula for addition X r(x + y) = xffr00ff(y) for all r: k ! m, (10:20) ff where for each ff, the operation r00ff: k ! m + deg(cff) is defined as having t* *he func- tional = (-1)deg(cff)(m+deg(cff)) for all c 2 E*(E_k);* *(10:21) (b) If, similarly, r(y) has the standard form X r(y) = yfi; (10:22) fi JMB, DCJ, WSW - 53 - 23 Feb 1995 Unstable cohomology operations then we have the full Cartan formula for addition, X X r(x+y) = (-1)deg(xff) deg(yfi) xffyfi(10:23) ff fi for all r: k ! m; (c) Assume a; b 2 E*(E_k). Let cffrunPthrough a basis ofPE*(E_k), and denote* * by r0ffthe operation dual to cff. Let a = i ffai;ffand b = j ffbj;ffbe the i* *terated coproducts of a and b as in eq. (10.18), where in both cases, we ignore those f* *f for which r0ff*ai;ff= (fflai;ff)1 for all i (10:24) (see the Remark following). Then the homology homomorphism r*: E*(E_k) ! E*(E_m ) satisfies X X r*(a * b) = * r0ff*ai;ffOr00ff*bj;ff in E*(E_m ),(10:25) i j ff where r00ffis defined by eq. (10.21) and the only signs come from shuffling the* * factors to form (a x b). Remark The formula (10.25) demands some explanation. The proof will show that the relevant set of ff is in fact finite, so that the iterated coproducts a and* * b are defined. If ff satisfies eq. (10.24), we have r0ff*ai;ffOr00ff*bj;ff= (fflai;ff)1 Or00ff*bj;ff= (fflai;ff)fflr00ff*bj* *;ff= (fflai;ff)(fflbj;ff)1m : In the usual (and sufficient) case when ffla = fflb = 0, we can easily arrange * *for each ai;ffand bj;ffto be 1 or lie in Ker ffl, by breaking up terms and shuffling as * *necessary. Then the ij-term contributes nothing to r*(a * b) unless ai;ff= 1 and bj;ff= 1 * *for all ff 2 that satisfy eq. (10.24). Such an index ff may be omitted from the *-prod* *uct in eq. (10.25) and the iterated coproducts a and b. Proof We first assume that the cffform a basis of E*(E_k), so that xff= r0ff(x* *) as in eq. (10.4). By the K"unneth homeomorphism, we can write X *kr = r0ffx r00ff in E*(E_kxE_k), (10:26) ff for uniquely determined elements r00ff2 E*(E_k). In other words, in the diagram u ______-r0ffxr00ff X _________E_kx-E_k E_?x E_? Q Q | | Q Qx+y ||k ||OE (10:27) Q Q | | Qs |? r |? E_k ___________E_m- the map r Ok is expressed as the sum of the maps gff= OE O(r0ffxr00ff), and is * *the universal example for computing r(x+y), where u: X ! E_k x E_k has coordinates JMB, DCJ, WSW - 54 - 23 Feb 1995 x10. Enriched Hopf rings x: X ! E_kand y: X ! E_k. Evaluation on cffx c identifies r00ffas in eq. (10.21* *), with the help of <*kr; cffxc> = = : Then eq. (10.20) is induced from eq. (10.26). To deduce (b), we substitute eq. * *(10.22) in eq. (10.20) and watch the signs. To remove the requirement that the cffform a basis, we note that by lineari* *ty, eq. (10.20) is preserved by each of the replacements listed in Lemma 10.6. (The operation r0ffis no longer defined, but appears only in (c).) For (c), we apply homology everywhere. We have to add the homomorphisms gff* in the sense of eq. (10.17), using the iterated coproduct (a x b), which is obt* *ained from a x b by shuffling. We note that any a 2 E*(E_k) comes from some finite subcomplex Y of E_k. All but finitely many of the r0ffvanish on Y , by the stro* *ng duality for E_k; these ff satisfy eq. (10.24), as we see by computing the iterated copr* *oduct a first in Y , since the zero operation 0: k ! m induces 0*c = (fflc)1m . | Similarly, the zero map !k: T ! E_kand inversion map k: E_k! E_kof E_kyield the useful formulae r(0k) = 1M in M (for all r) (10:28) and X r(-x) = xff in M (for all r): (10:29) ff For some applications, it is useful to cut out the finiteness argument in t* *he proof of Thm. 10.19(c) and work directly in a finite space Y . Proposition 10.30 Let f: Y ! E_k be a map, where E*(Y ) is a free E*-module of finite rank, with basis elements zff. Let yff2 E*(Y ) be dual to zff. Then f* *or any a 2 E*(Y ), b 2 E*(E_k), and operation r: k ! m, X X r*(f*a * b) = * yff*ai;ffOr00ff*bj;ff in E*(E_m ), i j ff where r00ff: k ! m + deg(zff) denotes the operation having the functional = (-1)deg(zff)(m+deg(zff)); a and b are computed as in eq. (10.18), and we use yff*: E*(Y ) ! E*(E_?). Proof By Thm. 1.18(a), E*(Y ) is dual to E*(Y ) and yffis defined. We modify t* *he proof of the Theorem by composing the square in diag. (10.27) with f x 1: Y x E* *_k! E_kPx E_k. We work in E*(Y xE_k) instead of E*(E_kxE_k) and write (f x1)**kr = ffyffx r00ff. We evaluate this on zffx c to determine r00ff. | Remark The commutativity of *-multiplication ensures that r(x+y) = r(y+x). Conversely, one could say that x + y = y + x in M requires *-multiplication to be commutative. The universal example has M = E*(E_kxE_k), x = k x 1, and y = 1 x k, and cffand dfirun through bases of E*(E_k). Then r(x+y) = r(y+x) for all r implies that cff* dfi= dfi* cfffor all ff and fi. The commutativity * *of * in general follows by linearity. JMB, DCJ, WSW - 55 - 23 Feb 1995 Unstable cohomology operations Similar discussions hold for other laws in a ring. In particular, x + 0 = x* * corre- sponds in this way to c * 1k = c, -(-x) = x to OOc = c, -(x + y) = (-x) + (-y) * *to O(a * b) = Oa * Ob, and the associativity of + to the associativity of *. Given a prime p, we can iterate eq. (10.23) to get X r(px) = r(x+x+ : :+:x) = xff1xff2: :x:ffp: If the indices ffi are not all the same, we can permute them cyclically and obt* *ain p distinct terms which by commutativity are all equal, with the same sign. This l* *eaves only the terms with ffi= ff for all i, and we find X r(px) F xffmod p: (10:31) ff This is particularly useful when E* has characterisitic p, so that px = 0, beca* *use comparison with eq. (10.28) then yields X F xff= 1M in M (for all r):(10:32) ff Multiplicative structure The multiplication maps OE: E_kx E_m ! E_k+m induce both the multiplication in UM and the O-multiplication in E*(E_*). This allows * *us to translate the axiom that aeM is multiplicative, aeM (xy) = aeM (x)aeM (y) in U* *M. Theorem 10.33 Let M be an unstable algebra, and assume that E*(E_k) is a fr* *ee E*-module for all k. Take x 2 Mk and y 2 Mm and assume that r(x) is in the standard form (10.3). Then: (a) We have the Cartan formula for multiplication X r(xy) = xffr00ff(y) for all r: k + m ! h, (10:34) ff where for each ff, the operation r00ff: m ! h + deg(cff) is defined as having t* *he func- tional = (-1)deg(cff)(h+deg(cff)) for all c 2 E*(E_m()* *;10:35) (b) If, similarly, r(y) is given by eq. (10.22), we have the full Cartan for* *mula for multiplication, X X r(xy) = (-1)deg(xff) deg(yfi) xffyfi(10:36) ff fi for all r: k + m ! h; (c) Take a 2 E*(E_k) and b 2 E*(E_m ). Assume that cffruns throughPa basis of E*(E_k),Pand denote by r0ffthe operation dual to cff. Let a = i ffai;ffa* *nd b = j ffbj;ffbe the iterated coproducts of a and b as in eq. (10.18), where * *in both cases, we ignore all ff that satisfy eq. (10.24). Then the homology homomorphi* *sm r*: E*(E_k+m ) ! E*(E_h) satisfies X X r*(a Ob) = * r0ff*ai;ffOr00ff*bj;ff in E*(E_m ),(10:37) i j ff where r00ffis defined by eq. (10.35) and the only signs come from shuffling the* * factors to form (a x b). JMB, DCJ, WSW - 56 - 23 Feb 1995 x10. Enriched Hopf rings The Remark following Thm. 10.19 applies. Proof This is formally identical to the proof of Thm. 10.19, with k: E_kx E_k!* * E_k replaced everywhere by OE: E_kx E_m ! E_k+m. | By naturality, we can adapt eq. (10.36) to x-products. Corollary 10.38 Given spaces X and Y and elements x 2 Ek(X) and y 2 Em (Y ), assume that r(x) and r(y) are given by eqs. (10.3) and (10.22). Then we have t* *he Cartan formula X X r(xxy) = (-1)deg(xff) deg(yfi) xffxyfi(10:39) ff fi in E*(X xY ), for any operation r: k + m ! h. | We have also the analogue of Prop. 10.30. Proposition 10.40 Let f: Y ! E_k be a map as in Prop. 10.30. Then for any a 2 E*(Y ), b 2 E*(E_k), and operation r: k + m ! h, X X r*(f*a Ob) = * yff*ai;ffOr00ff*bj;ff in E*(E_h), i j ff where r00ff: m ! h + deg(zff) denotes the operation having the functional = (-1)deg(zff)(m+deg(zff)) and a and b are computed as in eq. (10.18). | Since the unit element of UM is jM Oj*: E*(E_0) ! E* ! M, the axiom aeM (1M* * ) = 1UM translates easily into r(1M ) = 1M = 1M = 1M in(M10:41) for all r. Just as with addition, certain laws in the Hopf ring correspond to laws in * *an E*- algebra M. For example, associativity of O-multiplication corresponds to associ* *ativity of multiplication in M. Commutativity is slightly trickier: given x 2 Mk and y * *2 Mm , r(yx) = r((-1)kmxy) leads to the identity (10.10), thus explaining the signs an* *d the appearance of O. The comonad structure Finally, we translate the two axioms which state that aeM makes M a U-coalgebra. Since we have in effect returned to the First Answe* *r of x8, these are the usual axioms for an action, (sr)(x) = s(r(x)) and kx = x. The second is easily handled. From Prop. 6.11, we can use (6.41) to express* * the identity operation k as the functional = Q(ffl) Oqk: E*(E_k) --! Q(E)k*--! E*(E; o) --! E*: (10:42) When we put r = k, eq. (10.3) expands easily to yield the axiom X (Q(ffl)qkcff) xff= x in M (10:43) ff JMB, DCJ, WSW - 57 - 23 Feb 1995 Unstable cohomology operations for x 2 Mk. We have thus interpreted the counit natural transformation fflM: UM* * ! M of the comonad U, which was defined by (fflM)f = fk. The functional Q(ffl) Oq* *k = fflS Ooek*is not part of the Hopf ring structure as given so far, so we add it.* * (It is unrelated to the counit ffl: E*(E_k) ! E* of the Hopf algebra E*(E_k).) It is easy to recover the functional from r*, as in eq. (6.50), in t* *he form r* oem* fflS * : E*(E_k) --! E*(E_m ) ---! E*(E; o) --! E ; (10:44) by writing = = and using eq. (10.42). In the additiv* *e context, the reverse construction of r* from was neatly encoded in the comultipli* *cation Q( ) on Q(E)**. Here, we have no such map and must rely on the definition of M, which explicitly uses r*. In effect, we dualize and use r* instead. The first is the most complicated of all the axioms. When we substitute sr * *and r in eq. (10.3) and use = = , the axiom (sr)(x) * *= s(r(x)) expands to X X xff= s(r(x)) = s xff in M, (10:45) ff ff for all r, s. The right side is to be expanded using eqs. (10.20) and (10.16), * *and in general is extremely complicated. Our conclusion is that we need to know the induced homology homomorphism r*: E*(E_k) ! E*(E_m ) for every operation r: Ek(-) ! Em (-). This is the final* * piece of structure to add to the Hopf ring. To compute it successfully, we need r*c f* *or each O-generator c of E*(E_*), and then use formulae (10.25) and (10.37) for r*(a * * *b) and r*(a Ob). Summary We collect the various formulae to form the main theorem of this secti* *on. In addition to the Hopf ring structure on E*(E_*), we need: (i)The element [v] 2 E0(E_*) for each v 2 E* (see Defn. 10.8); (ii)The augmentation (see eq. (10.42)) Q(ffl) Oqk: E*(E_k) --! Q(E)k*--! E*(E; o) --! E* (10.46) which may be written fflS Ooek*; (iii)The homomorphism r*: E*(E_k) ! E*(E_m ) induced by each opera- tion r: k ! m. These constitute what we mean by an enriched Hopf ring structure. Theorem 10.47 Let M be an object of FAlg, i. e. a complete Hausdorff filte* *red E*-algebra, and assume that E*(E_k) is a free E*-module for all k (which is tru* *e for E = H(F p), BP , MU, KU, or K(n) by Lemma 4.17(a)). Then an unstable algebra structure on M consists of a value r(x) 2 M for all x 2 M and all r 2 E*(E_k) (* *where k = deg(x) and r(x) 2 Mm if r 2 Em (E_k)), which is E*-linear in r and therefo* *re (for fixed x) expressible in the standard form (10.3) X r(x) = xff in M (for all r): ff These values are subject to the following axioms: JMB, DCJ, WSW - 58 - 23 Feb 1995 x11. The E-cohomology of a point (a) For fixed x 2 Mk, r(x) satisfies the three consistency conditions: X X X (i) cfl xfl = (-1)deg(xff) deg(xfi)cff cfi xffxfi fl ff fi in (E*(E_k) E*(E_k)) b M; X (ii) (fflcff)=xff1M in M ; ff X (iii) (fflSoek*cff)=xffx in M; ff (b) As x varies, r(x) satisfies the following identitiesPin M for all r, whe* *re we assume similarly (as in eq. (10.22)) that r(y) = fi yfi: X X (i) r(x + y)= (-1)deg(xff) deg(yfi) xffyfi; ff fi X (ii) r(vx) = xff; ff X X (iii) r(xy) = (-1)deg(xff) deg(yfi) xffyfi; ff fi (iv) r(1M )= 1M ; (c) The composition law X X xff= s(r(x)) = s xff in M ff ff holds for all r, s, and all x 2 M; (d) For each of the ideals F aM in the filtration of M: (i)For fixed x 2 M, all except finitely many of the xfflie in F aM; (ii)There exists F bM such that r(x) 2 F aM for all x 2 F bM and all based operations r. Proof The equations in (a) are (10.14), (10.15), and (10.43). Those in (b) are* * (10.23), (10.16), (10.36), and (10.41). The equation in (c) is (10.45). In (d), (i) stat* *es that aeM (x): E*(E_k) ! M is continuous for each x, while (ii) states that aeM : M !* * UM is continuous. | Remark By (b), an unstable algebra structure on M is determined by the values r(x) on a set of (topological) E*-algebra generators x of M. Moreover, the Hopf* * ring laws imply that it is sufficient to verify axioms (a) and (d)(i) on these gener* *ators. In practice, the topological conditions (d) rarely cause us any distress. 11 The E-cohomology of a point In this section, we study the first of our test spaces, the one-point space* * T , for which E*(T ) is by definition the coefficient ring E*. Its unstable structure * *is com- pletely determined by eqs. (10.41) and (10.16) as r(v) = in E* = E*(T ) (for all r); (11:1) JMB, DCJ, WSW - 59 - 23 Feb 1995 Unstable cohomology operations which may be taken as an alternate definition of the Hopf ring elements [v], in* *stead of Defn. 10.8. It is easy to deduce how [v] interacts with each piece of the structure on * *E*(E_*). Much of this can be read off from the Hopf ring structure in x10. In particular* *, jR is still in some sense a ring homomorphism. Proposition 11.2 The Hopf ring elements [v] 2 E0(E_h) for each v 2 Eh have t* *he properties: (a) [v] = [v] [v]; (b) ffl[v] = 1; (c) [v + v0] = [v] * [v0] for v02 Eh; (d) [-v] = O[v]; (e) [vv0] = [v] O[v0] for v02 Ek; (f) 1m O[v] = 1m+h ; (g) r*[v] = [] (for all r); (h) r*1h = []; (i) qh[v] = jRv in Q(E)h0; (j) oeh*[v] = jRv in E-h(E; o), under stabilization. Proof For (a) and (b) we substitute eq. (11.1) in eqs. (10.14) and (10.15). Fo* *r (c) and (e), we write down the Cartan formulae (10.23) and (10.36) for r(v+v0) and r(vv* *0) and compare with eq. (11.1). For (d), we write down r(-v) from eq. (10.29) and compare with eq. (11.1). For (g), we use eq. (11.1) to compute s(r(v)) = ]>; by eq. (10.45), this must agree with for all s. Since [0n] = 1n, (f)* * and (h) are special cases of (e) and (g). For (i) and (j), we compare eq. (11.1) with eq. (* *6.43) and [8, eq. (11.23)], and use eqs. (6.14) and (6.13). | Constant operations Constant operations were introduced briefly in x8. Although they are of no real interest and contain no information, they are undeniably na* *tural and we have to be able to recognize them in their several disguises. Proposition 11.3 Let r: k ! m be the constant operation defined by r(x) = v1X for all x 2 Ek(X), where v 2 Em . Then: (a) As a class, r = v1k 2 E*(E_k); (b) As a map, r is the composite v Oq: E_k! T ! E_m; (c) As a functional, = (fflc)v in E* for all c 2 E*(E_k); (d) r*: E*(E_k) ! E*(E_m ) is given by r*c = (fflc)[v] for all c 2 E*(E_k). * * | Based operations Given a based space (X; o), we consider the naturality of an operation r: k ! m with respect to the inclusion of the basepoint. We augment Lemma 2.3. Proposition 11.4 The following conditions on an operation r: k ! m are equiv* *a- lent: (a) r(0) = 0 in E*(T ) = E*, i. e. r is based in the sense of Defn. 2.2; JMB, DCJ, WSW - 60 - 23 Feb 1995 x12. Spheres, suspensions, and additive operations (b) r(0) = 0 in E*(X) for all spaces X; (c) The operation r induces r: Ek(X; o) ! Em (X; o) for all X; (d) The class r lies in Em (E_k; o) Em (E_k); (e) The map r: E_k! E_m is based (up to homotopy); (f) The linear functional satisfies = 0; (g) The homomorphism r*: E*(E_k) ! E*(E_m ) satisfies r*1k = 1m . Proof Part (b) is equivalent to (a) by naturality. Because r(0) = 1X * * by eq. (10.28), (f) is equivalent to (b), and with the help of Prop. 11.2(h), to (* *g). | We can generalize (f). Lemma 11.5 Let (X; o) be a based space. Then for any x 2 Ek(X; o) and any operation r: k ! m, we have r(x) 1X mod E*(X; o). Proof We use eq. (10.28) and the naturality of r in diag. [8, eq. (3.2)]. | It is sometimes useful to be more specific. If we choose a basis of E*(E_k)* * consisting of 1k and elements cff2 Ker ffl, then for any x 2 Ek(X; o), eq. (10.4) takes th* *e form X r(x) = 1X + xff in E*(X)^ (for all;r)(11:6) ff where the elements xff2 E*(X; o)^. Formulae are often simpler for based operations, but the case of general r * *can be recovered easily enough by decomposing as in Lemma 8.1. Lemma 11.7 If we write r(x) = s(x)+v1X , where s is a based operation and v * *2 Em , the homology homomorphism r*: E*(E_k) ! E*(E_m ) is given by r*c = s*c * [v], w* *here we recognize v = r(0) = . Proof We write r as the composite 1xq sxv m E_k--! E_k x E_k---! E_k x T ---! E_m x E_m ---! E_m and take E-homology. The first two maps just give E_k~= E_kx T . | 12 Spheres, suspensions, and additive operations So far, except for adding an extra grading, our additive results are formal* *ly very similar to the stable case discussed in [8]. What is new is that suspension is * *no longer an isomorphism, but defines a new element e. The stable results can all be reco* *vered by stabilizing, which consists merely of setting e = 1. We assume throughout that E*(E_k), QE*(E_k), and E*(E; o) are free E*-modul* *es, so that we have available the machinery of comodule algebras of xx6, 7 as well * *as the stable results of [8]. In particular, the coaction aeX : E*(X) ! E*(X) b Q(E)** **is a homomorphism of E*-algebras for any X. Spheres Our second test space, after the one-point space T , is the circle S1.* * Its cohomology E*(S1; o) is a free E*-module with the basis {1S; u1}, where the can* *onical generator u1 2 E1(S1; o) is provided by [8, Defn. 3.23]. Thus aeS: E*(S1) ! E*(* *S1) Q(E)**is determined by aeSu1. JMB, DCJ, WSW - 61 - 23 Feb 1995 Unstable cohomology operations Definition 12.1 We define the suspension element e = eQ 2 Q(E)11by the identi* *ty aeSu1 = u1 e in E*(S1; o) Q(E)1*~=Q(E)1*. (12:2) It has degree zero. More generally, for the k-sphere Sk, E*(Sk) is free on the basis {uk; 1S}, * *where uk 2 Ek(Sk; o). Proposition 12.3 The suspension element e 2 Q(E)11has the following properti* *es, where k 0: (a) aeSuk = uk ek in E*(Sk) Q(E)k*; (b) ruk = uk in E*(Sk) for any additive operation r: k ! m; (c) The class uk 2 Ek(Sk), regarded as a map uk: Sk ! E_k, induces qkuk*z = ek 2 Q(E)kk, where z 2 Ek(Sk) is dual to uk; (d) In the coalgebra structure on Q(E)**, Q( )e = e e and Q(ffl)e = 1; (e) Q( )(vekw) = vek ekw in Q(E)** Q(E)**, for any v 2 E* and w 2 jRE*; (f) Given v 2 E* and w 2 jREh, the homomorphism Q(r): Q(E)k+h*! Q(E)m* induced on homology by any operation r: k + h ! m satisfies Q(r)(vekw) = vekjR in Q(E)m*; (g) Under stabilization, Q(oe)e = 1 in E*(E; o). Proof We prove (a) for k > 0 by induction on k, starting from eq. (12.2). If i* *t holds for k and m, the multiplicativity of ae gives ae(ukxum ) = (ukxum ) ek+m in E*(SkxSm ). The projection map q: Sk x Sm ! Sk+m induces q*uk+m = uk x um , which gives (a) for k+m. The result is true also for k = 0, if we make the obvious identificat* *ion e0 = 1. Then (b) follows by eq. (6.39) and (c) is an application of Prop. 6.44. To prove (d), we evaluate both axioms (6.33) for M = E*(S1) on u1. Part (e) follows immediately from (d) and the fact that Q( ) is a homomorphism of algebr* *as and of E*-bimodules. Then (f) follows from (e) and Lemma 6.51(c). For (g), we apply 1 Q(oe) to eq. (12.2) and compare with the stable coaction aeSu1 = u1 1* * in [8, eq. (11.24)]. | Remark As v, k, and w vary, the elements vekw span Q(E)** Q as a Q -module. (In fact, Q(oe) induces Q(E)k* Q ~=E*(E; o) Q if E is (-k-1)-connected.) Thus in the important case when Q(E)k*has no torsion, the innocuous formulae in (e) * *and (f) are powerful enough to determine Q( ) and Q(r) completely. Corollary 12.4 Let r: k ! m be an additive operation, regarded as a map of H-spaces r: E_k! E_m. Then the induced homomorphism on homotopy groups r* * E* ~=ss*(E_k; o) --! ss*(E_m ; o) ~=E is given on v 2 E-h by r*v = . JMB, DCJ, WSW - 62 - 23 Feb 1995 x13. Spheres, suspensions, and unstable operations Proof We reinterpret r* as the action of the operation r on Ek(Sk+h; o). The e* *lement v corresponds to the class vuk+h. From Prop. 12.3(b) and eq. (6.40), r(vuk+h) = uk+h in E*(Sk+h; o) .| Suspensions More generally, the action of the operations on the suspension X of a based space X is easily deduced from the action on X. Lemma 12.5 Given a based space (X; o) and x 2 Ek(X; o), the coaction aeX x * *is the image of aeX x under e: E*(X; o) b Q(E)k*--! E*(X; o) b Q(E)k+1*; where e denotes multiplication by the element e 2 Q(E)1*. Proof The projection map S1 x X ! X embeds E*(X; o) in E*(S1xX; S1xo). Here, x corresponds to u1 x x, whose coaction is known. | We can mimic this algebraically. We defined the formal suspension M of any E*-module M in [8, Defn. 6.6], merely by shifting all the degrees up one. Definition 12.6 Given any unstable comodule M, we make the suspension M of M an unstable comodule by equipping it with the coaction aeM defined by the commutative square aeM Mk __________M-b Q(E)k* ~ | | = | |e |? ae |? M (M)k+1 pppppppM-b Q(E)k+1* The axioms on aeM are readily verified. 13 Spheres, suspensions, and unstable operations In this section, we continue x12 by computing all the unstable operations on E*(Sk) for the spheres Sk, which requires one new Hopf ring element, the suspen* *sion element e. This leads to the unstable structure of E*(X) in terms of E*(X). We recall that E*(Sk) is a free E*-module with basis {1S; uk}, where uk is * *the standard generator. The algebra structure is given by u2k= 0, except that of co* *urse u20= u0. By the Remark after Thm. 10.47, we have only to find r(uk). Lemma 11.5 gives partial information. We assume that E*(E_k) is a free E*-module for all k. Definition 13.1 We define the suspension element e = eU 2 E1(E_1) by the iden- tity r(u1) = 1S + u1 in E*(S1) (for all r):(13:2) Here and in similar definitions, we use the freeness of E*(S1) and the dual* *ity FMod *(E*(E_k); E*) ~=E*(E_k) to ensure that e exists and is well defined. We * *note that ffle = 0 from eq. (10.15). Rather than develop all the properties of e no* *w, we include them below in Prop. 13.7 as the special case e1 = e. JMB, DCJ, WSW - 63 - 23 Feb 1995 Unstable cohomology operations Suspensions We deduce from eq. (13.2) the behavior of unstable operations under the suspension isomorphism : E*(X; o) ~= E*(X; o). We take an element x 2 Ek(X; o) Ek(X) and assume that r(x) is given by eq. (11.6), so that fflcff= 0.* * The quotient map q: S1 x X ! X embeds E*(X) in E*(S1xX) ~=E*(S1) E*(X); under this embedding, x corresponds to u1 x x. We compute r(u1xx) from the Cartan formula (10.39) and find X r(x) = 1X + (-1)deg(xff) xff (13:3) ff for all r. The other terms drop out because 11O cff= fflcff= 0 and e O1k = ffle* * = 0. This suggests how the suspension of an unstable algebra should be defined. * *The treatment is slightly different from the additive version in x12. First, we ne* *ed a basepoint. Definition 13.4 We call the unstable algebra M based_if we are given an augme* *n- tation M ! E* of unstable algebras._Then the kernelM is an invariant ideal, a* *nd we have the splitting M = E* M as E*-modules. We define the unstable suspension U M of M as the subalgebra __ * 1 U M = (1SE*) (u1M ) E (S ) M: (13:5) __ The action of r is given on u1 M by eq. (13.3) and on 1S E* by eq. (11.1). For example, if (X; o) is a based space, we have the augmentation E*(X) ! E*(o) = E*, with kernel E*(X; o) (as in [8, eq. (3.2)]). Inspection shows that * *much of the structure on M is not used. The multiplication on M is totally ignored. Ind* *eed, we do not need an unstable structure on M at all. __ __ Theorem 13.6 Given an additively unstable moduleM , we can make E* M an_unstable algebra, with_1 2 E* as the unit element and trivial multiplication* * on k P M , as follows. If x 2M and r(x) = ffxfffor additive operations * *r, where cff2 Q(E)k*, we lift each cffto "cff2 E*(E_k) such that qk"cff= cff, and define* * the action of unstable operations r on x by X r(x) = 1 + (-1)deg(xff) xff: ff Proof Because e O1 = 0 and e O(b*c) = 0 whenever fflb = 0 and fflc = 0, r(x) is independent of the choices of the "cff. The definition_(with signs) has been_ch* *osen so that: (a) the additive unstable structure on E* M restricts to that on M giv* *en by Defn. 12.6, and (b) it includes eq. (13.5) for a based unstable algebra M. (* *For (a), we note that diag. (6.16) gives qk+1(eU O"cff) = (-1)keQ cff.) Verification of * *the axioms of Thm. 10.47 is tedious but routine. | The elements ek It is convenient to use eq. (13.3) to find the structure of E** *(Sk). We deduce the fundamental properties of the Hopf ring element e. Proposition 13.7 We define the Hopf ring elements ek 2 Ek(E_k) for k 0 in t* *erms of e 2 E1(E_1) by ek = (-1)k(k-1)=2eOk for k > 0 (so that e1 = e) and e0 = [1] * *- 10. They have the following properties: JMB, DCJ, WSW - 64 - 23 Feb 1995 x13. Spheres, suspensions, and unstable operations (a) In E*(Sk) we have, for any k 0: r(uk) = 1S + uk (for all r); (13:8) (b) The class uk, regarded as a map uk: Sk ! E_k, induces uk*z = ek 2 Ek(E_k* *) in homology, where z 2 Ek(Sk) is dual to uk; (c) ek Oem = (-1)kmek+m if k > 0 or m > 0; (d) ek = ek1 + 1ek for all k > 0; (e) fflek = 0 2 E* for all k 0; (f) Oek = -ek for all k > 0; (g) ek O[] = ek for all rational numbers 2 E0 and all k > 0; (h) r*ek = [] * []Oek for all k 0 and all r: k ! m; (i) qkek = ekQ= ek in Q(E)k*, for all k 0, for additive operations; (j) oek*ek = 1 in E*(E; o), for all k 0, under stabilization. Remark The results make it clear that the correct interpretation of eO0is [1] * *- 10 = [1] - [00], as in [28] and elsewhere, rather than just the element [1]. Proof We give extensive details of this proof (only), as a good example of our* * ma- chinery in action. We establish eq. (13.8) for k > 0, and thus (a), by induction on k. It hol* *ds for k = 1 by definition. We recognize Sk as Sk+1 and uk as uk+1; then by eq. (13.3), eq. (13.8) holds for k + 1 if it holds for k, provided that ek+1 = (-1)ke Oek. * * Our definition of ek is designed to do exactly this. More generally, we have (c). For k = 0, we write E*(S0) = E* E*. In Alg, this is a product of algebras,* * with the projections induced respectively by the inclusions of the basepoint and the* * other point. In this presentation, u0 = (0; 1), and of course 1S = (1; 1). By eq. (11* *.1), the action on u0 is r(u0) = r((0; 1)) = (; ) = (1S - u0) + * *u0; which gives (a) if we define e0 = [1] - 10. Then (b) is an application of Prop. 10.5. When we substitute eq. (13.8) in* *to eq. (10.14), we find, for k > 0, 1k1S + ekuk = 1k1k1S + 1kekuk + ek1kuk; since u2k= 0. This gives (d). (But e0 acquires the extra term e0e0, because u2* *06= 0; this is obvious anyway from Prop. 11.2. Also, (c), (d), and (g) are clearly fa* *lse for k = m = 0.) Similarly, eq. (10.15) yields 1S + (fflek)uk = 1S (even for k = 0),* * which gives (e). For (g), which includes (f) as the special case = -1 (by Prop. 10.12(a) and Prop. 11.2(d)), the distributive law (10.11) and (d) yield ek O[+] = ekO[] + ek* *O[] for all ; 2 E0. Since ek O[1] = ek, (g) follows. (We are in effect expanding r* *(uk).) For (h), we substitute eq. (13.8) into eq. (10.45). On the left, we have (sr)(uk) = 1S + uk; while on the right, iteration of eq. (13.8) yields, after simplification, s(r(uk)) = ]> 1S + ] * []Oek> uk; JMB, DCJ, WSW - 65 - 23 Feb 1995 Unstable cohomology operations with the help of eqs. (10.16) and (10.23). Comparison of these gives r*ek. For k = 1 in (i) and (j), we stabilize eq. (13.2) by Lemma 10.7 and compare* * with Defn. 12.1 and [8, eq. (11.24)]. For general k, we use the multiplicative prope* *rties in diag. (6.16) of qk and oek*. | We have the analogue of Cor. 12.4. By Lemma 2.3(d), a based operation r: k * *! m is represented by a based map r: (E_k; o) ! (E_m ; o). We need to know its effe* *ct on homotopy groups. Lemma 13.9 Given a based operation r: k ! m, the induced homomorphism on homotopy groups r* m-h Ek-h ~=ssh(E_k; o) --! ssh(E_m ; o) ~=E is given on v 2 Ek-h for any h 0 by r*v = in Em-h . Proof Viewed cohomologically, the element v 2 Ek-h or map v: Sh ! E_k corre- sponds to vuh 2 Ek(Sh; o). From eqs. (10.16) and (13.8), we compute r(vuh) = uh, which simplified because r is based, so that = 0. | 14 Complex orientation and additive operations In this section, we study the effect of a complex orientation on additive o* *perations. The relevant test space is C P 1, for which E*(C P 1) = E*[[x]] by [8, Lemma 5.* *4], where x = x() is the Chern class of the Hopf line bundle . All the stable resu* *lts carry over, almost without change, except that now b1 = e2 instead of 1. We assume that E*(E_k), Q(E)k*, and E*(E; o) are free E*-modules. Definition 14.1 We define elements bi2 Q(E)22ifor all i 0 by the identity 1X aex = b(x) = xi bi in E*(C P 1) b Q(E)2*~=Q(E)2*[[x]]; (14:2) i=0 where b(x) is a convenient formal abbreviation that rapidly becomes essential. We use eq. (6.39) to convert eq. (14.2) to the equivalent form 1X rx = xi in E*(C P 1) = E*[[x]], for all r . (14:3) i=0 Since the Hopf bundle is universal, eqs. (14.2) and (14.3) hold for the Chern c* *lass x = x() of any complex line bundle over any space X (after completion, if necessar* *y). Proposition 14.4 The elements bi2 Q(E)22ihave the following properties: (a) b0 = 0 and b1 = e2, so that b(x) = xe2 + x2b2 + x3b3 + : :;: (b) The Chern class x 2 E2(C P 1), regarded as a map of spaces x: CP 1 ! E_2, induces q2x*fii= bi2 Q(E)22i, where fii2 E2i(C P 1) is dual to xi; JMB, DCJ, WSW - 66 - 23 Feb 1995 x14. Complex orientation and additive operations (c) Q( )bk is given by Xk Q( )bk = B(i; k) bi in Q(E)** Q(E)2*, i=1 where B(i; k) denotes the coefficient of xk in b(x)i, or formally, 1X Q( )b(x) = b(x)i bi; i=1 (d) Q(ffl)bk = 0 for k > 1, or formally, Q(ffl)b(x) = x; (e) The stabilization Q(oe): Q(E)2*! E*(E; o) sends the element bi 2 Q(E)22i* *to the stable element bi2 E2i-2(E; o) of [8, Defn. 13.1]. Proof For (a), we restrict eq. (14.2) to C P 1~=S2 and compare with eq. (12.2)* *. For (b), we apply Prop. 6.44. For (c) and (d), we substitute ae into diags. (6.33)* * and evaluate on x. For (e), we compare Defn. 14.1 with [8, Defn. 13.1]. | Still following the stable strategy, we next apply ae to the multiplication* * map : CP 1 x CP 1 ! CP 1, to obtain the formal identity X b(F (x; y)) = FR(b(x); b(y)) = b(x) + b(y) + b(x)ib(y)jjRai;j(14:5) i;j in Q(E)**[[x; y]], which looks exactly like the stable version [8, eq. (13.6)].* * Again, FR(X; Y ) is a convenient abbreviation. The consequences are the same. The p-local case Lemma 14.6 Assume that E* is a p-local ring. Then the generator bk of Q(E)*** *is redundant unless k is a power of p. Proof The proof of [8, Lemma 13.7] applies without change. | We therefore reindex the b's. Definition 14.7 When E* is a p-local ring, we define b(i)= bpifor each i 0. As in [8, x13], we obtain X b([p](x)) = [p]R(b(x)) = pb(x) + b(x)i+1jRgi (14:8) i>0 in Q(E)2*[[x]], which looks exactly like the stable version [8, eq. (13.11)] bu* *t is in a k different place. Again, we extract the coefficient of xp . Definition 14.9 For each k > 0, we define the k th main (additively unstable) relation as (Rk) : L(k) = R(k) in Q(E)2*, (14:10) k where L(k) and R(k) denote the coefficient of xp in the left and right sides of* * eq. (14.8) respectively. JMB, DCJ, WSW - 67 - 23 Feb 1995 Unstable cohomology operations 15 Complex orientation and unstable operations In this section, we extend our study of the test space C P 1 to all unstabl* *e op- erations. Everything we did in x14 carries over, with a lot more complication * *but no essential difficulty. Again, it is enough to know r(x) for all operations r* *, where x = x() 2 E2(C P 1) is the Chern class. We assume that E*(E_k) is a free E*-module for all k. Definition 15.1 We define elements bi2 E2i(E_2) for i 0 by the identity 1X r(x) = xi= in E*(C P 1) = E*[[x]] (15:2) i=0 P i for all r, where we take xi inside the < ; > and write formally b(x) = ibix . We first determine how the elements bk interact with the Hopf ring structur* *e. Proposition 15.3 The elements bk 2 E2k(E_2) of the Hopf ring E*(E_*) have the properties: _ (a) b0 = 12 and b1 = e2 = -eO2, so that b(x) = 12 + b(x) if we define _ 1X i b(x) = bix in E*(E_2)[[x]]; (15:4) i=1 (b) The universal Chern class x 2 E2(C P 1), regarded as a map x: CP 1 ! E_2, induces x*fik = bk 2 E2k(E_2), where fik 2 E2k(C P 1) is dual to xk (as in [* *8, Lemma 5.4]); P (c) bk = i+j=kbi bj, or formally, b(x) = b(x) b(x); (d) fflbk = 0 if k > 0, and fflb0 = 1, or formally, fflb(x) = 1; _ _ _ _ (e) Ob(x) = (12 + b(x))*(-1)= 12 - b(x) + b(x)*2- b(x)*3+ : :;: (f) For all rational numbers 2 E0, _ * _ (-1) _ *2 b(x) O[] = (12 + b(x)) = 12 + b(x) + _______b(x) + : :;: (15:5) 2 (g) For all r, r*bk is given as the coefficient of xk in the formal identity 1 r*b(x) = [] * * b(x)OjO[] in E*(E_*)[[x]]; j=1 (h) q2bk = bk 2 Q(E)22k, the additively unstable element in Defn. 14.1; (i) oe2*bk = bk 2 E2k-2(E; o), the stable element in [8, Defn. 13.1]. Remark The sign in (a) is absent from [23, Prop. 2.4]. The commutativity of diag. (6.16) requires Q(OE)(q1q1)(ee) = -(q1e)(q1e) = -q2b(0)= -q2e2 = q2(eOe); bearing in mind that deg(q1) = 1. The unexpected sign first appeared in Prop. 1* *3.7(c). JMB, DCJ, WSW - 68 - 23 Feb 1995 x15. Complex orientation and unstable operations Proof Naturality for the inclusion S2 ~=C P 1 CP 1 gives (a), by comparison wi* *th Prop. 13.7. Part (b) comes from Prop. 10.5. We read off (c) and (d) from eqs. (* *10.14) and (10.15). Part (e) is the special case = -1 of (f). For (f), eq. (10.11) an* *d (c) give b(x) O[+] = b(x)O[] * b(x)O[] for all ; 2 E*. Since b(x) O[1] = b(x) and * *we are working in the *-multiplicative group of formal power series over E*(E_2) o* *f the form 1 + : :,:which has no n-torsion if 1=n 2 E*, the result follows. (We are i* *n effect expanding r(x); cf. eq. (10.16).) For (g), we apply eq. (10.45) to x 2 E2(C P 1* *) and expand. For (h) and (i), we stabilize eq. (15.2) by Prop. 6.11 and compare with* * the additive and stable versions, eq. (14.3) and [8, eq. (13.3)]. | From (c) and eq. (10.11), we deduce the convenient distributive law (a * c) Ob(x) = aOb(x) * cOb(x); (15:6) where a and c are allowed to involve x. This formal device will prove extremely* * useful for computations in Hopf rings. We have one immediate application to the Froben* *ius operator F defined by F c = c*p. Corollary 15.7 For any element c in the Hopf ring E*(E_*), ae F (c Obn) mod p;if k = pn; (F c) Obk 0 mod p; if k is not divisible by p. Proof By iterating eq. (15.6) we have (F c) Ob(x) = F (c Ob(x)). We pick out * *the coefficient of xk, working mod p. | We next study the naturality of operations with respect to the multiplicati* *on : CP 1 x CP 1 ! C P 1. We expand *r(x) = r(*x) by the formal group law [8, eq. (13.5)] and the Cartan formulae, to obtain the analogue of eq. (14.5). * * The complicated result is best expressed formally as i j b(F (x; y)) = FR(b(x); b(y)) = b(x) * b(y) * * b(x)OiOb(y)OjO[ai;j](15:* *8) i;j as in [23, Thm. 3.8(i)], where FR(X; Y ) = X * Y * *i;jXOiOY OjO[ai;j], in the * *sense that the O- and *-multiplications apply only to Hopf ring elements, not to x and y. The p-local case Lemma 14.6 carries over. Lemma 15.9 Assume that E* is a p-local ring. Then the O-generator bk of the * *Hopf ring E*(E_*) is redundant unless k is a power of p. Proof As before, we takeithejcoefficient of xiyj in eq. (15.8), where i + j = * *k. On the left, there is a term kibk, from bk(x+y)k, and this is the highest b that occu* *rs; on the right,inojb beyond bi or bj occurs. We choose i and j as in [8, Lemma 13.7], to* * make k i not divisible by p and therefore invertible, which shows that bk is redunda* *nt. | We therefore reindex the b's as usual. Definition 15.10 When E* is a p-local ring, we define b(i)= bpifor each i 0. JMB, DCJ, WSW - 69 - 23 Feb 1995 Unstable cohomology operations We extend standard multi-index notation slightly by defining bOI= bOi0(0)ObOi1(1)ObOi2(2)ObOi3(3)O: : : (15:11) for any multi-index I = (i0; i1; i2; : :):. We also need a shift operation. Definition 15.12 Given a multi-index I = (i0; i1; i2; : :):, we define the * *shifted multi-index s(I) = (0; i0; i1; i2; : :):. We iterate this process h times, for* * any h 0, to form sh(I) = (0; : :;:0; i0; i1; i2; : :):. We even undo it, by defining* * s-1(I) = (i1; i2; i3; : :):, provided i0 = 0; our convention is that this is undefined i* *f i0 6= 0. This notation allows us to iterate Cor. 15.7 neatly in the form ( O s-1(I)) mod pif i = 0; (F c) ObOI F (c Ob 0 (15:13) 0 mod p if i0 6= 0. We follow the stable plan and study instead of the much simpler p-th power map i: CP 1 ! CP 1. Naturality of the general operation r is expressed by i*r(x* *) = r(i*x). When we substitute the p-series [8, eq. (13.9)] and expand, we obtain, * *as in [23, Thm. 3.8(ii)], X b px + gixi+1 = b(x)*p* * b(x)Oi+1O[gi] (15:14) i i in E*(E_*)[[x]], or, in condensed notation, b([p](x)) = [p]R(b(x)). Definition 15.15 For each k > 0, we define the kth main unstable relation as (Rk) : L(k) = R(k) in E*(E_2), (15:16) k where L(k) and R(k) denote respectively the coefficient of xp in the left and * *right sides of eq. (15.14). k Thus L(k) is the coefficient of xp in b([p](x)), exactly as in Defn. 14.9.* * However, R(k) is vastly more complicated than before, and we study it in more detail in * *x19 in the case E = BP . The work of Ravenel-Wilson [23], which we review in x17, impl* *ies that, despite appearances, the relations (Rn) contain all the information prese* *nt in eq. (15.8), with the understanding that we use eq. (15.8), by way of Lemma 15.9* *, only to express the redundant bj's (which still appear in b(k), b(k)O[], and r*b(k)* *) in terms of the b(i). 16 Examples for additive operations In x5, we developed a comonad to express all the structure of additive unst* *able E-cohomology operations, for favorable E. In x6, we developed a bigraded algeb* *ra Q(E)**that contains equivalent information, where Q(E)kihas degree k - i. In th* *is section, we describe Q(E)**for each of our five cohomology theories E*(-), name* *ly E = H(F p), MU, BP , KU, and K(n). (The first example splits into two, and we break out the degenerate special case K(0) = H(Q ).) As stably in [8], our purp* *ose is to exhibit the structure of the results, not to derive them. JMB, DCJ, WSW - 70 - 23 Feb 1995 x16. Examples for additive operations All the results here are formally very close to the stable results. By Prop* *. 12.3(g), Q(oe)e = 1. As E*(E; o) = colimkQ(E)k*by eq. (4.8), where the suspensions Q(E)k* **! Q(E)k+1*have been revealed in Lemma 12.5 as simply multiplication by e, we stab* *ilize everything merely by setting the suspension element e = 1. In this way, we reco* *ver all the corresponding stable results. Indeed, in the case E = KU, we have to ob* *tain the stable structure this way. All four answers of x5 are of course available, but the Second Answer remai* *ns the most practical, consisting as in Thm. 7.7 of the coactions aeX : Ek(X) --! E*(X) b Q(E)k*: These coactions are automatically additive, multiplicative (for cup products an* *d x- products), and unital (aeX 1X = 1X 1). (We simplify notation by suppressing re* *dun- dant completions and suffixes.) We use exactly the same test spaces and test maps as we did stably. The poi* *nt remains that complete knowledge of the behavior of E*(-) on these is sufficient* * to suggest the correct structure of Q(E)**(except that the case E = K(n) requires * *some extra work). By Prop. 6.42(b), the one-point space T in effect defines the righ* *t unit jR, and the circle S1 defines e 2 Q(E)11by eq. (12.2). As all our examples have complex orientation, we have available the elements bi of Defn. 14.1. In each case, we list the generators and relations for the bigraded E*-alge* *bra Q(E)**, describe the right unit jR, and give the values of the algebra homomorp* *hisms = Q( ): Q(E)**! Q(E)** Q(E)**and ffl = Q(ffl): Q(E)**! E* on each generator. In some cases, we can express the universal property of Q(E)**very simply. The stabilization Q(oe) maps each generator to its stable namesake, except that of * *course Q(oe)e = 1. Example: H(F 2) We take E = H = H(F 2), the Eilenberg-MacLane spectrum. Our test space is R P 1, for which H*(R P 1) = F2[t], a polynomial algebra on t* *he generator t 2 H1(R P 1). We define elements ci2 Q(H)1*by the identity 1X aet = ti ci in H*(R P 1) b Q(H)1*~=Q(H)1*[[t]]. i=0 Restriction to S1 = RP 1shows that c0 = 0 and c1 = e. As stably, the multiplica* *tion : RP 1 x RP 1 ! R P 1 implies that ci = 0 unless i is a power of 2. We therefore write i= c2i2 Q(H)12ifor each i 0, so that 1X i aet = t2 i in H*(R P 1) b Q(H)1*~=Q(H)1*[[t]]; (16:1) i=0 which looks just like the stable version [8, eq. (14.1)], except that now 0 = e. Theorem 16.2 For the Eilenberg-MacLane ring spectrum H = H(F 2): (a) Q(H)**= F2[0; 1; 2; 3; : :]:, a polynomial algebra over F2 on generators* * i 2 Q(H)12ifor i 0, where 0 = e; (b) In the complex orientation for H(F 2), b(i)= 2ifor all i 0, and bj = 0 * *if j is not a power of 2; JMB, DCJ, WSW - 71 - 23 Feb 1995 Unstable cohomology operations (c) is given by Xn i n = 2n-i i in Q(H)** Q(H)1*; i=0 (d) ffl is given by ffln = 0 for n > 0 and ffl0 = 1. Proof Part (a) is of course a reformulation of classical results. For fixed k* *, the stabilization Q(oe): Q(H)k*! H*(H; o) is the monomorphism that is dual (with a shift in degree) to the well-known epimorphism oe*k: H*(H; o) ! P H*(H_k) that * *tells which Steenrod operations can act nontrivially on Hk(-). The proof of (b) is t* *he same as stably. We prove (c) and (d) by taking M = H*(R P 1) in diags. (6.33) a* *nd evaluating on t. | As stably in [8, x14], we combine the universal property of the polynomial * *algebra F 2[0; 1; 2; : :]:with Thm. 7.7(b). Corollary 16.3 Let B be a discrete commutative graded F2-algebra. Assume that the ring homomorphism : H*(X) ! H*(X) b B is natural for spaces X. Then on t 2 H1(R P 1), has the form 1X i t = t2 0i in H*(R P 1) b B ~=B[[t]], i=0 i-1) where the elements 0i2 B-(2 determine uniquely for all X and may be chosen arbitrarily. | Example: H(F p) (for p odd) We write H = H(F p), the Eilenberg-MacLane spec- trum. The complex orientation defines elements i= b(i)for i 0, and, just as st* *ably, bj = 0 whenever j is not a power of p. The only difference now is that 0 = b1 =* * e2 instead of 1. The other test space is the lens space L = K(F p; 1), for which H*(L) = Fp[* *x] (u). As x is a Chern class, aeLx is given by eq. (14.2). This leaves only aeLu,* * which reduces (as stably) to 1X i aeLu = u e + xp oi in H*(L) b Q(H)1*, (16:4) i=0 for certain elements oi that it defines. Theorem 16.5 For the Eilenberg-MacLane ring spectrum H = H(F p), with p odd: (a) Q(H)**is the commutative algebra over Fp with generators: e 2 Q(H)11, a polynomial generator; i2 Q(H)22pifor all i 0, a polynomial generator for i > 0; oi2 Q(H)12pifor all i 0, an exterior generator; JMB, DCJ, WSW - 72 - 23 Feb 1995 x16. Examples for additive operations subject to the relation 0 = e2; (b) is given by e = e e, Xk i k = pk-i i in Q(H)** Q(H)2*, (16:6) i=0 and Xk i ok = ok e + pk-i oi in Q(H)**bQ(H)1*; i=0 (c) ffl is given by ffle = 1, ffli= 0 for i > 0, and ffloi= 0 for all i. Proof Part (a) is again a reformulation of classical results, which may be rec* *overed in this form from [27, Thm. 8.5], in somewhat different notation, by taking the* * inde- composables. We obtain (b) and (c) by substituting aeL in diags. (6.33) and eva* *luating on x and u. | We have the analogue of Cor. 16.3. Corollary 16.7 Let B be a discrete commutative graded Fp-algebra. Assume that the ring homomorphism : H*(X) ! H*(X) b B is natural for spaces X. Then on H*(L) = Fp[x] (u), has the form 1X i x = xe02+ xp 0i i=1 1X i u = ue0+ xp o0i i=0 i-1) 0 -(2pi-1) where the elements e02 B0, 0i2 B-2(p , and oi 2 B determine uniquely for all X and may be chosen arbitrarily. | Example: H(Q ) We write E = H = H(Q ), the Eilenberg-MacLane spectrum. As always, there is the suspension element e 2 Q(H(Q ))11, whose properties we know from Prop. 12.3. There is nothing else. Theorem 16.8 For the ring spectrum H = H(Q ): (a) Q(H)**= Q[e], a polynomial algebra on e 2 Q(H)11; (b) The coalgebra structure is given by e = e e and ffle = 1. | Example: MU The coefficient ring is MU* = Z[x1; x2; x3; : :]:, with a pol* *y- nomial generator xi in degree -2i for each i. These give rise to the elements jRxi2 Q(MU)-2i0. We have complex orientation, almost by definition, and therefo* *re the elements bi 2 Q(MU)22i, with b0 = 0 and b1 = e2. We have the relations (14.* *5) between the b's and the jRv, but unlike the stable case, because e is no longer* * invert- ible, they do not render the generators jRxi redundant. Implicit in [23, Cor. 4* *.6(a)] is that this is the whole story. Theorem 16.9 (Ravenel-Wilson) For the unitary Thom ring spectrum MU: (a) Q(MU)**is the commutative algebra over MU* with generators: JMB, DCJ, WSW - 73 - 23 Feb 1995 Unstable cohomology operations jRxi2 Q(MU)-2i0(for i > 0); e 2 Q(MU)11; bi2 Q(MU)22i(for i 1); all of even degree, subject to the relations (14.5) and b1 = e2; (b) is given by e = e e and Xk bk = B(i; k) bi in Q(MU)** Q(MU)2*, i=1 where B(i; k) denotes the coefficient of xk in b(x)i; (c) ffl is given by ffle = 1 and fflbk = 0 for k > 1. | Although we no longer have a polynomial algebra, part of Cor. 16.3 carries * *over. It applies equally well to the two following cases, which we include here. Corollary 16.10 Let B be a discrete commutative E*-algebra, where E = MU, BP , or KU. Then a ring homomorphism : E*(X) ! E*(X) b B that is natural for spaces X is uniquely determined by its values on E*(S1) and E*(C P 1). | Example: BP The coefficient ring is now BP *= Z(p)[v1; v2; v3; :::], with pol* *ynomial generators vn in degree -2(pn-1). We have complex orientation, but because BP * is p-local, we need only the generators b(i)2 Q(BP )22pi, where b(0)= e2. Again* *, [23, Cor. 4.6(b)] implies that this is all there is; in particular, eq. (14.5) is re* *dundant, except to express the other bj in terms of the b(i)and the elements vi and wi= * *jRvi. Theorem 16.11 (Ravenel-Wilson) For the Brown-Peterson ring spectrum BP : (a) Q(BP )**is the commutative algebra over BP *with generators: i-1) wi= jRvi2 Q(BP )-2(p0 (for i > 0); e 2 Q(BP )11; b(i)2 Q(BP )22pi(for i 0); subject to the main relations (Rk) (from eq. (14.10)) for k > 0 and b(0)= e2; (b) is given by e = e e and pkX b(k)= B(i; pk) bi in Q(BP )** Q(BP )2*, i=1 k i where B(i; pk) denotes the coefficient of xp in b(x) ; (c) ffl is given by ffle = 1 and fflb(k)= 0 (for k > 0). | We discuss the structure of Q(BP )**in more detail in x18. Remark Alternatively, we could use the generator hiinstead of b(i)as in [6]; h* *owever, Quillen's element ti (see [21] or Adams [1, II.16]) does not exist in this cont* *ext for i > 1, for lack of conjugation in Q(BP )**. JMB, DCJ, WSW - 74 - 23 Feb 1995 x16. Examples for additive operations Example: KU We take E = KU, the complex Bott spectrum, with the coefficient ring KU* = Z[u; u-1] (where u 2 KU-2), right unit jR: KU* ! Q(KU)**given by jRu = v, and Chern class x given by [8, eq. (5.2)]. The simple form [8, eq. (5.* *16)] of the formal group law reduces eq. (14.5) to b(x + y + uxy) = b(x) + b(y) + b(x)b(y)v; (16:12) which looks like the stable version [8, eq. (14.13)], with b(x) = b1x + b2x2+ b* *3x3+ : :,: except that now b1 = e2 6= 1. The coefficient of xiyj yields the relation min(i;j)Xi+j-k i bibj = ukbi+j-kv-1; (16:13) k=0 i k like [8, eq. (14.15)], except that the case i = 1 now gives the reduction formu* *la b1bi= (i+1)bi+1v-1 + iubiv-1 for i > 0. (16:14) The results here are much clearer than in the stable case, and there is some ov* *erlap with the work of tom Dieck [10]. Theorem 16.15 For the complex Bott spectrum KU: (a) Q(KU)**is generated as an algebra over KU* = Z[u; u-1] by the elements: v = jRu 2 Q(KU)-20; v-1 = jRu-1 2 Q(KU)20; e 2 Q(KU)11, the suspension element; bi2 Q(KU)22ifor i > 0; subject to the relations b1 = e2 and (16.13) for i > 0, j > 0; (b) Q(KU)**is a free KU*-module, with a basis consisting of all monomials of* * the forms vn, bivn, evn, and ebivn, for i > 0 and n 2 Z; (c) is given by e = e e and Xk bk = B(i; k) bi in Q(KU)** Q(KU)2*, i=1 where B(i; k) denotes the coefficient of xk in b(x)i; (d) ffl is given by ffle = 1 and fflbk = 0 for all k > 1. Proof We start with (b). We take the Hopf line bundle over CP 1 and regard the element u-1[] 2 KU2(C P 1) as a map f: CP 1 ! KU__2= Z x BU. By Lemma 4.6, f induces an isomorphism of KU*-modules KU*(C P 1) --! QKU*(Z xBU) ~=KU* QKU*(BU); which we compute. By the definition [8, eq. (5.2)] of the Chern class x, u-1[] * *= u-1+x in KU2(C P 1); geometrically, the components of f are the map CP 1 ! Z with ima* *ge 1, and x: CP 1 ! BU. Thus q2f*fi0 = v-1 and q2f*fii = q2x*fii = bi for i > 0, with the help* * of Prop. 14.4(b); we have the desired basis of Q(KU)2*. For Q(KU)2n*, we multiply by v-n+1, an isomorphism. JMB, DCJ, WSW - 75 - 23 Feb 1995 Unstable cohomology operations For the odd case, the description of KU*(U) in [8, Cor. 5.12] in terms of t* *he Bott map b: (Z xBU) ! U shows that multiplication by e induces an isomorphism Q(KU)2n*~=Q(KU)2n+1*. We have specified enough relations to reduce any monomial in the b's, e, v,* * and v-1 to a linear combination of the elements in (b), which proves (a). Parts (c)* * and (d) are included in Props. 14.4 and 12.3. | Now that we know the additive situation, we return to finish off the stable* * case. We may discard the odd spaces in eq. (4.8) and write KU*(KU; o) = colimnQ(KU)2n*: Corollary 16.16 In the stable algebra KU*(KU; o): (a) Every element of KU*(KU; o) of even degree can be written in the form c = uq(1u-1 + 2u-2b2 + : :+:nu-nbn)v-m for some integers q, m, n, and i; (b) This element c = 0 if and only if i= 0 for all i. Proof By Thm. 16.15(b), we can write the general element of Q(KU)2m+22quniquely in the form Xn c = uq 0v-1 + iu-ibi v-m i=1 with integer coefficients. Since e2 = b1, eq. (16.14) yields Xn Xn e2c = uq+1 0u-1b1 + (i + 1)iu-i-1bi+1+ iiu-ibi v-m-1 i=1 i=1 in Q(KU)2m+42q+2, which gives (a). Further, e2c = 0 only if c = 0, which implie* *s (b). | Example: K(n) The coefficient ring is K(n)* = Fp[vn; v-1n], where vn* * 2 n-1) K(n)-2(p . We write wn = jRvn, as we did for BP . Obviously, wn and vn are no longer equal as they were stably, because they lie in different groups. We have a complex orientation, and therefore the usual elements bj. Because K(n)* is p-local, we need only the b(i)for i 0. (In fact, bj = 0 if j is not a* * power of p and j < pn, for dimensional reasons, but not in general if j > pn.) When * *we n apply ae to the p-th powernmap i: CP 1 ! CP 1 , which induces i*x = vnxp as in* * [8, eq. (14.26)], we obtain bpjwn = vjnbj, and therefore n pi -1 2pn bp(i)= vn b(i)wn in Q(K(n))* (16:17) for i 0. This stabilizesnto [8, eq. (14.27)]. In particular, bp(0)= vnb(0)w-1n. As always, b(0)= e2. A more sophisticat* *ed analysis, involving other cohomology theories as in [28, Prop. 1.1(j)], shows t* *hat this relation can be desuspended once to give n-1 -1 2pn-1 ebp(0) = vnewn in Q(K(n))* . (16:18) JMB, DCJ, WSW - 76 - 23 Feb 1995 x16. Examples for additive operations n The other test space is the skeleton Y = L2p -1 of the lens space L, for w* *hich n K(n)*(Y ) = (u) K(n)*[x: xp = 0]. We know aeY x, because x is inherited from C P 1. As stably, we define elements ai; ci2 Q(K(n))1*by the coaction pn-1X pn-1X aeY u = xi ai+ uxi ci : (16:19) i=0 i=0 By restriction to S1 Y , we see that a0 = 0 and c0 = e. Then eq. (16.18) is eq* *uivalent n-1 * to the statement aeY y = y e, where y = vnuxp 2 K(n) (Y ); in other words, y behaves like u1 2 K(n)1(S1). The same partial multiplications : L2k+1x L2m ! Y as in [8, x14] show that ci= 0 for all i > 0 and that ai= 0 for i not a power o* *f p. We therefore reindex, as usual. Definition 16.20 We define a(i)= api2 Q(K)12pi, for 0 i < n. In the new notation, n-1X i aeY u = ue + xp a(i) in K(n)*(Y ) Q(K(n))1*. (16:21) i=0 Having odd degree, the a(i)are exterior generators of Q(K(n))**. This is not al* *l; we again appeal to [28, Prop. 1.1(i)] to find that one more factor e can be squeez* *ed out of eq. (16.18) if we first multiply by a(0), to give the relation n-1 -1 2pn-1 a(0)bp(0) = vna(0)wn in Q(K(n))* . (16:22) Theorem 16.23 For the Morava K-theory ring spectrum K(n): (a) Q(K(n))**is the commutative bigraded algebra over K(n)* = Fp[vn; v-1n], * *where n-1) vn 2 K(n)-2(p , with generators: n-1) wn = jRvn 2 Q(K(n))-2(p0 ; w-1n= jRv-1n; e 2 Q(K(n))11; a(i)2 Q(K(n))12pi(for 0 i < n); b(i)2 Q(K(n))22pi(for i 0); subject to the relations b(0)= e2, (16.17), (16.18), and (16.22); (b) is given by e = e e, Xk i a(k)= a(k)e + bp(k-i)a(i) in Q(K(n))** Q(K(n))1*, (16:24) i=0 and pkX b(k)= B(i; pk) bi in Q(K(n))** Q(K(n))2*, (16:25) i=1 k i where B(i; pk) denotes the coefficient of xp in b(x) (and Lemma 14.6 is used * *to express b(x) in terms of the b(j), vn, and wn); (c) ffl is given by ffle = 1, ffla(k)= 0 (for k 0), and fflb(k)= 0 (for k >* * 0). JMB, DCJ, WSW - 77 - 23 Feb 1995 Unstable cohomology operations Proof The algebra structure (a) is implicit in the main theorem of [28], by ta* *king indecomposables. As always, we obtain a(i)and ffla(i)by evaluating the coacti* *on axioms (6.33) on u 2 K(n)*(Y ). The rest of (b) and (c) can be obtained similar* *ly, or by appealing to Props. 12.3 and 14.4. | Corollary 16.26 Let B be a discrete commutative K(n)*-algebra. Then a ring homomorphism : K(n)*(X) ! K(n)*(X) b B that is natural for spaces X is uniquely determined by its values on K(n)*(C P 1) and K(n)*(Y ). | Remark If k n, eq. (16.25) simplifies just as in [8, Thm. 14.32] to Xk i b(k)= bp(k-i) b(i) in Q(K(n))** Q(K(n))2*, i=1 which resembles eq. (16.6). 17 Examples for unstable operations In this section, we discuss the enriched Hopf ring for each of our five coh* *omology theories E*(-), namely for E = H(F p), MU, BP , KU, and K(n). According to x10, this is what we need to handle general unstable operations. As in x16, we divid* *e the case H(F p) in two and treat K(0) = H(Q ) separately. Even more than before, our intent is to exhibit the structure of the results, not to reestablish them. Our strategy is the same as in the stable and additive contexts, using exac* *tly the same test spaces and test maps. Each E has a complex orientation, which provides by Defn. 15.1 the elements bi of the Hopf ring, in addition to e and the [v]. W* *e have O[1] = [-1] by Prop. 11.2(d), and its properties were listed in Prop. 10.12. As pointed out in (10.46), we need more than just the Hopf ring and the ele* *ments [v]. The elements Q(ffl)qkc = fflSoek*c are given by x16. We also need r*c fo* *r each operation r; by Thms. 10.19(c) and 10.33(c), it is in principle enough to know * *these for each O-generator c. Our presentation changes somewhat from x16. Each family of O-generators has* * its own Proposition, which lists all the pertinent information. It is therefore suf* *ficient to describe each Hopf ring by listing its O-generators and the defining relations,* * and to refer to these propositions for further details. We recover all the results for* * additive operations merely by taking the indecomposables. Example: MU We recall that MU* = Z [x1; x2; x3; : :]:, where deg(xi) = -2i, * *is better described as generated by the elements ai;j, as in [8, x14]. We have the* * elements bi, as well as e and [v] = jR(v). Stably, [8, eq. (13.6)] gave an inductive for* *mula for jRai;jin terms of MU* and the bi. Unstably, eq. (15.8) is only a relation betw* *een these elements. Corollary 4.6(a) of [23] says in effect that this is all there * *is. Theorem 17.1 (Ravenel-Wilson) For the unitary cobordism ring spectrum MU, MU*(MU__*) is the Hopf ring over MU* = Z[x1; x2; x3; :::] with O-generators: [xi] 2 MU0(MU__-2i) for each i > 0 (see Prop. 11.2); e 2 MU1(MU__1) (see Prop. 13.7); JMB, DCJ, WSW - 78 - 23 Feb 1995 x17. Examples for unstable operations bi2 MU2i(MU__2) for i 1 (see Prop. 15.3); subject to the relations eO2= -b1 and eq. (15.8). | Example: BP The main reference is still [23]. As BP *is p-local, Lemma 15.9 a* *nd Defn. 15.10 apply, to define the elements b(i)of the Hopf ring. We have as alwa* *ys e and the elements [v] for each v 2 BP *. Theorem 17.2 (Ravenel-Wilson) For the Brown-Peterson ring spectrum BP , BP*(BP__*) is the Hopf ring over BP *= Z(p)[v1; v2; v3; : :]:with the O-generat* *ors: [] 2 BP0(BP__0), for each 2 Z(p)(see Prop. 11.2); [vi] 2 BP0(BP__-2(pn-1)), for i > 0 (see Prop. 11.2); e 2 BP1(BP__1) (see Prop. 13.7); b(i)2 BP2pi(BP__2) for i 0 (see Prop. 15.3); subject to the relations [] O[0] = [0], [] * [0] = [ + 0], e O[] = e, b(i)O[] =* * : : : (see Prop. 15.3(f)), eO2= -b(0), and the main relations (Rn) for n > 0 as in eq* *. (15.16). We implicitly use eq. (15.8), but only to express inductively the bj, for j* * not a power of p, in terms of the b(i), v, and [v]; this is needed for computing b(i* *), Ob(i), b(i)O[], and r*b(i). Proof This is the content of [23, Cor. 4.6(b)]. By Prop. 11.2, each [v] for v * *2 BP * can be expressed in terms of the [] and [vi]; we have enough generators. The li* *sted relations come from Props. 11.2, 13.7, and 15.3, and eq. (15.16). This reduce* *s the *-generators (see x10) to three types: (i)bOIO[vJ]; (ii)e ObOIO[vJ]; (17.3) (iii)[vJ]; in terms of the multi-index notation bOIintroduced in eq. (15.11). For each k, the *-generators that lie in BP*(BP__k) generate it as a BP *-a* *lgebra. Assume first that k is even, so that we have only types (i) and (iii). We write BP__k= BP kx BP__0kas in Lemma 4.17; then BP*(BP__k) ~=BP*(BP k) BP*(BP__0k); (17:4) where we recognize the first factor as the group ring over BP *of the abelian g* *roup BP kwith basis elements [v] for v 2 BP k. The type (i) generators lie in BP*(BP* *__0k) and the type (iii) in BP*(BP k), which is described by Lemma 4.4. Because [vJ]*[0vJ* *] = [(+0)vJ], we have enough relations for the type (iii) generators. The work of [* *23] reduces the type (i) generators to certain allowable generators bOIO[vJ], which* * form a system of polynomial generators BP*(BP__0k). Since this reduction (see x19) use* *s only the relations (Rn), we have enough relations. If k is odd, only generators of type (ii) occur. These reduce similarly to* * the allowable generators of type (ii), which are exterior generators of BP*(BP__k).* * | Example: H(Q ) This example is of course classical. JMB, DCJ, WSW - 79 - 23 Feb 1995 Unstable cohomology operations Theorem 17.5 For the ring spectrum H = H(Q ), H*(H_*) is the Hopf ring over Q with generators: [] 2 H0(H_0) for each 2 Q (see Prop. 11.2); e 2 H1(H_1) (see Prop. 13.7); subject to the relations [] O[0] = [0], [] * [0] = [+0], and e O[] = e. Proof For k < 0, H_k = T , and we have only the Q -basis element 1k. For k = 0, H_0 = Q, regarded as a discrete group, and the grou* *p ring H*(H_0) = Q [Q ] has a basis consisting of the elements []. The first two rela* *tions, from Prop. 11.2, show how these multiply. For k > 0, the third relation, from Prop. 13.7(g), reduces us to the single* * *- generator eOk 2 Hk(H_k) of H*(H_k). We have the polynomial algebra Q [eOk] if * *k is even, or the exterior algebra (eOk) if k is odd. | Example: H(F 2) We write H = H(F 2). As H*(H_*) is a Hopf ring over F2, we have the Frobenius operator F and the Verschiebung V . We imitate Defns. 15.1 and 15.10 in a mod 2 version, using the same test sp* *ace R P 1 = K(F 2; 1) as before, for which H*(R P 1) = F2[t]. We define ci 2 Hi(H_1* *) = Hi(R P 1) for i 0 by the identity 1X r(t) = ti= in H*(R P 1) (for all r);(17:6) i=0 P i where we write formally c(t) = icit as in Defn. 15.1. In other words, ci is * *dual to ti and the elements ci form an F2-basis of H*(H_1). We are primarily interested in the accelerated elements c(i)= c2i. As befor* *e, we have the suspension element e. The complex orientation provides elements bi whi* *ch are redundant, as in x16. Proposition 17.7 The Hopf ring elements ci2 Hi(H_1) (for i 0) and c(i)= c2i2 H2i(H_1) (for i 0) have the following properties: (a) c0 = 11 and c(0)= c1 = e; P (b) ck = i+j=kci cj, or formally, c(t) = c(t) c(t); (c) V c(i)= c(i-1)for i > 0, and V c(0)= 0; (d) fflck = 0 if k > 0, and fflc0 = 1, or formally, fflc(t) = 1; (e) Oc(t) = c(t)*(-1), expanded as in Prop. 15.3(e); i j (f) ci* cj = i+jici+j; (g) F c(i)= c(i)* c(i)= 0; (h) bi= ciOci in H2i(H_2); (i) For all r, r*ck is the coefficient of tk in the formal identity 1 r*c(t) = * c(t)OjO[] in H*(H_*)[[t]]; j=0 (j) q1c(i)= i in Q(H)1*, and q1cj = 0 if j is not a power of 2; (k) oe1*c(i)= i in H*(H; o), and oe1*cj = 0 if j is not a power of 2. JMB, DCJ, WSW - 80 - 23 Feb 1995 x17. Examples for unstable operations Proof The naturality of r for the multiplication : RP 1 x RP 1 ! R P 1, which induces *t = tx1 + 1xt, yields the identity X X X (tx1 + 1xt)k = tixtj k i j in H*(R P 1xR P 1) = F2[tx1; 1xt], with the help of the Cartan formula (10.23).* * The coefficient of tix tj gives (f). The special case (g) of (f) also follows from * *eq. (10.32). We expand r(t2) for the Chern class t2 by eqs. (17.6) and (10.36) and compare w* *ith eq. (15.2); most terms cancel, to give (h). The other parts are formally as in Prop. 15.3, with all degrees halved, exc* *ept that (c) is immediate from (b). | Just as in Lemma 15.9, except that everything is now explicit in (f), cj is* * redundant unless j is a power of 2. This leads to the following elegant description of th* *e Hopf ring, which is a reformulation of classical results. Theorem 17.8 For the Eilenberg-MacLane ring spectrum H = H(F 2), H*(H_*) is the Hopf ring over F2 with generators c(i)2 H2i(H_1) for i 0 (see Prop. 17.7), subject to the relation [1]*2= 10. Proof By Prop. 17.7(c), we can write cOI = V cOs(I)for any multi-index I = (i0; i1; i2; : :):. Then F cOI = F ([1]OcOI) = F [1] OcOs(I)= 0 by eq. (10* *.13), as in eq.P(15.13), and H*(H_k) is an exterior algebra on those generators cOI for whi* *ch OI i1;i2;::: tit = k. Here, c is dual to the primitive element Sq k in cohomology (* *in terms of the Milnor basis [18] of H*(H; o)). (The index i0 serves only as paddi* *ng, to ensure that i1 + i2 + i3 + : : :k.) | Example: H(F p) (for p odd) We write H = H(F p). We have, as always, the suspension element e. The complex orientation defines elements bi for all i 0;* * but Lemma 15.9 shows that only the b(i)= bpifor i 0 are needed. Also, b0 = 12 and b1 = e2 =i-eO2.j However, the bj for j not a power of p do not vanish, but sati* *sfy bi* bj = i+jibi+j, which is all that survives from eq. (15.8). In particular, * *b*p(i)= 0 for all i > 0, as is also clear from eq. (10.32) applied to x. For the other test space L = K(F p; 1), we have H*(L) = Fp[x] (u). We only need to know r(u). We define elements ai2 H2i(H_1) and ci2 H2i+1(H_1) by 1X 1X r(u) = xi+ uxi in H*(L), i=0 i=0 P i which wePcondense formally to + u by writing a(x) = iaix * * and c(x) = icixi. Thus ai is dual to xi, ci is dual to uxi, and the ai and ci for* *m a basis of H*(H_1). Again, we accelerate the indexing by defining a(i)= apifor i 0. Proposition 17.9 The Hopf ring elements ai 2 H2i(H_1), a(i)= api2 H2pi(H_1), and ci2 H2i+1(H_1), (for i 0), have the following properties: (a) a0 = 11 and c0 = e; JMB, DCJ, WSW - 81 - 23 Feb 1995 Unstable cohomology operations P (b) ak = i+j=kai aj; (c) V a(i)= a(i-1)for i > 0, and V a(0)= 0; (d) fflak = 0 for all k > 0; (e) Oa(x) = a(x)*(-1), expanded as in Prop. 15.3(e); i j (f) ai* aj = i+jiai+j; (g) F a(i)= a*p(i)= 0; (h) ci= e * ai; (i) For all r, r*ak is the coefficient of xk in the formal identity 1 1 r*a(x) = * b(x)OiO[] * * a(x)Ob(x)OiO[] in H*(H_*)[[x* *]]; i=0 i=0 (j) q1a(i)= oi in Q(H)1*, and q1aj = 0 if j is not a power of p; (k) oe1*a(i)= oi in H*(H; o), and oe1*aj = 0 if j is not a power of p. Proof We consider naturality of operations with respect to the multiplication * *: L x L ! L, for which *u = ux1 + 1xu. In condensed notation, we compare *r(u) = + (ux1 + 1xu) with r(*u), which we expand by eq. (10.23) as r(*u) = + ux1 + 1xu + uxu: The coefficient of xix xj gives (f), which implies (g). (Alternatively, (g) fol* *lows from eq. (10.32) applied to u.) The coefficient of u x xi gives (h). The other parts* * require no new ideas. | In particular, all the ci and most of the ai are redundant. We trivially ha* *ve the relation [1]*p = [p] = [0] = 1, from which it follows, as in the previous examp* *le, that (aOI)*p = 0 and (bOI)*p = 0 for all I. Once again, this is the whole stor* *y. A detailed exposition by Ravenel and Wilson from this point of view is presented * *in [27, Thm. 8.5] (with slightly different notation: a(i)is written ff(i), and b(i)is w* *ritten fi(i)). Theorem 17.10 (Ravenel-Wilson) For the Eilenberg-MacLane ring spectrum H = H(F p), H*(H_*) is the Hopf ring over Fp with the O-generators: e 2 H1(H_1) (see Prop. 13.7); a(i)2 H2pi(H_1), for i 0 (see Prop. 17.9); b(i)2 H2pi(H_2), for i 0 (see Prop. 15.3); subject to the relations [1]*p= 10 and eO2= -b(0). | Example: KU We recall that KU* = Z[u; u-1]. The complex orientation defines elements bi for i > 0. As before, these, along with elements [un] = [] O[un] an* *d e, are all we need. In view of the formal group law F (x; y) = x+y+uxy, the relation (15.8) bec* *omes _ _ _ _ _ 1 + b(x+y+uxy) = (1 + b(x)) * (1 + b(y)) * (1 + b(x)Ob(y)O[u]) (17:11) JMB, DCJ, WSW - 82 - 23 Feb 1995 x17. Examples for unstable operations which is more complicated than the additive analogue (16.12), but still managea* *ble. Again, we take the coefficient of xiyj. The left side is the same as before. * *On the right, we may choose xsyt with s > 0 and t > 0 from the third factor, which for* *ces us to take xi-sfrom the first factor and yj-t from the second; or we can take a* *ll of xiyj from the first two factors. The result, after some rearranging, is min(i;j)Xi+j-k i biObj = ukbi+j-kO[u-1] k=0 i k i-1Xj-1X - bi-sO[u-1] * bj-tO[u-1] * bsObt s=1t=1 (17:12) i-1X j-1X - bi-sO[u-1] * bsObj - bj-tO[u-1] * biObt s=1 t=1 - biO[u-1] * bjO[u-1] This serves as an inductive reduction formula for biObj, for any i > 0 and j > * *0. In particular, the suspension formula becomes b1O bj =(j+1)bj+1O[u-1] + jubjO[u-1] j-1X (17:13) - bj-kO[u-1] * b1Obk - b1O[u-1] * bjO[u-1] k=1 Theorem 17.14 For the complex K-theory ring spectrum KU, KU*(KU__*) is the Hopf ring over KU* = Z[u; u-1] with the O-generators: [u] 2 KU0(KU__-2) (see Prop. 11.2); [u-1] 2 KU0(KU__2) (see Prop. 11.2); e 2 KU1(KU__1) (see Prop. 13.7); bi2 KU2i(KU__2) for i > 0 (see Prop. 15.3); subject to the relations [u] O[u-1] = [1], Oe = -e, Obi = : : :(see Prop. 15.3(* *e)), eO2= -b1, and eq. (17.12). Explicitly, for the even spaces we have M KU*(KU__2n) = [mu-n] * KU*[b1O[u-n+1]; b2O[u-n+1]; b3O[u-n+1]; : :]:; m2Z a direct sum (over m) of polynomial algebras, and for the odd spaces KU*(KU__2n+1) = (eO[u-n]; eOb1O[u-n+1]; eOb2O[u-n+1]; : :):; an exterior algebra over KU* (where we use [mu-n] = [m] O[u-n], [un] = [u]On, [* *u-n] = [u-1]On, [u0] = [1], [n] = [1]*n, and [-n] = [-1]*n = (O[1])*n). Proof We computed KU*(BU) in [8, Lemma 5.6]. By Prop. 15.3(b), the Chern class x: CP 1 ! KU__2 induces x*fii = bi, so that we may write KU*(0 x BU) = KU*[b1; b2; : :]:. For the copy KU*(m x BU), we *-multiply this by [m]. This gi* *ves KU*(KU__2). For other even spaces, we apply the *-isomorphism - O[un]. JMB, DCJ, WSW - 83 - 23 Feb 1995 Unstable cohomology operations For the odd spaces, we quote [8, Cor. 5.12]. To see that we have specified enough relations, we note that every *-genera* *tor reduces to e O[un] or e ObiO[un] on the odd spaces, or biO[un] or [] O[un] on t* *he even spaces, where 2 Z . We allow n = 0 and = 1 and use [um ] O[un] = [um+n ] and [] O[0] = [0]. In the even case, we need at most one *-factor of the form [] O[* *un], and we may always insert the redundant factor [0] O[un] = 1. Thus we can reduce* * any expression in the generators to standard form. | Example: K(n) We use the same test spaces as before, C P 1 and the finite lens n-1 space L2p , and follow the same strategy. The main reference is [28]. Some of* * the algebra resembles the case E = H(F p). As usual, the complex orientation determines Hopf ring elements bi, where b* *0 = 12 and b1 = e2 = -eO2. As K(n) is p-local, Lemma 15.9 shows that the bj other than the b(i)= bpiarePredundant. If we apply eq. (10.32) to the Chern class x, we ob* *tain the identity j xpj = 1. This shows that F bj = 0 for all j * *> 0; in particular, b*p(i)= 0. n Next, we apply the general operation r to i*x = vnxp by eq. (15.2) to obta* *in n Opn pn+i b(vnxp ) = b(x) O[vn]. Equating coefficients of x yields the relation n pi -1 bOp(i)= vn b(i)O[vn ]; (17:15) the obvious analogue of eq. (16.17). n-1 * * pn For the other test space Y = L2p , we have K(n) (Y ) = (u) K(n) [x: x = 0]. The class x is a Chern class, which we know all about. Parallel to eq. (16.* *19), we use u 2 K(n)1(Y ) to define elements ai; ci 2 K(n)*(K(n)_1) for 0 i < pn by the identity pn-1X pn-1X r(u) = xi+ uxi in K(n)*(Y ) (for all r): i=0 i=0 Proposition 17.16 The Hopf ring elements ai 2 K(n)2i(K(n)_1) (for 0 i < pn), a(i)= api2 K(n)2pi(K(n)_1) (for 0 i < n), and ci 2 K(n)2i+1(K(n)_1) (for 0 i < pn) have the following properties: (a) a0 = 11 and c0 = e; P (b) ak = i+j=kai aj; (c) V a(i)= a(i-1)for 0 < i < n, and V a(0)= 0; (d) fflak = 0 for all k > 0; (e) Oak is the coefficient of xk in a(x)*(-1), expanded as in Prop. 15.3(e); i j (f) ai* aj = i+jiai+jif i + j < pn; (g) F a(i)= a*p(i)= 0 for 0 i < n - 1; (h) ci= e * ai; (i) For all r, r*ak is the coefficient of xk in the formal identity pn-1 pn-1 r*a(x) = * b(x)OiO[] * * a(x)Ob(x)OiO[] i=0 i=0 JMB, DCJ, WSW - 84 - 23 Feb 1995 x17. Examples for unstable operations n in K(n)*(K(n)_*)[x : xp = 0]; (j) q1a(i)= a(i)2 Q(K(n))1*, and q1aj = 0 if j is not a power of p; (k) oe1*a(i)= a(i)2 K(n)*(K(n); o), and oe1*aj = 0 if j is not a power of p. Proof All the proofs are formally identical to those of Prop. 17.9, except tha* *t we use the space Y instead of L. As in x16, the partial multiplications : L2k+1x L2m !* * Y yield (f) and (h). P pi For (g), we apply eq. (10.32) to u and obtain i>0 x = 0. But be* *cause n *p n-1 xp = 0 already, we are able to deduce that ai = 0 only for 0 < i < p . (We s* *hall see in a moment that a*p(n-1)6= 0.) | We have to rely on [28, Prop. 1.1] for two facts, just as in x16. The first* * is that when i = 0, eq. (17.15) desuspends once, exactly as eq. (16.18) suggests, to n-1 -1 e ObOp(0)= vne O[vn ]: (17:17) n-1 1 * *1 1 In other words, the class y = vnuxp 2 K(n) (Y ) still behaves like u1 2 K(n)* * (S ) and satisfies eq. (13.2). The second is that when we take account of decomposab* *les, eq. (16.22) acquires an extra term, n-1 a*p(n-1)= vna(0)- a(0)ObOp(0)O[vn]: (17:18) This complements (g). We have the material for the main theorem of [28]. Theorem 17.19 For the Morava K-theory ring spectrum K(n), K(n)*(K(n)_*) is the Hopf ring over K(n)* = Fp[vn; v-1n] with the O-generators: [vn] 2 K(n)0(K(n)_-2(pn-1)) (see Prop. 11.2); [v-1n] 2 K(n)0(K(n)_2(pn-1)) (see Prop. 11.2); e 2 K(n)1(K(n)_1) (see Prop. 13.7); a(i)2 K(n)2pi(K(n)_1), for 0 i < n (see Prop. 17.16); b(i)2 K(n)2pi(K(n)_2), for i 0 (see Prop. 15.3); subject to the relations [1]*p = 10, [vn] O[v-1n] = [1], eO2 = -b(1), (17.15), * *(17.17), and (17.18). | Thus we have the *-generators: (i)aOIObOJO[vkn] in even degrees; (ii)e OaOIObOJO[vkn] in odd degrees; where I = (i0; i1; : :;:in-1), with each ir = 0 or 1, and J = (j0; j1; j2; : * *:):, with 0 j < pn, and k 2 Z . In (ii), we may assume j0 < pn - 1 by eq. (17.17). The relations a*p(i)= 0 (for i < n - 1) and b*p(i)= 0 (for all i) follow from [1]*p* * = 10 by eq. (10.13), as in Thm. 17.10. JMB, DCJ, WSW - 85 - 23 Feb 1995 Unstable cohomology operations 18 Relations for additive BP -operations In this section, we discuss relations in the bigraded algebra Q**= Q(BP )*** *, follow- ing [23], in preparation for discussing additive unstable operations in BP -coh* *omology. In view of Thm. 16.11(a), Q**is spanned as a BP *-module by the monomials efflbIwJ = efflbi0(0)bi1(1):w:j:11wj22: :;: (18:1) where ffl 1 and we use standard notation with multi-indicesPI = (i0; i1; i2; :* * :):andP J = (j1; j2; : :):. We define the length of I as |I| = tit, and similarly |J|* * = tjt. We also need the special multi-index 0 = (1; 0; 0; : :):. The main relations For E = BP , we easily compute the first main relation from Defn. 14.9 and [8, eq. (15.4)] (or equivalently from eqs. (14.5) and [8, eq. (1* *5.3)]) as (R1) : v1b(0)= pb(1)+ bp(0)w1 in Q(BP )2*. (18:2) (Indeed, this is the only candidate that stabilizes correctly to [8, eq. (15.6)* *].) We still have bi = 0 whenever i-1 is not a multiple of p-1. We can use the p-serie* *s [8, eq. (15.5)], just as stably, to simplify the higher relations (Rk) by neglectin* *g enough. Denote by V and W respectively the ideals (p; v1; v2; : :):and (p; w1; w2; : * *:):in Q**, which correspond to the left and right actions of the ideal I1 . We also need t* *he ideal M = (e; b(0); b(1); b(2); : :): Q**, so that M + V is the obvious augmentati* *on ideal consisting of all the Qkifor i > 0. In particular, bi2 M + V for all i. From De* *fn. 14.9 and [8, eq. (15.7)], the right side of (Rk) has the form k-1X i k R(k) bp(k-i)wi+ bp(0)wk mod V + M W 2, (18:3) i=1 while the left side L(k) 2 V and will not much concern us here. The new feature* * is k 2 that because wk appears in the form bp(0)wk, where b(0)= e is no longer 1, (Rk* *) fails to express wk in terms of the other generators, and W 6= V ; this made it nece* *ssary to add wk as a new generator of Q**in Thm. 16.11. The Ravenel-Wilson basis The relations (Rk) show that many of the monomials (18.1) are redundant. In defining the basis, it is easier to specify which mon* *omials are not wanted. Definition 18.4 We disallow all monomials of the form 2 pn bp(i1)bp(i2):b:(:in)wnc (i1 i2 : : :in, n > 0), (18:5) where c stands for any monomial in the b(i), wi, and e (c = 1 is permitted). A* *ll monomials (18.1) not of this form are declared to be allowable. Nevertheless, we need a positive construction of the allowable monomials, a* *nd we need to know how they behave under suspension. Given any indices 0 = k0 k1 k2 : : :kn; where n 0, (18:6) we define the monomial 2 pn L- bL = b(k0)bp(k1)bp(k2):b:(:kn)= b(0)b 0 : (18:7) JMB, DCJ, WSW - 86 - 23 Feb 1995 x18. Relations for additive BP -operations It is easy to see that every allowable monomial can be written uniquely in the * *canonical form 2 n c = efflbL-0 bM wJ = efflbp(k1)bp(k2):b:p:(kn)bM wJ;(18:8) where ffl = 0 or 1 and M and J satisfy the conditions: (i)t < ku implies mt< pu, for 0 < u n; (ii)t kn implies mt< pn+1; (18.9) (iii)jt= 0 for all t n; as well as (18.6). In detail, we choose, by induction on u, the smallest ku suc* *h that 2 pu bp(k1)bp(k2):b:(:ku)divides c, to make (i) hold for u. If no such ku exists, we* * set n = u-1 and have (ii). Since c is allowable, it can have no factor wu, which gives (iii* *) for t = u. (In case n = 0, we have merely c = efflbM wJ, (i) and (iii) are vacuous, and (i* *i) says only that mt< p for all t.) The main technical result is that there is only one way the suspension ec o* *f c can fail to be allowable. (This is in effect equivalent to the discussion in [23, * *x5].) We recall from Defn. 15.12 the shifted multi-index s(I). 2 pn+1 L M Lemma 18.10 Assume that the monomial bH = bp(i0)bp(i1):b:(:in)divides b b ,* * where bL (with the same n) is as in eq. (18.7), i0 i1 : : :in, and M satisfies cond* *itions (i) and (ii) of (18.9). Then: (a) iu = ku for 0 u n, so that H = pL; (b) We can write M = (p-1)L + s(M0), where M0 again satisfies (i) and (ii). Proof We show first that iu ku for all u. For any t < ku, we have mt< pu by (* *i). Then the exponent of b(t)in bLbM is at most (1 + p + p2 + : :+:pu-1) + (pu - 1) < pu+1; which shows that t 6= iu. We proceed by induction on n. For n = 0, bp(i0)divides b(0)bM , where mt < * *p for all t. We must have i0 = 0 and m0 = p - 1, which gives M the required form. For n > 0 we must have in = kn, since in > kn is forbidden by (ii). Let ff * * 0 be the smallest index such that kff= kn; then we must have iff= iff+1= : :=:in = k* *n. From lkn = pff+ pff+1+ : :+:pn and mkn < pn+1 we deduce mkn-(p-1)lkn < pn+1-(p-1)(pff+ : :+:pn) = pn+1-(pn+1-pff) = pff: (18:11) If kn = 0 we clearly have ff = 0 and hence m0 = (p-1)l0, and can write M = u+1 (p-1)L + s(M0). If kn > 0, we have ff > 0. We delete the factors bp(iu)for ff * *u n from both sides of our hypothesis, as well as any factors b(t)for t > kn, to de* *duce 2 pff p pff-1 M00 00 that bp(i0)bp(i1):b:(:iff-1)divides b(k0)b(k1): :b:(kff-1)b , where M satisf* *ies (i) and (ii) for the sequence (k0; k1; : :;:kff-1). By induction, we deduce that H = pL and * *that M has the form (p-1)L + s(M0). If t kn, we have m0t= mt+1 < pn+1, which gives (ii) for M0. To establish (i), assume that t < ku. If also t + 1 < ku, we have m0t mt+1 < pu, as desired. Otherwise, ku = t + 1. Let fi be the smallest index such that kfi> t + 1, so t* *hat JMB, DCJ, WSW - 87 - 23 Feb 1995 Unstable cohomology operations ku = ku+1 = : :=:kfi-1= t + 1. Then mt+1< pfiand lt+1 pu + pu+1 + : :+:pfi-1. As in eq. (18.11), we find m0t= mt+1- (p-1)lt+1< pu. | Lemma 18.12 In the bigraded algebra Q**= Q(BP )**: (a) Every allowable monomial can be written uniquely in the form (18.8), sub* *ject to the conditions (18.6) and (18.9), and conversely, every monomial of this for* *m is allowable; (b) The suspended monomial from eq. (18.8) n M J b(0)c = efflbLbM wJ = efflb(0)bp(k1):b:p:(kn)b w is disallowed if and only if jn+1 > 0 and b(p-1)Ldivides bM , in which case we * *can write 0 M (p-1)Ls(M0) L- M0 J0 wJ = wn+1wJ and b = b b , with b 0 b w allowable; (c) Every allowable monomial can be written uniquely in the extended canonic* *al form 2(L)+:::+sh-1(L))sh(M)hJ c = efflbL-0 b(p-1)(L+s(L)+s b wn+1w (18:13) with L as in eq. (18.7), where h 0, either b(p-1)Ldoes not divide bM or jn+1 * *= 0 (or both), and conditions (18.6) and (18.9) hold; (d) In (c), the monomial bLbM wJ is allowable. Proof In (a), we need to establish the converse. If c is disallowed, so is b(* *0)c. By Lemma 18.10(a), b(0)c can be disallowed only if H = pL; but by Lemma 18.10(b), * *the necessary factors b(0)are not present in c. Moreover, b(0)c is disallowed if and only if it contains bpLwn+1 as a facto* *r (using 0) L- M0 J0 the same n). If so, we write bM = b(p-1)Lbs(M , where b 0 b w is allowab* *le by (a). This proves (b). Parts (c) and (d) follow by induction on h. We take h maximal. | Lemma 18.14 In the stable range defined by i pk, every allowable monomial in Qki= Q(BP )kihas the form efflbi0(0)bi1(1):,:w:ith no factors of the form wjt. Proof For each monomial c 2 Q**, we define g(c) = i - pk, where c 2 Qki. We compute g from g(b(n)) 0 if n > 0, g(wn) = 2p(pn-1), g(e) = -(p-1), and g(b(0)) = -2(p-1), using g(ac) = g(a) + g(c). Thus if c contains wn as a facto* *r, g(c) > 0 unless c contains at least {2p(pn-1) - (p-1)}=2(p-1) factors b(0), whi* *ch disallows it. | Theorem 18.15 (Ravenel-Wilson) The allowable monomials (18.1) form a basis of the free BP *-module Q**= Q(BP )**. This is proved in [23, Thm. 5.3, Prop. 5.1]. We content ourselves with show* *ing, as part of Thm. 18.16, that the allowable monomials span Q**, assuming that it * *is spanned by all the monomials (18.1). We shall obtain for each disallowed monomi* *al (18.5) a reduction formula that expresses it in terms of other monomials. A fin* *iteness argument then implies that the allowable monomials must span. A counting argume* *nt is needed to show they in fact form a basis. As only the relations (Rk) and e2 * *= b(0) are used in the reduction, they constitute sufficient relations in Thm. 16.11. JMB, DCJ, WSW - 88 - 23 Feb 1995 x18. Relations for additive BP -operations Knowing that the allowable monomials form a basis of Q**is not enough. In o* *rder to work with this basis, we need to know how the ideal W looks in terms of the* * basis. We therefore define Am for any m 0 as the BP *-submodule of Q**spanned by all * *the allowable monomials efflbIwJ that have I 6= 0 and |J| m. Although A m is not an ideal for m > 0, it is convenient for computation, because when an element c 2 * *Q** is expressed in terms of the basis, it is obvious whether or not it lies in A m* *. We shall prove the following parts of the structure of Q**, after developing the n* *ecessary reduction formula. Theorem 18.16 In the bigraded algebra Q**= Q(BP )**: (a) Am + V = M W m + V for any m > 0 (or_A0 + V = M + V if m = 0), so that * the image of Am in the quotient algebraQ *(see eq. (18.17)) is an ideal; (b) The allowable monomials span Q**= Q(BP )**as a BP *-module. Lemma 15.2 of [8] allows us to work mod V , in the quotient Fp-algebra __* * Q *= Q*=V ~=QH*(BP__*; Fp) : (18:17) __even Better yet, we may ignore e and work in the subalgebraQ * . Higher order relations As they stand, the relations (Rk) are not very practica* *l. j We derive a more useful relation by eliminating the terms that come from b(x)p * *wj for j < n in eq. (18.3) from the n relations (Rk1), (Rk2), . . . , (Rkn), as in* * [23, Lemma 5.13]. The result is of course a determinant. For ulterior purposes, we make * *the elimination totally explicit. Definition 18.18 Given any positive integers i1; i2; : :;:in, where n 1, we * *define i1pi2 pin-* *1pin L(i1; i2; : :;:in) and R(i1; i2; : :;:in) as the coefficient of xp1x2 : :x:n-1* *xn in 2 pn-1 b(x1)pb(x2)p : :b:(xn-1) b([p](xn)) and 2 pn-1 b(x1)pb(x2)p : :b:(xn-1) [p]R(b(xn)) respectively. By eq. (14.8), these are equal in Q**. Then given any integers 0 < k1 < k2 < : :<:kn, where n > 1, we deduce the n* *th order derived relation X X (Rk1;k2;:::;kn) : fflssL(i1; i2; : :;:in) = fflssR(i1;(i2;1:8* *:;:in):19) ss ss in Q**by summing over all permutations ss 2 n, where fflssdenotes the sign of s* *s and we permute the n entries in (i1; i2; : :;:in) = ss(k1; k2; : :;:kn). (For n = 1* *, it reduces to L(k1) = R(k1), which is just (Rk1).) We note that this relation lies in Qf(n)*, where the numerical function 2(pn - 1) | deg(vn)| f(n) = 2(1 + p + p2 + : :+:pn-1) = _________= _________ p - 1 p - 1 was introduced in eq. (1.4). JMB, DCJ, WSW - 89 - 23 Feb 1995 Unstable cohomology operations The left side of (Rk1;k2;:::;kn) lies in V and will be of little interest h* *ere. By (18.3), the right side reduces to X p p2 pn-1 pj 2 fflssb(i1-1)b(i2-2): :b:(in-1-n+1)b(in-j)wj mod V + M W ,(18:20) ss;j where we sum over all permutations ss and all j > 0, and adopt the convention t* *hat b(i)= 0 for i < 0. However, we have arranged matters so that no (explicit) term* *s in wj with j < n survive; when we interchange ij and in, we find identical terms h* *aving opposite signs. The term of most interest is the leading term with ss = id, 2 pn-1 pn bLwn = bp(k1-1)bp(k2-2):b:(:kn-1-n+1)b(kn-n)wn ; (18:21) which is thereby expressed in terms of other monomials and hence redundant. (The multi-index L serves only as a convenient abbreviation, unrelated to eq. (18.7)* *. The indices ku are different, too.) To make this more precise, we note that all terms bIwj in the sum (18.20) h* *ave |I| = |L| = p + p2 + : :+:pn if j = n, or |I| > |L| if j > n. We order the terms thatPcontain wn by defining the weight of any multi-index I = (i0; i1; i2* *; : :):as wt (I) = ttit (which is not the weight used in [23]). This makes bLwn the hea* *viest term with its length, because if we improve the ordering of the indices of any * *other term in (18.20) by interchanging ir and is, where r < s and ir > is, we increas* *e its weight by (is-r)pr + (ir-s)ps - (ir-r)pr - (is-s)ps = (ir-is)(ps-pr) > 0 : Thus (Rk1;k2;:::;kn) provides a reduction formula 2 pn X I bLwn = bp(k1-1)bp(k2-2):b:(:kn-n)wn b wj (18:22) I;j __* 2 inQ *mod M W , where the sum is taken over certain pairs (I; j) with j n, f* *or which |I| > |L|, or |I| = |L| and wt(I) < wt(L). The first nth order relation (R1;2;:::;n) is particularly important, as onl* *y one term of the sum (18.20) is meaningful, namely bpm(0)wn, where m = f(n)=2. We observe* * that this monomial lies just inside the stable range of Lemma 18.14. In this simple * *case, we can do better with a little more attention to detail, to obtain the direct a* *nalogue of [8, Lemma 15.8]. Lemma 18.23 In Qf(n)*= Q(BP )f(n)*we have the relation bpm(0)wn vnbm(0) mod InQ(BP )f(n)* for each n > 0, where m = f(n)=2 = 1 + p + p2 + : :+:pn-1. Proof We proceed by induction on n, starting from eq. (18.2), and work through* *out n mod InQ**. On the left side of eq. (14.8) we have b([p](x)) b(vnxp + : :):, * *by [8, eq. (15.5)]. Then R(j) = L(j) 0 for all j < n, and the only surviving terms in (R1;2;:::;n) are bh(0)L(n) bh(0)R(n), where h = p + p2 + : :+:pn-1. On the le* *ft, we clearly have L(n) vnb(0). On the right, bh(0)wj 0 for all j < n, by the induc* *tion JMB, DCJ, WSW - 90 - 23 Feb 1995 x18. Relations for additive BP -operations hypothesis;nby eq. (18.3) and dimensional reasons, the only surviving term in R* *(n) is bp(0)wn. | __* Proof of Thm. 18.16 We work entirely in the quotient algebra Q * defined by eq. (18.17). We first generalize (18.22) to show that M W m Am + M W m+1 (18:24) for any m 1. As an Fp-module, M W m is generated by those monomials efflbIwJ that have |J| m. These lie in Am or M W m+1 except for the disallowed monomi* *als that have |J| = m. On comparing the monomial (18.5) with eq. (18.21), we see that each such monomial has the form bLwnc, where L is given by eq. (18.21) and c = efflbLwN , with |N| = m-1. When we multiply eq. (18.22) by c, both ordering* *s are preserved, and we express the general disallowed monomial bLwnc as a signed sum* * of monomials with greater length, or the same length and lower weight, mod M W m+* *1. Because there are only finitely many monomials in each bidegree, eq. (18.24) fo* *llows by induction. For any i > m, eq. (18.24) gives Am + M W i Am + Ai+ M W i+1= Am + M W i+1: Then by induction on i, starting from eq. (18.24), M W m Am + M W i for all i > m. In any fixed bigrading, M W iis zero for large i. Thus M W m * *A m and we have (a) for m > 0. For m = 0, we note that every monomial in M either * *lies in M W A1 or is automatically allowable and so lies in A0. On reinstating the monomials_of the form wJ, which are all allowable, we se* *e that * the allowable monomials spanQ *. Then (b) follows by Nakayama's Lemma in the form [8, Lemma 15.2(d)]. | The ideals Jn Just as the ideal I1 BP *led to the introduction of the ideal W Q**, the ideal Jn, needed for our splitting theorems, leads to an ideal in * *Q**. Definition 18.25 We define the ideal Jn = (wn+1; wn+2; wn+3; : :): Q**. We need to know how Jn sits inside Q**. The answer is remarkably clean, in* * a certain range. Lemma 18.26 Assume n 0. Then: (a) If k < f(n+1), Qk*\ Jn is the left BP *-submodule of Qk*spanned by all t* *he allowable monomials efflbIwJ 2 Qk*that contain an explicit factor wt for some t* * > n; (b) If k = f(n+1), Qk*\ Jn is the left BP *-submodule of Qk*spanned by all t* *he allowable monomials as in (a), together with all disallowed monomials of the fo* *rm 2 pn+1 bp(i1)bp(i2):b:(:in+1)wn+1, where 0 i1 i2 : : :in+1. Remark The first disallowed monomial in (b) is bpm(0)wn+1, where m = f(n+1)=2. Lemma 18.23 shows it definitely does not lie in the submodule described in (a). Proof The stated elements obviously lie in Jn. To show the converse, we fix k * *and a large integer m, and prove by downward induction on h that all elements in Qkio* *f the JMB, DCJ, WSW - 91 - 23 Feb 1995 Unstable cohomology operations form cwh lie in the indicated submodule whenever i < m. This statement is vacuo* *us for sufficiently large h (depending on m and k). We therefore fix t > n, assume* * the statement for all h > t, and prove it for h = t. We ignore efflthroughout and a* *ssume k is even. Case 1: c = bI. The number |I| of b -factors in c is k=2 + pt - 1. In (a)* *, as k < f(n+1) and t > n, this is always less than p + p2 + : :+:pt, which makes cwt = bIwt automatically allowable. The same holds in (b), except in the extre* *me case bIwn+1, which may be allowable or disallowed; either way, it is in. Case 2: c = bIwhwJ allowable, with h t. Then cwt= bIwhwtwJ remains allow- able, by the form of Defn. 18.4. Case 3: c = awh, with h > t, any a. Then cwt = (awt)wh is in by induction, provided i < m. t-1) By Thm. 18.15, these c generate Qk+2(p* as a BP *-module. | 19 Relations in the Hopf ring for BP In this section, we develop the unstable analogues of the results of x18, w* *orking in the Hopf ring BP*(BP__*) for BP . By taking account of *-decomposable elemen* *ts, we can improve many of these results by one. The structure of the Hopf ring was described briefly in x17. Before we can even state some of our results precisel* *y, it is necessary to clarify the concept of ideal in a Hopf ring. Hopf ring ideals As it is obviously impractical to retain everything in typica* *l Hopf ring calculations (the preceding sections should convince), we need to control * *carefully what is thrown away. There is an obvious relevant concept, valid in any Hopf ri* *ng H. We concentrate on the structure of H as a *-algebra, treating O-multiplication * *chiefly as a means of creating new *-generators from old. Definition 19.1 We call a bigraded R-submodule I of any Hopf ring H over R a Hopf ring ideal if the quotient H=I inherits a well-defined Hopf ring structure* * from H (over the possibly smaller ground ring R=fflI). If we ignore the O-multiplication and coalgebra structure, I must obviously* * be a *-ideal in the ordinary sense, i. e. an R-submodule for which b * c 2 I wheneve* *r b 2 H and c 2 I. Lemma 19.2 Let H be a Hopf ring over R and I R an ideal. Let I be the *-ide* *al in H generated by the elements cff. Then I is a Hopf ring ideal, with quotient * *a Hopf ring over R=I, if and only if: (i) cff2 IH + H I for all ff; (ii)fflcff2 I for all ff; (iii)a Ocff2 I for all a 2 H and all ff; (iv)IH I. Proof The conditions are evidently necessary. Conditions (i) and (ii) ensure * *that H=I inherits a comultiplication and counit ffl. Condition (iv) shows that H=* *I is JMB, DCJ, WSW - 92 - 23 Feb 1995 x19. Relations in the Hopf ring for BP defined over R=I. For any a; b 2 H, eq. (10.11) and (iii) show that a O(b * cff* *) 2 I; this is enough to furnish H=I with a O-multiplication. All the necessary identities * *in H=I (see x10) are inherited from H. | Remark It is clear from the Lemma that the sum I + J of two Hopf ring ideals is another Hopf ring ideal. However, their *-product ideal I * J (defined as the u* *sual product of ideals) need not be a Hopf ring ideal, as (i) can fail. We note that* * (ii) and (iii) nevertheless continue to hold for I * J, with the help of eq. (10.11). When R = Fp, we can define a rather more useful ideal. Definition 19.3 Given an ideal I in a Hopf ring over F p, we define F I as the *-ideal generated by {F x : x 2 I}. The ideal F Iis far smaller than I*p, and clearly is a Hopf ring ideal by L* *emma 19.2 whenever I is. (We use eq. (10.13) to verify (iii).) The redundant generators We proved in Lemma 15.9 that the generator bi is redundant unless i is a power of p. As in (17.3), this implies that BP*(BP__*)* * is *-generated as a BP *-algebra by O-monomials of the forms (cf. eq. (18.1)) bOIO[vJ]= bOi0(0)ObOi1(1)ObOi2(2)O:O:[:vj11vj22: :]: (i) Oi Oi Oi Oj Oj = b(00)Ob(11)Ob(22)O: :O:[v1] 1O[v2] 2O: :;: (19:4) (ii) e ObOIO[vJ]; (iii)[vJ] = [] O[v1]Oj1O[v2]Oj2O: :;: in the notation of eq. (15.11). To carry out computations, we need to express * *the redundant bi in terms of these *-generators. In_order to make the finiteness of_our computations apparent, we write b(x)* * = 12 + b(x) as in eq. (15.4) and use 1 Ob(x) = 0. Then eq. (15.8) expands to _ X i j 12 + b x + y + ai;jx y i;j (19:5) _ _ n _ Oi _ Oj o = (12 + b(x)) * (12 + b(y)) * * 12 + b(x) Ob(y) O[ai;j] i;j As in Lemma 15.9, if n is not a power of p, we take s as the largest power of p* * less than n, and the coefficient of xsyn-s then yields a reduction formula for bn. F* *or the low bn's we can be explicit; they are no longer trivially zero, as in x18. Lemma 19.6 For 1 i < p we have bi= b*i(0)=i! . Proof All that is left of eq. (19.5) in this range is b(x+y) = b(x) * b(y). He* *nce b(x) must be the exponential series exp(b1x), expanded using *-multiplication. | Beyond this range, we must settle for inductive formulae in terms of O-mono* *mials of the form bi1Obi2O: :O:birO[vJ] : (19:7) JMB, DCJ, WSW - 93 - 23 Feb 1995 Unstable cohomology operations We expand the formal group law F (x; y) fully, in the form X F (x; y) = x + y + vIxiyj; ;I;i;j summing over appropriate quadruples (; I; i; j) consisting of a coefficient 2 * *Z(p), a multi-index I, and exponents i and j. The right side of eq. (19.5) becomes i _ j i _ j n _ _ * o 12 + b(x) * 12 + b(y)* * 12 + b(x)OiOb(y)OjO[vI]; ;I;i;j where {1 + : :}:* is expanded by the binomial series as in eq. (15.5). Every el* *ement of the Hopf ring that appears here is a *-product of elements of the form (19.7* *). This is still not enough! To make the induction succeed, we really need a r* *eduction formula for every O-monomial (19.7) that contains a O-factor bn with n not a po* *wer of p, without relying on iterated appeals to the distributive law (10.11). A re* *duction formula for bn Obh1Obh2O: :O:bhq, whenever n is not a power of p and the hi are* * any positive integers, will suffice, as - O[vI] is a *-homomorphism and [vI] O[vJ] * *= [vI+J]. We therefore O-multiply eq. (15.8) by b(z1) Ob(z2) O: :O:b(zq) (and thus wo* *rk in the (q+2)-fold product (C P 1)q+2). On the right, we use the distributive law (15.* *6) to move all the b(-)'s inside the *-factors, to obtain _ X I i j _ _ 12q+2+ b x + y + v x y Ob(z1) O: :O:b(zq) n ;I;i;j _ _ _ o = 12q+2+ b(x)Ob(z1)O : :O:b(zq) n _ _ _ o (19:8) * 12q+2+ b(y)Ob(z1)O : :O:b(zq) n _ _ _ _ * o * * 12q+2+ b(x)OiOb(y)OjOb(z1)O : :O:b(zq)O[vI] ;I;i;j The coefficient of xsyn-szh11: :z:hqqyields the desired reduction formula. Insp* *ection of the O-monomials that appear on the right shows that they are all simpler, so th* *at the induction makes progress. (In detail, they all have lower height, or thePsame h* *eight but more b -factors, if we define the height of the monomial (19.7) as rir.) None of this is necessary for the other generators, (19.4)(ii). For these * *it is far simpler to start from Lemma 14.6, work in Q(BP )**, and suspend by applying e O* *-. The main relations As given in Defn. 15.15, the main relations are particularly opaque. We make eq. (15.14) more useful in our situation by first expanding the p-series [8, eq. (13.9)] for BP in full as X [p](x) = px + vIxm ; (19:9) ;I;m much as we just did for F (x; y), and summing over appropriate combinations of coefficient 2 Z(p), multi-index I, and exponent m. Then eq. (15.14) becomes _ X I m n _ o*p n _ Om I o* 12 + b px + v x = 12 + b(x) * * 12 + b(x) O[v ] (19:10) ;I;m ;I;m JMB, DCJ, WSW - 94 - 23 Feb 1995 x19. Relations in the Hopf ring for BP where we again expand {12 + : :}:* by the binomial series as in eq. (15.5). The first main relation, the coefficient of xp, simplifies (with the help o* *f [8, eq. (15.4)] and Lemma 19.6) to b*p(0) (R1) : v1b(0)= pb(1)+ bOp(0)O[v1] - _______ in BP*(BP__2) (19:11) (p-1)! (although it is far easier to extract this as the coefficient of xp-1y in eq. (* *15.8), using [8, eq. (15.3)]). Subsequent relations rapidly become extremely complicated an* *d can be handled only by neglecting terms wholesale. We need some ideals. Let V be the ideal (p; v1; v2; : :):in BP*(BP__*) (more accurately, genera* *ted as a graded *-ideal by all the elements p1k and vn1k for each k). We need the unsta* *ble analogue of the ideals M W m of x18, coming from the right action of I1 on Q** **. It is obvious how to handle the generators vi of I1 . For the generator p, eq. (1* *0.13) shows that in the quotient Hopf ring BP*(BP__*)=V over Fp = BP *=I1 , we may w* *rite c O[p] = c O(F [1]) = F (V cO[1]) = F V c. Indeed, it is even more convenient t* *o ignore e and work in the Hopf subring __ H = BP*(BP__even)=V ~=H*(BP__even; Fp); (19:12) using only those elements that do not involve the O-generator e (though of cour* *se we keep b(0)= -eO2). __ Definition 19.13 We define M 0 as the *-ideal inH generated by all the elem* *ents bOIO[vJ] with I 6= 0, whether allowable or not. For m > 0, we define M m induct* *ively as the *-ideal generated by F M m-1 and all elements bOIO[vJ] with I 6= 0, whet* *her allowable or not, that have |J| m. Equivalently, M m is the *-ideal generated by all elements F h(bOIO[vJ]) wi* *th I 6= 0 and h + |J| m. (Thus M m is roughly, but not quite, the Hopf ring analogue of * *the right BP *-action of the ideal Im1.) We thus have the decreasing sequence of id* *eals __ H M 0 M 1 M 2 : ::: __ We note that M 0 is just the obvious augmentation ideal inH consisting of all* * the Hi(BP__even; Fp) with i > 0. Lemma 19.14 For all m 0: __ (a) M m is a Hopf ring ideal in the Hopf ringH = BP*(BP__even)=V ; (b) M m O[vn] M m+1 for all n > 0; (c) M m O[vJ] M m+|J|; (d) M m O[p] M m+1. Proof We first prove (b), from which (c) follows by induction. As - O[vn] is * *a *- homomorphism, it is enough to check that c O[vn] 2 M m+1 for the generators c * *of M m. For c = bOIO[vJ], we use [vJ] O[vn] = [vJvn]. For c = F a = a*p, where a * *2 M m-1, we have c O[vn] = F (a O[vn]) by eq. (10.13). This lies in F M m, by induction * *on m. We next apply Lemma 19.2 to prove (a). Clearly, fflM m = 0. For a genera* *tor of the form c = bOIO[vJ], with |J| m, we have a Oc = (a ObOI) O[vJ] 2 M m by JMB, DCJ, WSW - 95 - 23 Feb 1995 Unstable cohomology operations P (c), sincePa ObOI 2 M 0. Similarly, if we write bOI = iB0i B00i, we find t* *hat c = iB0iO[vJ] B00iO[vJ] has the required form, because for each i, either B* *0i2 M 0 or B00i2 M 0 for reasons of degree. For a generator F c with c 2 M m-1, we use induction on m. By eq. (10.13), a O(F c) = F (V a Oc) 2 F M m-1. Also, F c = (F F ) c has the required form. Because M m is now known to be a Hopf ring ideal, we have V a 2 M m for a* *ny a 2 M m. Then (d) is immediate from eq. (10.13), using [p] = F [1]. | __ We now have the tools to handle eq. (19.10). We work entirely_in H , so th* *at by [8, eq. (15.5)], the left side is trivial. By Lemma 19.14, b(x)Om O[vI] 2 M * *|I|. Most *-factors on the right side of eq. (19.10) are trivial mod M 2and we are left w* *ith only n _ o n _ j o __ 12 + F b(x) * * 12 + b(x)Op O[vj] inH [[x]] mod M 2. j>0 k When we pick out the coefficient of xp and neglect also certain products, we o* *btain Xk j __ R(k) F b(k-1)+ bOp(k-j)O[vj] inH mod M 2 + M 1*M 1, (19:15) j=1 analogous to eq. (18.3). Although the ideal here is not a Hopf ring ideal, (ii)* * and (iii) of Lemma 19.2 still hold, according to the Remark following that lemma. The Ravenel-Wilson generators We lift the allowable monomials of x18 via the canonical projections qk: BP*(BP__k) ! Q(BP )k*, so that multiplication is now * *to be interpreted as O-multiplication. Definition 19.16 We disallow all O-monomials of the form 2 Opn bOp(i1)ObOp(i2)O:O:b:(in)O[vn] Oc (i1 i2 : : :in, n > 0),(19:17) where c stands for any O-monomial in the b(i), [vj], and e (c = [1] is permitte* *d). All O-monomials (19.4)(i) and (ii) not of this form are declared to be allowable. It follows from Thm. 18.15 (and local finiteness) that the allowable O-mono* *mials generate BP*(BP__*), but far more is true, by [23, Thm. 5.3, Rk. 4.9]. Theorem 19.18 (Ravenel-Wilson) In the Hopf ring for BP : (a) If k is even, denote by BP__0kthe zero component of the space BP__k(so t* *hat BP__0k= BP__kif k > 0). Then BP*(BP__0k) is a polynomial algebra over BP *on th* *ose allowable O-monomials bOIO[vJ] with I 6= 0 that lie in it. If k 0, BP*(BP__k)* * = BP*(BP k) BP*(BP__0k) as in eq. (17.4). (b) If k is odd, BP*(BP__k) is an exterior algebra over BP *on those allowab* *le O-monomials e ObOIO[vJ] that lie in it. | As in x18, we need information on where the disallowed monomials lie. The difficulty with eq. (19.15) is that it is hard to tell whether a given element * *lies in M 2. We therefore define analogous ideals in terms of the polynomial generato* *rs in Thm. 19.18 for which this problem_does not exist. Again, we ignore e and neglec* *t V by working in the Hopf ringH over Fp (see eq. (19.12)). JMB, DCJ, WSW - 96 - 23 Feb 1995 x19. Relations in the Hopf ring for BP __ Definition 19.19 We define A 0as the *-ideal inH generated by all the allow* *able O-monomials bOIO[vJ] that have I 6= 0. For m > 0, we define Am inductively as * *the *- ideal generated by F Am-1 and all the allowable O-monomials bOIO[vJ] for which * *I 6= 0 and |J| m. In other words, Am is the *-ideal generated by all the elements F h(bOIO[vJ* *]), where bOIO[vJ] is allowable, I 6= 0, and h + |J| m. Theorem_19.20 For all m 0, A m = M m and is therefore a Hopf ring ideal in H = BP*(BP__even)=V ~=H*(BP__even; Fp). This result we shall prove in full. For m = 0, it is part of Thm. 19.18. Higher order relations As in x18, we derive a more useful relation by eliminat* *ion from the n relations (Rk1), (Rk2), . . . (Rkn), with multiplication now interpr* *eted as O-multiplication. We find it simpler to return to eq. (19.10) rather than try * *to deal directly with eq. (19.15). Definition 19.21 Given any positive integers i1, i2, . . . , in, where n 1, * *we define i1pi2 pin L(i1; i2; : :;:in) and R(i1; i2; : :;:in) as the coefficient of xp1x2 : :x:n in 2 Opn-1 b(x1)OpOb(x2)Op O: :O:b(xn-1) Ob([p](xn)) and 2 Opn-1 b(x1)OpOb(x2)Op O: :O:b(xn-1) OP (xn) (19:22) respectively, where P (x) denotes the right side of eq. (19.10). Then given any integers 0 < k1 < k2 < : :<:kn, where n > 1, we define the n* *th order derived relation X X (Rk1;k2;:::;kn) : fflssL(i1; i2; : :;:in) = fflssR(i1; i2; * *: :;:in) ss ss by summing over all permutations ss 2 n, where (i1; i2; : :;:in) = ss(k1; k2; :* * :;:kn). (For n = 1, we recover (Rk1).) This relation lies in BP*(BP__f(n)),_where f(n) denotes the usual numerical* * function (1.4). To study it, we work inH . The left side of (Rk1;k2;:::;kn) vanishes,* * as before. To handle the right side, we first rewrite (19.22) just as we did eq. (19.8), b* *y using eq. (15.6) to move all the O-factors b(-) inside the *-factors. The term px of* * [p](x) produces the *-factor n _ _ 2 _ n-1_ *p o 1 + b(x1)OpOb(x2)Op O: :O:b(xn-1)Op Ob(xn); (19:23) and the general term vIxm produces the *-factor n _ _ 2 _ n-1_ * o 1 + b(x1)OpOb(x2)Op O: :O:b(xn-1)Op Ob(xn)OmO[vI]; to be expanded as in eq. (15.5). By the form [8, eq. (15.5)] of the p-series, t* *he only *-factors of the latter kind that are not trivial mod M 2 are _ Op _ Op2 _ Opn-1_ Opj 1 + b(x1) Ob(x2) O: :O:b(xn-1) Ob(xn) O[vj] (19:24) JMB, DCJ, WSW - 97 - 23 Feb 1995 Unstable cohomology operations for j > 0. We can now efficiently extract the coefficient R(i1; i2;* * : :;:in) of i1pi2 pin xp1x2 : :x:n . From the factor (19.23) we have the term i O O 2 O n-1 j F b(in-1)Ob(pi1-2)Ob(pi2-3)O:O:b:p(in-1-n); after some shuffling, while the factor (19.24) yields 2 Opn-1 Opj bOp(i1-1)ObOp(i2-2)O:O:b:(in-1-n+1)Ob(in-j)O[vj]: (We continue the convention of x18 that meaningless terms, those involving any * *b(i) with i < 0, are treated as zero.) We now sum over ss and j, taking the opportun* *ity to permute the ir in the terms with F (which introduces a sign), to obtain (Rk1;k2* *;:::;kn) in the desired form X i Op Opn-1j (-1)n-1 fflssF b(i1-1)Ob(i2-2)O: :O:b(in-n) ssX 2 Opn-1 Opj (19:25) + fflssbOp(i1-1)ObOp(i2-2)O:O:b:(in-1-n+1)Ob(in-j)O[vj] 0 ss;j __ inH mod M 2 + M 1*M 1. As before, the terms involving [vj] for j < n cancel:* * when we interchange ij and in, we obtain two identical terms having opposite signs. * * We therefore sum only over j n. The terms of most interest are the two leading te* *rms with ss = id: i j i O O n-1j (-1)n-1F bOL = (-1)n-1F b(k1-1)Ob(pk2-2)O:O:b:p(kn-n) (19:26) and 2 n bOpLO[vn] = bOp(k1-1)ObOp(k2-2)O:O:b:Op(kn-n)O[vn]; (19:27) for a certain multi-index L (different from x18). The reduction formula We obtain a reduction formula for the general disallowed O-monomial (19.17) in BP*(BP__k). First, we assume k is even. For any n > 0, 0 < k1 < k2 < : :<:kn, and multi-indices M and J, the desired formula is: 2 Opn OM J bOp(k1-1)ObOp(k2-2)O:O:b:(kn-n)Ob O[vnv ] X Op Opn O - fflssb(i1-1)O: :O:b(in-n)Ob M O[vnvJ] ss6=id i n-1 Os-1(M) J j + (-1)nF b(k1-1)ObOp(k2-2)O:O:b:Op(kn-n)Ob O[v ] (19:28) X i Op Opn-1 O -1 j + (-1)n fflssF b(i1-1)Ob(i2-2)O: :O:b(in-n)Ob s (M)O[vJ] __ss6=id inH mod M h+2 + M h+1*M h+1, where we sum over permutations ss 2 n, (i1; i2; : :;:in) = ss(k1; k2; : :;:kn),* * and h = |J|. (Terms involving s-1(M) with m0 6= 0 are to be omitted.) To obtain thi* *s, we first apply - ObOM to eq. (19.25), using eq. (15.13) to rewrite the terms in* *volving F . The suppressed terms lie in M 2O bOM M 2 and (M 1 * M 1) ObOM M 1* M 1, as we know from Lemma 19.14(a) that M 2 and M 1 are Hopf ring ideals. Then we apply the *-homomorphism - O[vJ] and use Lemma 19.14(c). JMB, DCJ, WSW - 98 - 23 Feb 1995 x19. Relations in the Hopf ring for BP Remark Strictly speaking, this is only a reduction formula mod V , but it meet* *s our present needs. One can work modulo the slightly smaller ideal (v1; v2; : :):in* *stead and extract a more complicated reduction formula that is valid in BP*(BP__*) it* *self, without recourse to Nakayama's Lemma. For odd k, the reduction formula takes the far simpler form 2 Opn OM J e ObOp(k1-1)ObOp(k2-2)O:O:b:(kn-n)Ob O[vnv ] X Op Op2 Opn O - fflsse Ob(i1-1)Ob(i2-2)O: :O:b(in-n)Ob M O[vnvJ] ss6=id mod M h+2. To see this, one can suspend eq. (19.28) by applying e O-, which kil* *ls all *-products, including F c; but it is far simpler to suspend eq. (18.22) instead. Proof of Thm. 19.20 For m > 0, it follows from eq. (19.28) that __ M m Am + M m+1 + M m*M m+ F M m-1 inH , (19:29) by using exactly the same orderings of monomials (reinterpreted) as in the proo* *f of Thm. 18.16. For m = 0, we clearly have M 0 = A 0+ M 1 because the generators of M 0that are not in M 1 are all allowable. We show by induction on m that the term F M m-1 is not needed, that M m Am + M m+1 + M m*M m (19:30) for all m 1. This is clear for m = 0. If it holds for m-1, applying F yields F M m-1 F Am-1 + F M m + F M m-1*F M m-1 : Each term on the right is already included in the other terms of eq. (19.29) an* *d may be omitted. Next, we dispose of M m * M m. On *-multiplying eq. (19.30) by M *i1we have M m*M *i1 Am *M i1+ M m+1*M i1+ M m*M m*M i1 Am + M m+1 + M m*M i+11: It follows by induction on i that M m Am + M m+1 + M m*M *i1 for all i. Since M *i1is zero in each bigrading for large enough i, we must h* *ave M m Am + M m+1. As in the proof of Thm. 18.16, this implies M m = Am . | The suspension We can use eq. (19.28) to extract detailed information about the suspension homomorphism e O-: Qk*! P BP*(BP__k+1) when k is odd. (When k is even, there is nothing to discuss: the allowable monomial bIwJ 2 Qk*suspends to* * the allowable O-monomial e ObOIO[vJ] 2 P BP*(BP__k+1).) By Lemma 18.12(c), we can write every allowable monomial in Qk*uniquely in * *the extended canonical form h-1(L))sh(M)hJ c = e bL-(0) b(p-1)(L+s(L)+:::+s b wn+1w ; n where 0 = k0 k1 k2 : : :kn, n 0, bL = b(k0)bp(k1):b:p:(kn), M and J satisfy the conditions (18.9), and h 0 is maximal. What happens to e Oc is that if h >* * 0, JMB, DCJ, WSW - 99 - 23 Feb 1995 Unstable cohomology operations it is disallowed, as the derived relation (Rk0+1;k1+2;:::;kn+n+1) applies, and * *we pick out the leading term (19.26) mod V . If h > 1, we can repeat this cycle h times (al* *ways with the same indices ku). In all cases, e Oc has the leading term F h(bOLO bOM O[vJ]); (19:31) __ * * O where bOLO bOM O[vJ] is allowable by Lemma 18.12(d) and primitive inH because* * b L contains the factor b(0). In fact, one can show that every primitive allowable O-monomial in BP*(BP__* *k+1) can be written uniquely in the form bOLO bOM O[vJ], subject to the conditions (* *18.9). We have a computational verification mod V of the isomorphism Qk*~=P BP*(BP__k+* *1) induced by suspension. The first nth order relation The relation (R1;2;:::;n) is particularly importa* *nt, as only the two leading terms are meaningful. Bendersky has pointed out (during the proof of [3, Thm. 6.2]) that with a little more attention to detail, one ob* *tains a sharper version, the unstable analogue of Lemma 18.23. Lemma 19.32 (Bendersky) In BP*(BP__f(n)) we have the relation bOpm(0)O[vn] vnbOm(0)+ (-1)n(bOm(0))*p mod InBP*(BP__f(n)), (19:33) for each n > 0, where m = f(n)=2 = 1 + p + p2 + : :+:pn-1. Proof Although this result can be extracted from (R1;2;:::;n) by detailed exam* *ination, it is far simpler to return to (Rn). We proceed by induction on n, starting fr* *om eq. (19.11) for n = 1. For n > 1, we assume the result for all smaller n, and o* *btain it for n by evaluating bOph(0)O(Rn) mod In, where h = f(n-1)=2 = 1 + p + p2+ : :+:* *pn-2. n We recall that (Rn) is defined as the coefficient of xp in eq. (19.10). On* * the left, n * * Oph+1 we have bOph(0)Ob(vnxp + : :):by [8, eq. (15.5)], which provides only the term * *vnb(0) . The right side simplifies enormously, because h > 0 and b(0)O- kills *-decompos* *ables; we obtain _ X _ bOph(0)OP (x) = pbOph(0)Ob(x) + bOph(0)Ob(x)OmO[vI] : ;I;m By induction, bOph(0)O[vj] 0 mod In for all j < n - 1, since h = f(n-1)=2 > f(* *j)=2. n Thus the only terms of interest in [p](x) in our range of degrees are vnxp * * and n-1 vn-1xp , as it follows from [8, eq. (14.26)] and the map BP ! K(n-1) of ring spectra that any terms in eq. (19.9) of the form vin-1xm with i > 1 have divis* *ible n Oph Opn by p. The term vnxp yields b(0)Ob(0)O[vn], which is the leading term (19.27).* * By n-1 induction and eq. (15.13), vn-1xp yields n-1 n-1 Oh Opn-1 n-1 Oh Opn-1 bOph(0)ObOp(1)O[vn-1] (-1) F (b(0)) Ob(1) (-1) F (b(0)Ob(0) ); which is the other leading term, (19.26). | The ideals Jn For the unstable version of our splitting theorems we need the u* *n- stable analogue of the ideal Jn of Defn. 18.25. Definition 19.34 For n 0, we define Jn BP*(BP__*) as the *-ideal generated by all elements of the form c O([vj]-1), where j > n. JMB, DCJ, WSW - 100 - 23 Feb 1995 x19. Relations in the Hopf ring for BP Lemma 19.35 Jn is a Hopf ring ideal in BP*(BP__*). Proof We apply Lemma 19.2; only (i) requires any comment. It holds for [vj] - * *1, by the identity ([v] - 1) = ([v]-1)[v] + 1([v]-1); (19:36) P * *0 00 which is valid for any v 2 BP *by Prop. 11.2(a). We combine this with c = ic* *ici to obtain X X (cO([v]-1)) = c0iO([v]-1) c00iO[v] + c0iO1 c00iO([v]-1);(19:37) i i which shows that (i) holds for the typical *-generator of Jn. | Lemma 19.38 [v] 1 mod J nfor all v 2 Jn. Proof Suppose v = v0+ vjvK with j > n. As Jn is a Hopf ring ideal, we have [v] = [v0] * [vK ]O[vj] [v0] * [vK ]O1 = [v0] mod Jn . The result follows by induction on the number of terms in v. | The unstable analogue of Lemma 18.26 requires more detail but no new ideas. Lemma 19.39 For k f(n+1), Jn\BP*(BP__k) is the *-ideal in BP*(BP__k) genera* *ted by all elements that lie in BP*(BP__k) and have any of the following forms, whe* *re vJ contains a factor vj with j > n: (i)(if k is even) an allowable monomial bOIO[vJ]; (ii)(if k is odd) an allowable monomial e ObOIO[vJ]; (iii)(if k 0 and is even) [vJ] - 1k, with 2 Z(p); (iv)(if k = f(n+1)) a disallowed monomial 2 Opn+1 bOp(k1-1)ObOp(k2-2)O:O:b:(kn+1-n-1)O[vn+1] with 0 < k1 < k2 < : :<:kn+1. Remark To make (i) correct for I = 0, it is necessary to define bO0= eO0= [1] * *- 1 as in Prop. 13.7, so that bO0O[vJ] = [vJ] - 1. Proof Denote by I the *-ideal in BP*(BP__k) generated by the stated elements. * *It is clear from Lemma 19.38 that I Jn. To show the converse, we fix k and a large m, and prove by downward inducti* *on on h that all elements in BPi(BP__k) of the form c O([vh]-1) lie in I whenever i <* * m. This statement is vacuous for sufficiently large h (depending on m and k). We theref* *ore fix t > n and assume the statement holds for all h > t. Case 1: c = [vJ]. (This includes the degenerate cases [1] and 1k = [0k].) T* *hen c O([vt]-1) = [vJvt] - 1 is listed in (iii). Case 2: c = efflObOI. As in Lemma 18.26, c O([vt]-1) = efflObOIO[vt] has t* *o be allowable, except in the extreme case when k = f(n+1) and j = n + 1; either way* *, it is a listed generator of I. Case 3: c = efflObOIO[vhvJ] allowable, where h t. From the form of Defn. 1* *9.16, c O([vt]-1) = efflObOIO[vhvtvJ] remains allowable and is thus a listed generato* *r of I. JMB, DCJ, WSW - 101 - 23 Feb 1995 Unstable cohomology operations Case 4: c = efflObOIO[vhvJ], with h > t. We can write c O([vt]-* *1) = efflObOIO[vtvJ] O([vh]-1), which lies in I by induction, provided i < m. By Thm. 19.18, we have enough *-generators c. If c = a*d, eqs. (10.11) and * *(19.36) give c * ([vt]-1) = aO([vt]-1) * dO[vt] + aO1 * dO([vt]-1); which shows that the statement holds for c = a * d whenever it holds for a and * *d. | 20 Additively unstable BP -objects In this section, we discuss the additively unstable structures developed in* * xx5, 7 in the case E = BP , with particular attention to what becomes of the stable re* *sults of [8, x15]. We easily recover Quillen's theorem, that for any space X, the ge* *nera- tors of BP *(X) all lie in non-negative degrees. Our main result Thm. 20.11 say* *s in effect that there are no relations there either; more precisely, all relations * *follow from relations in non-negative degrees. We apply the theory to Landweber filtration* *s of an additively unstable module or algebra M, and find that the presence of addit* *ive unstable operations implies severe constraints on the degrees of the generators* * of M; this may be viewed as a better version of Quillen's theorem. By Thms. 6.35 and 7.11, module and comodule structures are equivalent, with or without multiplication. The most convenient context remains the Second Answer of x5, that an additively unstable BP -cohomology module (algebra) consists of a BP *-module (BP *-algebra) M equipped with coactions aeM : Mk --! M b Q(BP )k* (20:1) that (as k varies) form a homomorphism of BP *-modules (BP *-algebras) and sati* *sfy the usual coaction axioms (6.33). We continue to abbreviate Q(BP )**to Q**. T* *he bigraded algebra Q**was discussed in detail in x18. Connectedness The principle is that nothing interesting ever happens in negati* *ve degrees. The first result in this direction is due to Quillen [22, Thm. 5.1]. Theorem 20.2 (Quillen) For any space X, BP *(X)^is generated, as a BP *-modu* *le, (topologically if X is infinite) by elements of positive degree and exactly one* * element of degree 0 for each component of X. This will be an immediate consequence of Lemmas 4.10 of [8] and 20.5 (below* *). Quillen's proof is geometric; in contrast, x6 provides a global algebraic proof* * of the weak form of Quillen's theorem. Theorem 20.3 Given any integer k < 0, there exist for n 1: (i)additive unstable BP -operations rn defined on BP k(-), with deg(rn) !* * 1 and deg(rn) |k| for all n; (ii)elements v(n) 2 BP *; such that in any additively unstable BP -cohomology module M (e. g. BPP*(X)^for* * any space X), any x 2 Mk decomposes as the (topological infinite) sum x = nv(n)rn* *x, with deg(rnx) 0 for all n. In particular, M is generated (topologically) by elements of degree 0. JMB, DCJ, WSW - 102 - 23 Feb 1995 x20. Additively unstable BP -objects Proof Let {c1; c2; c3; : :}:be the Ravenel-Wilson (or any other) basis of the * *free BP *- module Qk*. By eq. (6.39) and the following Remark, we can write X x = kx = xn (20:4) n with xn = rnx, where rn denotes the operation dual to cn. If cn 2 Qkj, we must * *have j 0; then deg(rn) = - deg(cn) = j - k -k gives (i). We put v(n) = and note that deg(xn) = deg(rn) + deg(x) = j 0. | Remark The coefficients in eq. (20.4) are readily computed from eq. (6.41) as * *v(n) = Q(ffl)cn. Thus v(n) = vJ if cn = efflbm(0)wJ, and vanishes for monomials cn not* * of this form, so that many terms in eq. (20.4) are zero. If M is bounded above or X is finite-dimensional, the sum is finite and no * *topology on M is needed. To handle the generators in degree 0, we need a stronger hypothesis. Lemma 20.5 Let M be a connected (see Defn. 7.14) additively unstable algebra (e. g. BP *(X)^ for any connected space X). Then as a topological BP *-module, * *M is generated by 1M 2 M0 and elements of strictly positive degree. The generator 1* *M is never redundant. Remark Again, we may ignore the topology on M if M is bounded above or X is finite-dimensional. Proof We choose a basis {c1; c2; c3; : :}:of Q0*with c1 = 1; then given x 2 M0* *, we have eq. (20.4) with deg(xn) = - deg(cn) > 0 for all n > 1. Thus x <0; 1> x1 mod L, where L denotes the BP *-submodule of M generated (topologically) by the elements of positive degree. For the collapse operation 0 introduced in Defn. 7.13, we similarly have 0x <0; 1> mod L. But <0; 1> = <0; 1> = 1. As M is connected, 0x = 1M for some 2 Z(p), by Defn. 7.14. We deduce from Thm. 20.3 that M = L + (BP *)1M . Since L = 0 and (v1M ) = v1M for any v 2 BP *, this is a direct sum decomposition. | Primitive elements We generalize the theory of Landweber filtrations to the ad- ditive unstable context by following the same strategy as stably. We explore a * *general unstable comodule M by looking for morphisms f: BP *(Sk; o) ! M, for any k 0. As a BP *-module, BP *(Sk; o) is free on the canonical generator uk. Thus f is * *deter- mined, as a homomorphism of BP *-modules, by the element x = fuk 2 M. Since aeSuk = uk ek by Prop. 12.3(a), the condition we need is clear. Definition 20.6 Let M be any unstable comodule. If k 0, we call x 2 Mk additively unstably primitive if aeM x = x ek in M b Qk*. This obviously stabilizes to [8, Defn. 15.9], so that the additively unstab* *le primi- tives of M form a subgroup of the stable primitives of M. We do not define prim* *itives in negative degrees, for lack of a space Sk, and because ek is meaningless. In * *fact, for k < 0, x 1 does not in general lie in the image of the stabilization M Q(oe): M b Qk*--! M b BP*(BP; o) : JMB, DCJ, WSW - 103 - 23 Feb 1995 Unstable cohomology operations (Perhaps it never does?) Remark One might object that we have abolished primitives in negative degrees * *by simply defining them away, while some alternate definition might work. However,* * no such definition can be satisfactory. It is obvious from Defn. 12.6 that if x 2 M is primitive, so is x 2 M. On the other hand, we shall find (nontrivially) in Cor. 20.12 that the only primit* *ive in M of degree zero is 0 (at least, for the kind of comodule we discuss). It foll* *ows, by suspending enough, that no definition of primitive can have both these prope* *rties and produce anything interesting in negative degrees. It is immediate from the definition that if x 2 Mk is primitive, aeM (vx) = x ekjRv in M b Q** (for v 2 BP *). (20:7) We again recall from eq. (1.4) the numerical function 2(pn - 1) n-1 n-2 f(n) = _________= 2(p + p + : :+:p + 1) p - 1 and remind that deg(vn) = -(p-1)f(n) for n > 0. Lemma 20.8 Let x 2 Mk be a nonzero primitive element of the unstable BP - cohomology comodule M, and take n > 0. (a) If k < pf(n), then vinx 6= 0 for all i > 0 and is not additively unstabl* *y primitive; (b) If k pf(n) and Inx = 0, then vnx is additively unstably primitive. Corollary 20.9 If the additively unstably primitive element x 2 M satisfies * *Inx = 0 and is a vn-torsion element, then: (a) deg(vinx) pf(n) whenever vinx 6= 0; (b) vinx is additively unstably primitive or zero for all i. Proof We apply the Lemma to vinx by induction on i. Part (a) never applies (u* *nless vinx = 0); hence (b) must apply, to show that vi+1nx is primitive. | All this follows easily from Lemma 18.23. Proof of Lemma 20.8 From eq. (20.7) we have aeM (vinx) = x ekwin: In case (a), we note that by Defn. 18.4, ekwinis a basis element of Q**, so tha* *t aeM (vinx) is clearly nonzero. Even if k 2(pn-1)i, vinx is not primitive because aeM (vi* *nx) is different from n n vinx ek-2(p -1)i= x vinek-2(p -1)i: In case (b), we use the same formulae, with i = 1. The difference is that * *by Lemma 18.23, they now coincide, since e2 = b(0)and Inx = 0. | Remark For any x 2 Mk, where k 0, the coaction axiom (ii) of [8, eq. (8.7)] f* *orces aeM x to have the form X aeM x = x ek + xff cff; ff JMB, DCJ, WSW - 104 - 23 Feb 1995 x20. Additively unstable BP -objects where the cffare other Ravenel-Wilson basis elements and deg(xff) > k. Assuming that k < pf(n), so that ekwn is a basis element, let r be the operation (or fun* *ctional) dual to it. Proceeding as in the proof of the Lemma, we obtain X r(vnx) = x + xff; ff which shows that vnx 6= 0 if (for example) x is a module generator of M. Landweber filtrations The preceding results allow us to sharpen Thms. 15.10 and 15.11 of [8]. Theorem 20.10 Let M be the BP *-module with the single generator x 2 Mk and Ann (x) = In, so that M ~=k(BP *=In). (a) If n > 0, M admits an unstable comodule structure if and only if k f(n)* *-2, and it is unique. The additively unstably primitive elements are those of the * *form vinx, where 2 Fp, and k + deg(vin) f(n) if i > 0. (b) If n = 0, M ~= kBP *admits an unstable comodule structure if and only if k 0, and it is unique. The additively unstably primitive elements are those of* * the form x, with 2 Z(p). Remark Unlike the stable case, there are only finitely many primitives for n >* * 0. Of course, our definition forces this by requiring the degree of a primitive el* *ement to be non-negative. However, the theorem gives a much stronger condition. Proof By Thm. 20.3, we must have k 0, the canonical generator x is necessarily primitive, and ae must be given by eq. (20.7). Thus in (a), ae will be well de* *fined if and only if ek(jRv) 2 InQ**whenever v 2 In. Lemma 18.23 shows that this holds for v = vi for all i < n, since k f(n) - 2 pf(i); this is sufficient. On the * *other hand, if k < f(n) - 2 = pf(n-1), Lemma 20.8(a) (with n replaced by n-1) would contradict vn-1x = 0. Because ae is a BP *-module homomorphism (when it exists), the coaction axi* *oms [8, eq. (8.7)] need only be checked on x, where they are obvious. (Alternative* *ly, k(BP *=In) is a quotient of the geometric comodule BP *(Sk; o).) Since any additively unstably primitive element is also by design stably pr* *imitive, [8, Thm. 15.10] restricts the candidates for primitives to vinx. Lemma 20.8 sho* *ws, by induction on i 0, that vi+1nx is additively unstably primitive if and only * *if deg (vinx) pf(n). This is what we want, since | deg(vn)| = (p-1)f(n). The proof of (b) is similar, but far simpler. | With this restriction on the basic building blocks for an unstable module, * *we obtain the expected improvement in [8, Thm. 15.11]. Theorem 20.11 Let M be an unstable BP -cohomology comodule that is finitely presented as a BP *-module and has the discrete topology. Then there exists a f* *iltration by subcomodules 0 = M0 M1 : : :Mm = M; JMB, DCJ, WSW - 105 - 23 Feb 1995 Unstable cohomology operations where each Mi=Mi-1 is generated, as a BP *-module, by a single element xi, whose annihilator ideal Ann (xi) = Ini for some ni, and deg(xi) f(ni) - 2 (if ni > 0* *), or deg (xi) 0 (if ni= 0). If, further, M is a spacelike BP *-algebra (see Defn. 7.14), for example BP* * *(X) for any finite complex X, we can take each Mito be an invariant ideal in M. At the * *last stage, we may take xm = 1 and nm = 0 or 1. Unfortunately, although the statement of the Theorem is exactly as expected, Landweber's method fails; Lemma 2.3 of [16] does not appear to be available her* *e. (The BP *-submodule 0: In = {y 2 M: Iny = 0} of M is defined but does not appea* *r to be unstably invariant, owing to the dimensional restriction in Lemma 18.23.) I* *nstead, we are forced to construct a suitable primitive x1 2 M directly. We would have preferred Landweber's construction because it guarantees that Ann (x1) is maxim* *al, which is useful in applications. Proof We start with a nonzero element x 2 Mk of top degree; by Thm. 20.2, k 0 and x is automatically primitive. We construct a sequence of nonzero primiti* *ve elements ys 2 M such that Isys = 0, starting with y0 = x. (Here, it is conveni* *ent to write v0 = p.) We stop when we reach an element yn that is vn-torsion-free (vinyn 6= 0 for all i > 0) and put x1 = yn and n1 = n; this must occur eventual* *ly, by Lemma 20.8(a) (e. g. when 2pn > k). Assume we have ys, where s 0. If it is vs-torsion-free, we stop; this is yn. Otherwise, take the smallest exponent q s* *uch that vqsys = 0 and put ys+1 = vq-1sys, to get Is+1ys+1 = 0. By Cor. 20.9 (with s in * *place of n), ys+1 is primitive and deg(ys+1) pf(s). We have found a primitive x1 such that Inx1 = 0, x1 is vn-torsion-free, and deg (x1) pf(n-1) = f(n) - 2. (If n = 0, there was no induction, and deg(y0) = k 0.) As Ann (x1) is an invariant ideal (in the stable sense), its radical ide* *al must be a finite intersection of invariant prime ideals in BP *, therefore be Im for* * some m. That is, q ________ In Ann (x1) Ann (x1)= Im : q ________ Since vn 62 Ann (x1), we conclude that m = n and Ann (x1) = In. We finish as in the stable case, by setting M1 = (BP *)x1, observing that t* *his submodule is invariant by eq. (20.7), and replacing M by M=M1. The induction continues until M = 0, and must terminate (easily, unlike the stable case), bec* *ause each Mk is a finitely generated module over the Noetherian ring Z (p)and we need consider only k 0. Now assume that M is a spacelike algebra, i. e. a product of connected alge* *bras. This product is evidently finite, otherwise M would be uncountable. We easily r* *educe to the case when M is connected, which includes the case when M = BP *(X) for a connected finite complex X. By Lemma 20.5, the module (BP *)x is automatically * *an ideal in M; by induction, so is (BP *)ys for each s, in particular M1. At the l* *ast step, the module M=Mm-1 is also an algebra; we therefore have 1 = vxm and x2m= v0xm for some v,v0 2 BP *. Then xm = 1xm = vx2m = vv0xm = v01, which shows that Ann (1) = Ann (xm ) = Inm , and we may replace the generator xm by 1. This impl* *ies nm 1, since f(n) > 2 for n 2. | JMB, DCJ, WSW - 106 - 23 Feb 1995 x21. Unstable BP -algebras Corollary 20.12 For M as in Thm. 20.11, the suspension M contains no nonzero additively unstably primitive elements in degree zero. Proof We observe that 0 = M0 M1 : : :Mm = M is a Landweber filtration of M. By Thm. 20.10, the only unstable comodule of the form k(BP *=In) that has a nonzero primitive in degree zero is BP *, which does* * not occur as a Landweber factor Mi=Mi-1of M. | 21 Unstable BP -algebras In this section, we apply the theory of xx10, 19 to an unstable BP -cohomol* *ogy algebra M. Our main application is Thm. 21.12 on Landweber filtrations of M, which contains Thm. 1.5 and improves on Thm. 20.11 by one degree. Of course, we can always recover an additively unstable algebra from an uns* *table algebra simply by discarding the nonadditive operations. As a general rule, we * *can improve our results by one degree (but never more than one, in view of Thm. 13.* *6) by retaining all operations, at the cost of working in a far more complicated a* *nd unfamiliar environment. We developed the necessary machinery in x10. Primitive elements It is clear from x20 that the way to study a general unstab* *le algebra M is to look for unstable morphisms f: BP *(Sk) ! M from the (relativel* *y) well understood object BP *(Sk). Since BP *(Sk) is a free BP *-module with bas* *is {1S; uk}, f is uniquely determined, as a homomorphism of BP *-modules, by f1S =* * 1M and the element x = fuk 2 Mk. We extend the concept of primitive element to the unstable context, using Prop. 13.7 as a guide. Definition 21.1 We call x 2 Mk (where k 0) unstably primitive if r(x) = 1M + x for all r, (21:2) where we interpret e0 = [1] - 10 (as in Prop. 13.7). This is a necessary and sufficient condition for f to be a morphism of unst* *able algebras, by eqs. (10.41), (10.16), and the Cartan formula (10.23). Among the u* *nsta- ble operations is the squaring operation, defined by r(y) = y2 for all y, which* * implies that f is a homomorphism of BP *-algebras (even if k = 0). When we restrict to additive operations, x is automatically additively primitive, and we have avail* *able all the results of x20. Many elementary properties of primitives follow directly from the definitio* *n. Proposition 21.3 Let M be an unstable algebra. Then: (a) Unstable primitives are natural: if x 2 M is unstably primitive and f: M* * ! N is a morphism of unstable algebras, then fx 2 N is also unstably primitive; (b) The elements 0 2 Mk (for any k 0) and 1M are unstably primitive; (c) If x 2 Mk is unstably primitive, where k > 0, then x2 = 0; JMB, DCJ, WSW - 107 - 23 Feb 1995 Unstable cohomology operations (d) If x 2 Mk is unstably primitive, where k > 0, then x is unstably primiti* *ve for any 2 Z(p); (e) If k > 0 is odd, the unstable primitives in Mk form a Z(p)-submodule; (f) If k > 0 is even and x; y 2 Mk are unstably primitive, then x + y is uns* *tably primitive if and only if xy = 0; (g) The only nonzero unstable primitive in BP *= BP *(T ) is 1; (h) Any unstable primitive x 2 M0 is idempotent, x2 = x; (i) If x 2 M0 is unstably primitive (and therefore idempotent), then the con* *jugate idempotent 1M -x is also unstably primitive, but -x is never unstably primitive (unless -x = x). Proof Part (a) is trivial. Part (b) is clear from eqs. (10.41) and (10.28). As* * noted above, f is an algebra homomorphism, which gives (c) and (h). Then (g) follows * *from (b) and (h). In (d), eq. (10.16) gives r(x) = 1M + x : Since k > 0, Prop. 13.7(g) gives [] Oek = ek, which shows that x is primitive. We prove (e) and (f) together. If x; y 2 Mk are primitive, the Cartan form* *ula (10.23) yields r(x+y) = 1M + x + y + (-1)k xy; which is to be compared with eq. (21.2). The unwanted last term vanishes if k i* *s odd, because ek is then an exterior generator; but if k is even, ek * ek is a basis * *element of BP*(BP__k). For (e), we combine this with (d). For (i), we first use eq. (10.29) to compute r(-x) = 1M + x, which shows that -x is not primitive. We then use eqs. (10.23) and (10.41) to c* *ompute r(1M -x) = 1M + x, which shows that 1M - x is primitive.* * | We deduce that the Remark following Defn. 20.6 extends to show that unstable primitives cannot usefully be defined in negative degrees, even though the unst* *able suspension (see Defn. 13.4) had to be defined somewhat differently. __ Corollary 21.4 Let M = BP *M be a based unstable BP -algebra. __ __ (a) If x 2M is unstably primitive, so is x 2 BP * M ; (b) If M is the kind of algebra considered_in Thm. 20.11, there are no unsta* *ble primitives of degree zero in BP * M other than 0 and 1 2 BP *. * * __ Proof Part (a) is clear from eq. (13.3). For (b), take any primitive y 2 BP * M in degree 0. By Prop. 21.3(g), its augmentation in BP *must be_0 or 1; if_1,_we* * use Prop. 21.3(i) to replace y by 1 - y. Then y = x for some x 2M . As y 2 M is also additively primitive, Cor. 20.12 shows that y = 0. | ` * * If X is the disjoint union X1 X2 of two spaces, we have BP (X) = BP (X1) BP *(X2), a product of unstable algebras. By Prop. 21.3, the elements (1; 0) an* *d (0; 1) are primitive idempotents in BP *(X). The converse is also true, algebraically. JMB, DCJ, WSW - 108 - 23 Feb 1995 x21. Unstable BP -algebras Theorem 21.5 If x 2 M0 is an unstably primitive element in the unstable alge* *bra M, other than 0 and 1M , so that x and 1M - x are idempotents, we have the spli* *tting M ~=xM (1M -x)M of M as a product of unstable algebras. Proof By Prop. 21.3(i), both x and 1M - x are primitive and idempotent. We def* *ine the first projection pK : M ! K = xM by pK y = xy; since x is idempotent, pK is a homomorphism of BP *-algebras. We define pL: M ! L = (1M -x)M similarly, by pLy = (1M -x)y. These will give the desired splitting of M. Given y 2 M, we assume that rM (y) is in the standard form (10.22), where rM denotes the operation of r on M. By the Cartan formula (10.36), X X rM (xy)= yfi+ xyfi fi fi X = xrM (y) + (1M -x)yfi: fi Hence xrM (xy) = xrM (y), which shows that pK is an unstable morphism, provided we define the action rK : K ! K of r on K by rK (z) = xrM (z) for z 2 K M. All the necessary laws are inherited from M. We treat pL similarly. | Landweber filtrations We repeat the theory of x20, with an improvement of one in degree. If x 2 Mk is primitive in the unstable algebra M, where k > 0, we co* *mpute from eq. (10.16) that r(vx) = 1M + x (21:6) for any v 2 BP -h. Lemma 21.7 Let M be an unstable algebra, and x 2 Mk an unstably primitive element, where k > 0. Then the BP *-submodule (BP *)x generated by x is an unst* *ably invariant ideal in M, provided it is an ideal. Proof We apply Lemma 8.10, with the help of eq. (21.6). | It is still true that an element of positive top degree in M is automatical* *ly primi- tive, for lack of any other possible terms in r(x). We now use the additional structure of the unstable operations to sharpen Lemma 20.8. We recall once more from eq. (1.4) the numerical function 2(pn - 1) n-1 n-2 f(n) = _________= 2(p + p + : :+:1) : p - 1 Lemma 21.8 Let x 2 Mk be a nonzero unstably primitive element of the unstable algebra M, and n > 0. (a) If k pf(n), then vinx 6= 0 for all i > 0 and is not unstably primitive; (b) If k > pf(n) and Inx = 0, then vnx is unstably primitive. Corollary 21.9 If the unstably primitive element x 2 M satisfies Inx = 0 and* * is a vn-torsion element, where n > 0, then: (a) deg(vinx) > pf(n) whenever vinx 6= 0. (b) vinx is unstably primitive or zero for all i. JMB, DCJ, WSW - 109 - 23 Feb 1995 Unstable cohomology operations Proof This is formally the same as for Cor. 20.9. | Proof of Lemma Part (a) adds nothing to Lemma 20.8(a) unless k = pf(n), in which case we must take i = 1 if we are to have deg(vinx) 0. To test whether or not vnx is primitive, we have to compare r(vnx) = 1M + x from eq. (21.6) with 1M + vnx = 1M + x; where we write deg(vn) = -d. For (a), we take k = 2pm, where m = f(n)=2. Lemma 19.32 expands e2pm O[vn], to show that r(vnx) has the term x. As bOm(0)is a *-polynomial generator of BP*(BP__2m), we deduce that vnx cannot * *be primitive or zero, whatever Ann (x) is. Similarly, for i > 1, r(vinx) has the t* *erm x = x; which shows that vinx 6= 0. For (b), we apply a further suspension ek-2pm O- to eq. (19.33), which kill* *s de- composables, to yield ek-2pm Oe2pm O[vn] vnek-2pm Oe2m = vnek mod In , This shows that vnx is unstably primitive, a stronger statement than Lemma 20.8 provides. | As promised, these two results improve on Lemma 20.8 and Cor. 20.9 by one degree. We use them to deduce the main theorems, which likewise improve on Thms. 20.10 and 20.11 by one. Theorem 21.10 Let M be the BP *-module BP * (BP *)x, where the annihilator ideal Ann (x) = In and deg(x) = k > 0. If M is made an algebra by taking 1 2 BP* * * as the unit element and setting x2 = 0, then: (a) If n > 0, M admits an unstable algebra structure if and only if k f(n) * *- 1, and it is unique. The nonzero unstably primitive elements in M are 1M and the elements vinx, where 2 Fp ( 6= 0) and i satisfies i = 0 or deg(vinx) > f(n). (b) If n = 0, M admits a unique unstable structure. The nonzero unstably pri* *mi- tive elements in M are 1M and the elements x with 2 Z(p)( 6= 0). Proof In (a), we regard M as the quotient of the geometric unstable algebra BP* * *(Sk) with BP *-basis {1S; uk} by the ideal Inuk. The proof is formally the same as Thm. 20.10, except that we use Lemma 21.8 instead of Lemma 20.8, Cor. 21.9 inst* *ead of Cor. 20.9, and eq. (21.6) instead of eq. (20.7). To determine the primitives in positive degrees, we first note that x is pr* *imitive by Prop. 21.3(d) and apply Lemma 21.8 to vinx, by induction on i. The primitives in degree zero are given already by Prop. 21.3. | For completeness, we mention the analogous results for k = 0. JMB, DCJ, WSW - 110 - 23 Feb 1995 x22. Additive splittings of BP -cohomology Proposition 21.11 For the unstable algebra BP *(T ) = BP *: (a) BP *has no proper nonzero invariant ideals; (b) The unstable algebra BP *(S0) ~=BP * BP *has the two copies of BP *as its only proper nonzero invariant ideals. Proof In (a), assume J is a nonzero ideal, and take v 6= 0 in J. As the eleme* *nts [v] are linearly independent in the Hopf ring, we see from eq. (11.1) that ther* *e is an operation r such that r(v) = 1 and r(0) = 0. Thus if J is invariant, we must ha* *ve 1 2 J, and therefore J = BP *. In (b), the operations are given similarly by r((v; v0)) = (; ) 2 BP * BP *; from which it is easy to see that any invariant ideal J that contains an elemen* *t (v; v0) with both v and v0 nonzero must contain 1S = (1; 1) and therefore everything. F* *or other ideals J, we can apply (a). | Theorem 21.12 Given any spacelike (see Defn. 7.14) discrete unstable* * BP - cohomology algebra M that is finitely presented as a BP *-module (e. g. BP *(X)* * for any finite complex X), there is a filtration by unstably invariant ideals 0 = M0 M1 : : :Mm = M in which each quotient Mi=Mi-1 is generated, as a BP *-module, by a single elem* *ent xi, whose annihilator ideal Ann (xi) = Ini for some 0 ni < 1, and deg(xi) max (f(ni)-1; 0). At the last step, nm = 0 and we may take xm = 1M . Proof This is formally identical to the algebra case of the proof of Thm. 20.1* *1, except that we use the corresponding results from this section instead of x20. Lemma 2* *1.7 shows that M1 = (BP *)x1 is indeed an invariant ideal. | 22 Additive splittings of BP -cohomology Lemma 22.1 will construct idempotent operations n in BP -cohomology, from which Parts (a) of our splitting theorems 1.12 and 1.16 will follow. In fact, w* *e find a large class of n, among which none seems to be preferred. At the end of the sec* *tion, we give an example where no choice of n has the obvious image Z(p)[v1; : :;:vn]* * on homotopy groups. Lemma 22.1 Assume that k < f(n+1), where n 0. Then there exists an additive idempotent operation n: k ! k having the following properties: (i)The image of n: BP__k! BP__kcan be canonically identified with BP__* *k; _ (ii)The map n factors to yield an H-space splitting n: BP__k! BP__kof * *the canonical H-map ss: BP__k! BP__k; _ (iii)For all spaces X, nnaturally embeds BP k(X) BP k(X) as a summand, in the sense of abelian groups (but not as BP *-modules); (iv)If also k f(n), the H-space BP__kdoes not decompose further. JMB, DCJ, WSW - 111 - 23 Feb 1995 Unstable cohomology operations Remark This result is best possible, in the sense that no additive n exists wh* *en k f(n+1). (In more detail, choose m so that f(m) k < f(m+1); then m > n and m exists. Lemma 22.2 will show that if n exists, we automatically have n Om = n. The modified idempotent 0n= m On satisfies 0nOm = 0n= m O0nand therefore decomposes BP__k further, contrary to (iv).) For k > f(n+1) this is obvious, because H*(BP__k) then has torsion [26]. The borderline case k = f(n+1) will* * be discussed in x23, where we find that a nonadditive n does exist. Proof of Thm. 1.12(a) and Thm. 1.16(a) (assuming Lemma 22.1) The two Theo- rems are equivalent by_[8, Thm. 3.6(a)]. As indicated, we use the splittings pr* *ovided by Lemma 22.1, namely n: BP__k! BP__kand, for each j > n, the map _ j* vj fj: BP__k+2(pj-1)---!BP_k+2(pj-1)--!BP__k: _ This jexists because k + 2(pj-1) < f(n+1) + (p-1)f(j) pf(j) < f(j+1): _ On homotopy groups, n induces a splitting of BP *! BP *=Jn, while fj induces a splitting of Jj-1 ! Jj-1=Jj, in view of the commutative diagram _ j vj BP *=Jj_______-BP * ________-Jj-1 Q Q = | | Q Q || || Qs |? ~ |? BP *=Jj______Jj-1=Jj-= in which multiplication by vj induces the isomorphism. We use the H-space structure of BP__kto multiply the maps n and the fj toge* *ther to form a map f: W ! BP__kfrom the restricted product W (the union of the finite subproducts) of BP__kand the spaces BP__k+2(pj-1). The homotopy groups of* * W are the direct sums i j M i j sss(W ) = sss BP__k sss BP__k+2(pj-1): j>n We have enough information to conclude that f induces an isomorphism of filtered groups f*: ss*(W ) ~= ss*(BP__k). For connectedness reasons, the above sum is * *in fact a product of graded groups, which makes W homotopy equivalent to the desired product of spaces. Finally, Lemma 22.1 shows that all factors of W after the fi* *rst are indecomposable, since k + 2(pj-1) 2(pj-1) = (p-1)f(j) f(j): If k f(n), so is the first. | Construction of idempotent operations To complete the proof, we need an idempotent operation n. We actually construct the BP *-linear func* *tional : Qk*= Q(BP )k*! BP *that corresponds to it in the list (6.9). We recall * *the coalgebra structure (Q( ); Q(ffl)) on Q**and the ideal Jn introduced in Defn. 1* *8.25. JMB, DCJ, WSW - 112 - 23 Feb 1995 x22. Additive splittings of BP -cohomology Lemma 22.2 Assume the linear functional : Qk*! BP *defined by the addi- tive operation n: k ! k satisfies the conditions: (i) = 0; (22:3) (ii) Q(ffl)c mod Jn for all c 2 Qk*. Then: (a) The homology homomorphism Q(n): Qk*! Qk*satisfies (i) Q(n)Jn = 0; (ii) Q(n) id: Qk*! Qk*mod J n; (b) Q(n) induces a splitting of the short exact sequence 0 --! Qk*\ Jn--! Qk*--! Qk*=(Qk*\ Jn) --! 0 of left BP *-modules; (c) ss On = ss: BP__k! BP__k; (d) The operation n is idempotent and has the properties listed in Lemma 22.* *1. We shall write Qk*=Jn for the tedious but more accurate expression Qk*=(Qk** *\J n). Remark From_a more invariant point of view, Q(ffl) induces the quotient augmen- tation Q(ffl): Qk*=Jn ! BP *=Jn. The conditions (22.3) on are convenien* *tly expressed by the commutative diagram Qk* pppppppppBPp*p- | ppp || | pp | | p | |ss pp | (22:4) | pp | | pp | |?p Q___(ffl)|? Qk*=Jn ______BP-*=Jn in_which_the vertical arrows are the obvious projections. In words, we plan to* * lift Q(ffl)to a homomorphism of BP *-modules Qk*=Jn ! BP *and define as the composite. This is easy if Qk*=Jn is a free BP *-module (and in view of (b), im* *possible otherwise). Proof We enlarge diag. (22.4) to the commutative diagram Q( ) 1 R Qk*____________-Q** Qk* ppppppppppQ**pBP-* ___________-Qk* | | pppp3 | | | | ppp | | |ss |1ss ppp | |ss | | pp | | | | ppp | | |? Q___( ) |? pp 1Q___(ffl) |? _ |? R Qk*=Jn _________Q**-Qk*=Jn ______Q**-BP *=Jn ________-Q**=Jn JMB, DCJ, WSW - 113 - 23 Feb 1995 Unstable cohomology operations _____ _ of BP *-module homomorphisms, whereQ( ) and R are quotients of Q( ) and R. By Lemma 6.51(c), we recover Q(n) as the top row, while the bottom row reduces by diag. (6.31) to the identity homomorphism of Qk*=Jn. Thus the diagonal provi* *des a splitting j: Qk*=Jn ! Qk*such that j Oss = Q(n) and ss Oj = 1. This is enough to establish (a), that Q(n) is idempotent with kernel exactly Qk*\ Jn. Part (b) is merely a restatement of (a). It follows that n also is ide* *mpotent. By [8, Lemma 3.9], the idempotent operation n is represented in Ho by the idempotent map n = i2O p2 on the product W = W1 x W2 of H-spaces, where i2: W2 ! W and p2: W ! W2. Corollary 12.4 gives the effect of n on homotopy groups: eq. (22.3)(i) shows that n*v = 0 if v 2 Jn, while (ii) shows that n*v Q(ffl)(ek+hjRv) = v mod Jn in ss*(BP__k) ~=BP * for all v 2 BP -h. These two statements identify ss*(W2) with BP *=Jn; more pre* *cisely, the composite f = ss Oi2: W2 ! BP__k! BP__kinduces the desired isomorphism on homotopy groups and is thus an isomorphism of abelian group objects in Ho . We need (c) to be sure our identifications are correct. Now that we know BP* *__k is a summand of BP__k, it is enough to work in QBP*(-). By construction, Qss* kills Jn; this, with (a)(ii), gives Qss*O_Qn* = Qss*. We can_now define the splitting n = i2O f-1: BP__k! BP__kof ss, so t* *hat ss O n = 1. From (c), we_have ss_= ss On = ss Oi2O p2 = f Op2, whi* *ch shows that the idempotent nO ss = nOf Op2 = i2O p2 = n is as expected. Now we can read off properties (i), (ii), and (iii) of Lemma 22.1. Property (iv) was proved in [26], but also follows from Cor. 12.4. Suppose * *there is a splitting BP__k' W1 x BP__k' W1 x W x W 0 of H-spaces that induces the decomposition BP *= Jn G G0on homotopy groups, where 1 2 G, and let r be the idempotent that splits off W 0, so that = * *0 and = 0. Suppose that W 0is (k+h-1)-connected, where we must have h > * *0. Then = 0 for all c 2 Qkiwhenever i < k + h. Choose a nonzero element v 2 BP -h that lies in G0 and is not divisible by * *p. Then r*v = v in homotopy and v 62 I1 + Jn (recall that I1 = (p)). Obviously, v 2 I1 = In+1 + Jn. There must be some integer m, satisfyingP1 m n, such that v 2 Im+1 + Jn but v 62 Im + Jn. We write v = py0+ mj=1vjyj+ z, with z 2 Jn. Si* *nce k + h f(n) + 2(pj-1) = f(n) + (p-1)f(j) pf(j); we have enough factors e to apply Lemma 18.23 for each j m, in the form j-1) mod Ij j-1) = vj = 0 : By Cor. 12.4, r*v 0 mod Im , which contradicts our choices of v and m. | Proof of Lemma 22.1 Lemma 18.26(a) makes it obvious that linear functionals exist as in diag. (22.4), so that Lemma 22.2 applies. | Example Even in the simplest case, namely 1: BP__2! BP__2for p = 2, 1* never induces the obvious splitting on homotopy groups. (Presumably, this failure is * *com- JMB, DCJ, WSW - 114 - 23 Feb 1995 x23. Unstable splittings of BP -cohomology pletely general.) We compute 1*v31in terms of the Hazewinkel generators [11]. T* *he element b4(0)w312 Q2*is not allowable; instead, 12 3 4 10 2 4 4 8 b4(0)w31= -___v1b(0)b(1)w1 + v1 + __v2b(0)- ___v1b(1)- __b(0)w2 - __b(2); 7 7 7 7 7 as can be checked by stabilizing and working in BP*(BP; o). By construction, <1* *; -> takes the values 1 on b(0), v2 on b(2)for some 2 Z(2), and zero on the other a* *llowable monomials that appear. Thus by Cor. 12.4, 4 - 8 1*v31= v31+ ______v2 ; 7 which always contains a term in v2. Remark It is often useful to arrange the operations n: k ! k compatibly as n a* *nd k vary. However, we emphasize that Thm. 1.12 as stated requires no compatibili* *ty conditions whatever. For fixed n, compatibility in k is easily arranged. Given n: k ! k that sat* *isfies conditions (22.3), the looped operation n: k-1 ! k-1 has the functional e k * Qk-1*--! Q* -----! BP and clearly again satisfies (22.3). We may choose n: k ! k arbitrarily for k = f(n+1) - 1 and use this approach for all lower k. For fixed k, we have n for all sufficiently large n. The compatibility con* *dition n On+1 = n (equivalently, Ker n+1 Ker n) is automatic, from Lemma 22.2. The other condition, n+1 On = n (equivalently, Im n Im n+1), does not hold in general, but can be arranged for all n simultaneously by replacing each n by 0n= : :O:n+2 On+1 On. (The infinite composite presents no difficulty, as Q(n) = id: Qki! Qkifor i < k + 2(pn+1 - 1).) This results in a sequence of commuting idempotents n that satisfy n Om = m On = n whenever n < m. 23 Unstable splittings of BP -cohomology In this section, we improve the splitting in Lemma 22.1 by one by allowing * *the idempotent operation n to be nonadditive. We defer the proof until after stati* *ng Lemma 23.5. For this, we need the more detailed relations in the Hopf ring deve* *loped in x19. Lemma 23.1 Assume that k = f(n+1), where n 0. Then there is a nonadditive operation n: k ! k having the following properties: (a) It satisfies the axioms [8, eq. (3.11)] and so is idempotent; (b) It has a coimage Coim n which is represented by the H-space BP__k; _ (c) Its representing map n: BP__k! BP__kfactors to yield a section n: BP__k! BP__k(not an H-map) of the canonical H-map ss: BP__k! BP__k. JMB, DCJ, WSW - 115 - 23 Feb 1995 Unstable cohomology operations Proof of Thms. 1.12 and 1.16, for k = f(n+1) (assuming Lemma 23.1) This is almost identical to the proof given in x22 for k < f(n+1), except that we apply [8, Lemma 3.10]_instead of [8, Lemma 3.9]. The maps fj appearingQthere are sti* *ll H-maps; only n is not. We can still represent Ker n by j>nBP__k+2(pj-1). If any of the spaces decomposed as a product, we could apply the loop space functor to obtain an H-space decomposition of BP__k-1, using additive operatio* *ns, which would contradict the part of Thm. 1.12 already proved. | Of course, we know_from Lemma 22.1 that for k = f(n+1), n: k ! k can never be additive and that n is never an H-map. However, looping gives an additive idempotent operation n: k-1 ! k-1, which will be one of those provided by Lemma 22.1. We have the converse, which we prove after stating Lemma 23.5. Theorem 23.2 Let n: k-1 ! k-1 be any of the additive idempotent operations provided by Lemma 22.1. Then: (a) If k-1 is even, n can be delooped uniquely to an additive idempotent ope* *ra- tion k ! k as in Lemma 22.1; (b) If k-1 is odd, n can be delooped (not uniquely) to a nonadditive idempot* *ent operation k ! k as in Lemma 23.1. The next two lemmas constitute the unstable analogue of Lemma 22.2. They are far more complicated, because instead of Q( ), we have only the natural tra* *ns- formation : U ! UU. This requires knowledge of the homology homomorphisms r* induced by each operation r, which is provided by Thms. 10.19 and 10.33 and the properties of each O-generator of BP*(BP__*). We warn that as a consequenc* *e, the form of the proofs runs totally counter to traditional proofs involving coh* *omology operations. We abbreviate to , etc. Lemma 23.3 If the unstable operation r: k ! m satisfies = 0, then the homology homomorphism r*: BP*(BP__k) ! BP*(BP__m) satisfies r*Jn = 0. Proof Our plan is to show that r*c = 0 in three steps, depending on the form of c 2 Jn, simultaneously for all operations r: k ! m that satisfy = 0, wh* *ere c 2 BP*(BP__k) determines k and m is arbitrary. Case 1: c = [vj] - 1, where j > n. By hypothesis, = . The* *n by Prop. 11.2(g), r*([vj] - 1) = [] - [] = 0: Case 2: c = aO([vj]-1), where j > n. Thus c is a *-generator of Jn. We app* *ly Thm. 10.33(c); the operations r00ffdefined by eq. (10.35) satisfy our hypothesis = = 0 for all d 2 Jn because cffOd 2 Jn, Jn being a Hopf ring ideal by Lemma 19.35. Using eq. (19.36* *) to compute the iterated coproduct ([vj]-1), we see that every term of r*c in eq. (* *10.37) contains a factor r00ff*([vj]-1), which vanishes by Case 1. Case 3: c = a * b, with b as in Case 2. Since such elements span Jn as a BP* * *- module, this will complete the proof. We apply Thm. 10.19(c); the operations r* *00ff defined by eq. (10.21) satisfy our hypothesis = = 0 for all d 2 Jn JMB, DCJ, WSW - 116 - 23 Feb 1995 x23. Unstable splittings of BP -cohomology because Jn is a *-ideal. Using eq. (19.37) to compute the iterated coproduct b,* * we see that every term of r*c in eq. (10.25) contains a factor of the form r00ff*(* *b0O([vj]-1)), which vanishes by Case 2. | Lemma 23.4 Let r: k ! m be an unstable operation. (a) If r satisfies 2 Jn for all c 2 BP*(BP__k), then r*c (fflc)1m mo* *d Jn for all c 2 BP*(BP__k); (b) If r satisfies Q(ffl)qkc mod Jn for all c 2 BP*(BP__k), then r*c* * c mod J nfor all c 2 BP*(BP__k). Proof We prove (a) in five steps, depending on the form of c, simultaneously f* *or all r: k ! m that satisfy the hypothesis, where c 2 BP*(BP__k) determines k and m is arbitrary. We work throughout mod Jn, which is a Hopf ring ideal by Lemma 19.35. Case 1: c = [v], for any v 2 BP *. By Prop. 11.2(g) and Lemma 19.38, r*[v] = [] 1. This includes the special case c = 1 = [0]. Case 2: c = e. By Prop. 13.7(h) and Lemma 19.38, r*e 1 * 1Oe = 1 * 0 = 0. Case 3: c = bi, where i > 0. By Prop. 15.3, working formally in BP*(BP__m)[* *[x]], r*b(x) = [] * * b(x)OjO[] * b(x)OjO1 = * fflb(x)Oj= 1* * : j>0 j>0 j>0 The coefficient of xi gives r*bi 0. Case 4: c = a Ob, where b = e or b = bi for some i > 0. We apply Thm. 10.33* *(c); the operations r00ffdefined by eq. (10.35) satisfy the hypothesis = * * 2 Jn for all d. Then using Prop. 13.7(d) or Prop. 15.3(c) to compute the iterated coproduct b, we see that every term of r*c in eq. (10.37) contains a factor r00* *ff*e or r00ff*bj with j > 0, which lies in Jn by Case 2 or Case 3. This, with Case 1, t* *akes care of all the *-generators (19.4) of BP*(BP__*). Case 5: c = a * d, with d as in Case 4. We apply Thm. 10.19(c) and again fi* *nd that each r00ffsatisfiesPour hypothesis = 2 Jn for all g* *. In the iterated coproduct d = j ffdj;ff, every term contains a factor dj;ffto which* * Case 4 applies. Thus every term of r*c in eq. (10.25) has a factor r00ff*dj;ff 0. As every *-monomial in the O-generators of BP*(BP__*) is included in Cases * *1 and 5 (by writing [v] * [v0] = [v+v0]), this completes the proof of (a). For (b), we recall from eq. (10.42) that = Q(ffl)qkc, so that (a) ap* *plies to r - k.PWe apply eq. (10.17) to r = (r-k) + k to deducePthat for any c 2 E*(E_k), r*c i(fflc0i)c00i= c, where as usual we write c = ic0i c00i. | We need one more result before we prove Lemma 23.1 and Thm. 23.2. The structure of BP*(BP__k)=Jn is much more opaque when k = f(n+1). We defer the proof until after Lemma 23.12. Lemma 23.5 For k f(n+1), where n 0: (a) BP*(BP__k)=Jn is a free BP *-module; (b) The homomorphism Q(BP )k-1*=Jn ! BP*(BP__k)=Jn induced by suspension is a split monomorphism of BP *-modules. JMB, DCJ, WSW - 117 - 23 Feb 1995 Unstable cohomology operations Note that we have two different ideals Jn here. One is an ideal in the alge* *bra Q** in the ordinary sense, while the other is a Hopf ring ideal in BP*(BP__*). Proof of Lemma 23.1 (assuming Lemma 23.5) To apply the method of Lemma 22.1, we need an operation n: k ! k that satisfies *Jn = 0 and * idmod Jn. In view of Lemma 23.3 and Lemma 23.4(b), these conditions are ensured by (and in fact equivalent to) the following conditions on the linear functional : (i) = 0; (23.6) (ii) Q(ffl)qkc mod Jn for all c 2 BP*(BP__k). Therefore we need to fill in the diagram Qk-1* _________BP*(BP__k)-ppppppppppBPp*- | | ppp3| | | ppp | | | pp | | | ppp | (23:7) | | ppp | |? |? pp |? Qk-1*=Jn______-BP*(BP__k)=Jn______-BP *=Jn analogous to diag. (22.4) with a lifting BP*(BP__k)=Jn ! BP * of the homomor- phism BP*(BP__k)=Jn ! BP *=Jn induced by Q(ffl) Oqk, which then defines . Lemma 23.5(a) makes this easy to do. For (a), we must verify the axioms [8, eq. (3.11)] on n. The first holds tr* *ivially, for dimensional reasons. The second is the identityPn(x+z) = n(x), for z = y - n(y). We assume that the standard formPr(x) = ff xffholds for all r, as in * *eq. (10.3). Then by eq. (10.20), n(x+z) = ffxff00ff(z), where the operation 00ffis define* *d as having the functional <00ff; c> = . Because z = (k-n)(y), we have o* *nly to prove that (k-n)*00ff= 1 in BP *(BP__k) for each ff. We compute the associated linear functional as <(k-n)*00ff; c> = <00ff; (k-n)*c> = : By Lemma 23.4(a), (k-n)*c (fflc)1 mod Jn. As kills Jn by Lemma 23.3 and Jn is an ideal, this agrees with = fflc. Now we can* * apply [8, Lemma 3.10] to construct the coimage of n. For (b) and (c), we have to check that n acts as desired on homotopy groups* *. By Lemma 13.9, n* is given on v 2 BP -h~= ssk+h(BP__k) by n*v = . For v 2 Jn, we have [v] 1 mod J nby Lemma 19.38, so that n*v = 0 by (i). For any v, (ii) gives n*v Q(ffl)qk(eOk+hO[v]) = v mod Jn. | Proof of Thm. 23.2 (assuming Lemma 23.5) Part (a) is trivial and belongs in * *x22, as suspension induces an isomorphism Qk-1*~=Qk*and preserves the conditions (22* *.3). In (b), we must have k f(n+1) for n to exist. In effect, th* *e lifting BP*(BP__k)=Jn ! BP *in diag. (23.7) is prescribed on Qk-1*=Jn. As we have by Lemma 23.5(b) a split monomorphism with free cokernel, it is easy to extend the given lifting over BP*(BP__k)=Jn. | Resolutions Lemma 23.5 is easy to prove when k < f(n+1). In the borderline case k = f(n+1), the presence of the extra disallowed monomials in Lemma 19.39 makes it necessary to do some homological algebra. JMB, DCJ, WSW - 118 - 23 Feb 1995 x23. Unstable splittings of BP -cohomology Lemma 23.8 In the sequence of homomorphisms of BP *-modules @2 @1 ffl C2 --! C1 --! C0 --! M --! 0; (23:9) assume that: (i)Each Ci is free of finite type; (ii)We have exactness at C0 and M; (iii)@1O @2 = 0 (we do not assume exactness at C1); (iv)The sequence @2F p @1F p fflF p C2 Fp-----! C1 Fp-----! C0 Fp----! M Fp (23:10) is exact at C1 Fp (as well as at C0 Fp). Then: (a) The sequence (23.9) is split exact in the sense that: (i)C0 splits as C0 ~=M @1C1; (ii)C1 splits as C1 ~=@1C1 @2C2; (b) M is a free BP *-module; explicitly, if L0 is a free module and the modu* *le homomorphism g0: L0 ! C0 induces an isomorphism g0F p L0 Fp-----! C0 Fp--! Coker (@1 Fp) ~=M Fp; the composite ffl Og0: L0 ! M is an isomorphism. Proof We build the following commutative diagram, which includes the projectio* *ns from diag. (23.9) to diag. (23.10), L2 L1 L0 |g2 |g1 |g0 |? @2 |? @1 |? ffl C2 __________-C1 __________-C0 __________-M | | | | |? @2F p |? @1F p |? fflF p |? C2 Fp ______C1- Fp ______-C0 Fp ______M- Fp It is easy to construct g0 as in (b), by lifting a basis of Coker(@1F p) to C0.* * Similarly, we construct g1: L1 ! C1, with L1 free, that induces an isomorphism L1 Fp ~= Coker (@2 Fp) ~= Im(@1 Fp), and again g2: L2 ! C2, with L2 free, that induces L2 Fp~= Im(@2 Fp). Then by Nakayama's Lemma in the form [8, Lemma 15.2(a)], the homomorphism L0 L1 ! C0 with components g0 and @1O g1 is an isomorphism, and similarly L1L2 ~=C1. These allow us to write gi: Li Cifor i = 0; 1; 2, and the isomorphis* *ms simplify to C0 = L0 @1L1 and C1 = L1 @2L2. The latter gives @1C1 = @1L1, which shows that M ~= Coker[@1 Fp] ~= L0 is free. Moreover, because @1|L1 is monic, @2C2 = @2L2 and we have split exactness at C1. | JMB, DCJ, WSW - 119 - 23 Feb 1995 Unstable cohomology operations For our application, we take a polynomial algebra R = BP *[x1; x2; x3; : :;:y1; y2; y3; : :]: on generators of negative degree, with deg(xi) ! -1 and deg(yi) ! -1 as i ! 1, to make R a BP *-module of finite type. We consider the quotient ring M = R=J as a BP *-module, where the ideal J = (xp1- c1; xp2- c2; : :):. The elements ci ar* *e to be in some sense negligible. We construct what we hope is the beginning (or end) o* *f an R-free resolution of M, M @2 M @1 C2 = Ruiuj --! C1 = Rui--! C0 = R --! M --! 0; (23:11) i n; n (ii)Monomials of the form b(k0)ObOp(k1)O:O:b:Op(kn), where 0 k0 k1(23.13) : : :kn; (iii)All other allowable monomials bOIO[vJ]. The first type visibly lie in Jn, and we ignore them, by taking R in Lemma 23.1* *2 as the quotient polynomial ring (using *-multiplication, of course) on the second * *and third types, which serve as the xi and yi respectively. The interesting generat* *ors of J nthen have the form xpi- ci. There are five types of term in the reduction formula (19.28) for the monom* *ial 2 Opn+1 bOp(k1-1)ObOp(k2-2)O:O:b:(kn+1-n-1)O[vn+1]: JMB, DCJ, WSW - 120 - 23 Feb 1995 Index of symbols 2 n+1 (i)bOp(i1-1)ObOp(i2-2)O:O:b:Op(in+1-n-1)O[vn+1]; n (ii)F (b(k1-1)ObOp(k2-2)O:O:b:Op(kn+1-n-1)); n (iii)F (b(i1-1)ObOp(i2-2)O:O:b:Op(in+1-n-1)); (iv)Terms in A2; (v) Terms in A1 * A1; where (i1; i2; : :;:in+1) denotes any nontrivial permutation of (k1; k2; : :;:k* *n+1). Because the suffixes in (i) are out of order, (i) is an example of a type (* *i) generator in (23.13), which has been discarded. The term we want is (ii), which is xpi. W* *e can take care of (iii) and (iv) by filtering R by powers of the ideal (y1; y2; : :)* *:and working with the associated graded groups; if we have exactness in diag. (23.10) after * *filtering, we had exactness before. In effect, we may ignore the yi's. We take care of (* *v) by filtering again, this time by powers of the ideal A1+(u1; u2; : :):in (23.10). * *This done, we have effectively reduced ci to zero, when we have exactness. Thus BP*(BP__k)* *=Jn is a free BP *-module, and we have constructed a basis. For (b), we have only to show that we have a monomorphism mod V . By Lemma 18.26(a) and Lemma 18.12(c), Qk-1*=Jn is a free BP *-module with a basis consisting of the monomials of the extended canonical form (18.13) h-1(L)sh(M)h J e bL-0 bL+s(L)+:::+s b wm w ; m that lie in Qk-1*and have no factor wj with j > n, where bL = b(k0)bp(k1):b:p:(* *km,) 0 = k0 k1 : : :km , m 0, h 0, and the conditions (18.9) on M and J hold. After suspension, we find the leading term (19.31), namely F h(bOLO bOM O[vJ]),* * which by Lemma 18.12(d) is the phth power of an allowable monomial. There are two cases: Case m < n. The element bOLO bOM O[vJ] is a generator yi of type (iii) in (* *23.13), and therefore harmless. Case m n. Since jt = 0 for all t m and t > n, we must have J = 0. Also, h = 0. We must have m = n, otherwise we would have k > f(n+1). We have a generator xi of type (ii), but it is not raised to a power. By Lemma 23.12, the elements F hyi and xi (for certain i) map to part of a * *basis __k ofH *=Jn, which is sufficient. (Because k0 = 0, it is clear that these element* *s lie in __k __k PH *. In view of the suspension isomorphism Qk-1*=V ~=PH * in [23, Thm. 5.3], * *all __k we really need to know is that enough basis elements of PH * in each degree rem* *ain __k linearly independent inH *mod Jn.) | Index of symbols This index lists most symbols in roughly alphabetical order (English, then * *Greek), with brief descriptions and references. Several symbols have multiple roles. A additive comonad, Thm. 5.8. -A (subscript) additively unstable context. A0 additive comonad, (6.23). JMB, DCJ, WSW - 121 - 23 Feb 1995 Unstable cohomology operations __ A additive comonad, on modules, x9 DM dual of E*-module M, [8, __ (only). Defn. 4.8]. A augmentation ideal in algebra A. d duality homomorphism, [8, A etc. generic category. eq. (4.5)]. Aop dual category of A, [8, x6]. E generic ring spectrum. A = E*(E; o), Steenrod algebra for E,E* coefficient ring of E-(co)homology, x2. [8, xx3, 4]. Ak = E*(E_k), the operations on degreeE*(-) E-cohomology, [8, x3]. k, x2. E*(-)^ completed E-cohomology, [8, A m ideal in Q(BP )**, x18._ Defn. 4.11]. A m Hopf ring ideal inH , Defn. 19.19.E*(-) E-homology, [8, x4]. Ab , Ab* category of (graded) abelian E_n nth space of -spectrum E, [8, groups, [8, x6]. Thm. 3.17]. Alg category of E*-algebras, [8, x6]. e suspension element, Props. 12.3, ai, a(i)Hopf ring element for H(F p), 13.7. Prop. 17.9. ek unstable k-fold suspension element, ai, a(i)Hopf ring element for K(n), Prop.*13.7.p Prop. 17.16. F c = c , Frobenius operator, x10. a(i) additive element for K(n), (16.21).F I Hopf ring ideal, Defn. 19.3. ai;j coefficient in formal group law, [8,Fa(x; y)formal group law, [8, eq. (5.1* *4)]. eq. (5.14)]. F M generic filtration submodule, [8, BG classifying space of group G. Defn. 3.36]. * FAlg category of filtered E -algebras,* * [8, B(i; k) coefficient in b(x)i, Prop. 14.4. x6]. BP Brown-Peterson spectrum, [8, x2]. L F DM generic filtration submodule of BP modified BP , x1. DM, [8, Defn. 4.8]. bI etc. monomial. F M etc. corepresented functor, [8, x8]. bOIetc. O-monomial, (15.11). FMod , FMod * (graded) category of bi additive element, Prop. 14.4. filtered E*-modules, [8, x6]. bi Hopf ring element, Prop. 15.3. Fp field with p elements. b(i) accelerated bi, Defns. 14.7, 15.10.FR(X; Y ) right formal group law, b(x) formal power series, (14.2), (14.5), (15.8). _ Defn. 15.1. F sE*(X) skeleton filtration, [8, b(x) series b(x) without the 1 term, eq. (3.33)]. (15.4). f generic map or module C the field of complex numbers. homomorphism. C P n, CP 1 complex projective space. f*, f* homomorphism induced by map Coalg category of E*-coalgebras, [8, x6]. f, [8, eq. (6.3)]. c etc. generic Hopf ring element. f(n) numeric function, (1.4). ci, c(i)Hopf ring element for H(F 2), G generic group. Prop. 17.7. Gp(C) category of group objects in C, ci Hopf ring element for H(F p), [8, x7]. Prop. 17.9. gi coefficient in p-series, [8, eq. (1* *3.9)]. ci Hopf ring element for K(n), H, H(R) Eilenberg-MacLane Prop. 17.16. spectrum, [8, x2]. JMB, DCJ, WSW - 122 - 23 Feb 1995 Index of symbols __ H quotient Hopf ring, (19.12). -Q (subscript) additive unstable Ho , Ho 0homotopy category of (based) context, shifted degree. spaces, [8, x6]. QA the indecomposables of algebra A, I identity functor. [8, eq. (6.10)]. I etc. generic multi-index. QE*(X) the indecomposables of |I| length of multi-index I, x18. cohomology of space X, (3.5). In, I1 ideal in BP *, (1.1). QE*(X) the indecomposables of i1, i2 injection in coproduct, [8, x2]. homology of H-space X, Defn. 4.3. * id identity morphism or permutation. Q(E)* bigraded algebra, Defn. 6.1. Jn ideal in BP *, (1.6). Q(r) homology homomorphism induced J n ideal in Q(BP )**, Defn. 18.25. * by operation*r, (6.48). J n Hopf ring ideal, Defn. 19.34. Q*_*= Q(BP )*, abbreviation.* KC unit object in (symmetric) Q * quotient algebra*of Q*, (18.17). monoidal category C, [8, x7]. Q(ffl)counit of Q(E)*, (6.28).* K(n) Morava K-theory, [8, x2]. Q(j) unit morphism of Q(E)*,*(6.17). KU complex K-theory Bott spectrum, Q(oe) stabilization on Q(E)*,*(6.3). [8, x2, Defn. 3.30]. Q(OE) multiplication in Q(E)*, (6.16).* L infinite lens space. Q( ) comultiplication on Q(E)*, L(k) left side of main relation (Rk). (6.27). Q field of rational numbers. L(i1; : :;:in)coefficient, Defns. 18.18, q map to one-point space T . 19.21. k qk projection to Q(E)*, (6.2). M etc. generic (filtered) module or RP 1 real projective space. algebra. R(k) right side of main relation (R ). M^, cM completion of filtered M, [8, R(i ; : :;:i )coefficient, Defns. 18k.1* *8, Defn. 3.37]. 119.21. n M ideal in Q(BP )**, x18. (R k) kth main relation, (14.10), M n Hopf ring ideal, Defn. 19.13. (15.16). Mod , Mod * (graded) category of (Rk ;:::;k)nth order relation, E*-modules, [8, x6]. 1Defns.n18.18, 19.21. MU unitary Thom spectrum, [8, x2]. r generic cohomology operation. o generic basepoint, point spectrum. E*-linear functional defined by P A the primitives in coalgebra A, [8, operation r, (6.9), (10.1). eq. (6.13)]. S stable comonad, [8, Thm. 10.12]. P E*(E_k) the additive operations, -S (subscript) stable context. Prop. 2.7. S1 unit circle, as space or group. P E*(X) the primitives in homology of Sn unit n-sphere. space X, Defn. 4.13. S__ comonad S on modules, x9 (only). P E*(X) the primitives in cohomology Stab, Stab* (graded) stable homotopy of H-space X, Defn. 3.1. category, [8, x6]. P (n) modified BP spectrum, x1. Set category of sets, [8, x6]. p fixed prime number. SetZ category of graded sets, [8, x7]. p1, p2 projection from product, [8, x2].s(I), sh(I)shifted multi-index I, [p](x) p-series, [8, eq. (13.9)]. Defn. 15.12. [p]R(x) right p-series, (14.8), (15.14).T the one-point space. JMB, DCJ, WSW - 123 - 23 Feb 1995 Unstable cohomology operations T + 0-sphere, T with basepoint added. 0 = (1; 0; 0; : :):, multi-index, x18. T (n) torus group. ffl generic counit morphism. t 2 H1(R P 1), generator of i pth power map on CP 1, [8, H*(R P 1), (16.1). eq. (13.9)]. U unstable comonad, Thm. 8.8. iF pairing for (symmetric) monoidal -U (subscript) unstable context. functor F , [8, x7]. U, U(n) unitary group. j generic unit morphism. u 2 KU-2, generator. jR right unit, Defns. 6.19, 10.8. u 2 E1(L), exterior generator of generic anything. E*(L), xx16, 17. n idempotent cohomology operation u1 canonical generator of E*(S1), [8, _ on BP , Lemmas 22.1, 23.1. Defn. 3.23]. n splitting of ss, Lemmas 22.1, 23* *.1. un canonical generator of E*(Sn), [8, 2 E0(E; o), universal class, [8, x9* *]. x3]. n 2 En(E_n), universal class, [8, V generic (often forgetful) functor. Thm. 3.17]. V Verschiebung operator, x10. n collapse operation, Defn. 7.13. V ideal in Q(BP )**, x18. (-) exterior algebra. V ideal in BP*(BP__*), x19. generic action. v generic element of E*. numerical coefficient. v = jRu 2 KU2(KU; o), Thm. 16.15. L left E*-action on Q(E)**, x6. [v] 2 E0(E_*), Defn. 10.8. R right E*-action on Q(E)**, (6.21). vn Hazewinkel generator of BP *, addition or multiplication in gener* *ic K(n)*, [11]. group object, [8, x7]. W ideal in Q(BP )**, x18. inversion morphism in generic group w generic element of jRE*, Prop. 12.3. object, [8, x7]. wn = jRvn, x16. Hopf line bundle over CP n. wt (I) weight of multi-index I, x18. generic line or vector bundle. X etc. generic space. i element for H(F 2), (16.1). X+ space X with basepoint adjoined. i element for H(F p), Thm. 16.5. x generic cohomology class or module v action of v on E*-module, [8, element. eq. (7.4)]. x 2 E*(C P 1), Chern class of Hopf ss generic permutation in n. line bundle, [8, Lemma 5.4]. ss*(X) homotopy groups of space X. x() Chern class of line bundle , [8, ss: BP ! BP projection, (1.8). Defn. 5.1]. ae generic coaction. Y skeleton of lens space L, [8, x14].aeM coaction on module M. Z the ring of integers. aeX coaction on E*(X) or E*(X)^. Z =p the group of integers mod p. , k suspension isomorphism, [8, Z (p) Z localized at p. (3.13), Defn. 6.6]. zF morphism for a (symmetric) X, kX suspension of space X. monoidal functor F , [8, x7]. M, kM suspension of module M, [8, ff etc. generic index. Defn. 6.6]. fii 2 E2i(C P n), [8, Lemma 5.3]. n permutation group on {1; 2; : :;:n}. fli 2 E2i+1(U(n)), [8, Lemma 5.11]. oe: A ! S natural transformation of : X ! X x X diagonal map. comonads, Thm. 5.8. JMB, DCJ, WSW - 124 - 23 Feb 1995 References __ __ __ oe:A !S natural transformation of O canonical antiautomorphism of Hopf comonads, on modules, x9 (only). algebra. oek: E_k! E stabilization map, [8, iterated coproduct, (10.18). Defn. 9.3]. generic comultiplication. X loop space on based space X. o: U ! A natural transformation of r looped operation, Prop. 2.12. comonads, Thm. 8.8. ! zero morphism of generic group oi element for H(F p), (16.4). object, [8, x7]. OE generic multiplication. References [1] J. F. Adams, Stable Homotopy and Generalised Homology, Chicago Lectures in Math., Univ. of Chicago (1974). [2] N. A. Baas, On bordism theory of manifolds with singularities, Math. Scand. 33 (1973), 279-302. [3] M. Bendersky, The BP Hopf invariant, Amer. J. Math. 108 (1986), 1037- 1058. [4] M. Bendersky, E. B. Curtis, H. R. Miller, The unstable Adams spectral sequence for generalized homology, Topology 17 (1978), 229-248. [5] M. Bendersky, D. M. Davis, Unstable BP -homology and desuspensions, Amer. J. Math. 107 (1985), 833-852. [6] J. M. Boardman, The eightfold way to BP -operations, Canadian Math. Soc. Proc. 2 (1982), 187-226. [7] _______, Stable and unstable objects for BP -cohomology, (preprint) Johns * *Hop- kins Univ. (Mar. 1986). 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Algebra 3 (1973), 43-58. [17] R. J. Milgram, The mod 2 spherical characteristic classes, Ann. of Math. (* *2) 92 (1970), 238-261. [18] J. W. Milnor, The Steenrod algebra and its dual, Ann. of Math. (2) 67 (195* *8), 150-171. [19] J. W. Milnor, J. C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211-264. [20] J. Morava, Noetherian localisations of categories of cobordism comodules, * *Ann. of Math. (2) 121 (1985), 1-39. [21] D. G. Quillen, On the formal group laws of unoriented and complex cobordism theory, Bull. Amer. Math. Soc. 75 (1969), 1293-1298. [22] _______, Elementary proofs of some results of cobordism theory using Steen* *rod operations, Adv. in Math. 7 (1971), 29-56. [23] D. C. Ravenel, W. S. Wilson, The Hopf ring for complex cobordism, J. Pure Appl. Algebra 9 (1977), 241-280. [24] _______, The Morava K-theories of Eilenberg-MacLane spaces and the Conner- Floyd conjecture, Amer. J. Math. 102 (1980), 691-748. [25] _______, The Hopf ring for P (n), (in preparation). [26] W. S. Wilson, The -spectrum for Brown-Peterson cohomology. Part II, Amer. J. Math. 97 (1975), 101-123. [27] _______, Brown-Peterson Homology_An Introduction and Sampler, Conf. Board Math. Sci. 48 (1982). [28] _______, The Hopf ring for Morava K-theory, Publ. Res. Inst. Math. Sci., K* *yoto Univ. 20 (1984), 1025-1036. J. Michael Boardman, W. Stephen Wilson Department of Mathematics, Johns Hopkins University 3400 N. Charles St., Baltimore, MD 21218, U.S.A. E-mail: boardman@math.jhu.edu E-mail: wilson@math.jhu.edu David Copeland Johnson Department of Mathematics, University of Kentucky Lexington, KY 40506, U.S.A. E-mail: johnson@ms.uky.edu JMB, DCJ, WSW - 126 - 23 Feb 1995