Unspecified Journal Volume 00, Number 0, Pages 000-000 S ????-????(XX)0000-0 THE UNITS OF A RING SPECTRUM AND A LOGARITHMIC COHOMOLOGY OPERATION CHARLES REZK Abstract.We construct a "logarithmic" cohomology operation on Morava E- theory, which is a homomorphism defined on the multiplicative group of i* *nvertible elements in the ring E0(K) of a space K. We obtain a formula for this ma* *p in terms of the action of Hecke operators on Morava E-theory. Our formula is clos* *ely related to that for an Euler factor of the Hecke L-function of an automorphic fo* *rm. 1. Introduction Recall that if R is a commutative ring, then the set Rx R of invertible el* *ements of R is naturally an abelian group under multiplication. This construction is a fu* *nctor from commutative rings to abelian groups. In general, there is no obvious relation b* *etween the additive group of a ring R and the multiplicative group of units Rx . Howev* *er, under certain circumstances one can define a homomorphism from (a subgroup of) Rx to a suitable completion of R, e.g., the natural logarithm Qx>0! R, or the p-adic lo* *garithm (1 + pZp)x ! Zp. The "logarithmic cohomology operation" is a homotopy theoretic analogue of t* *he above, where R is a commutative S-algebra and "completion" is Bousfield localiz* *ation with respect to a Morava K-theory. The purpose of this paper is to give a formu* *la for the logarithmic operation (in certain contexts) in terms of power operations. Befor* *e giving our results we briefly explain some of the concepts involved. 1.1. Commutative S-algebra. A spectrum is a topological object which represents a generalized homology and cohomology theory. A commutative S-algebra is a spec- trum equipped with a commutative multiplication; such a spectrum gives rise to * *a coho- mology theory which has a commutative product, as well as power operations. Jus* *t as any ordinary commutative ring is an algebra over the ring Z of integers, so a c* *ommutative S-algebra is an algebra over the sphere spectrum S. The definition of commutative S-algebra is rather technical; it is the resul* *t of more than twenty years of effort, by many people. There are in fact several differen* *t models of commutative ring spectra; the commutative S-algebra in the sense of [EKMM97 ],* * or a symmetric commutative ring spectrum in the sense of [HSS00 ], or some other equ* *ivalent model. These models are equivalent, in the sense that they have equivalent homo* *topy theories; for the purpose of stating results it does not matter which model we * *use. ____________ Received by the editors June 9, 2005. 1991 Mathematics Subject Classification. 55N22; 55P43, 55S05, 55S25, 55P47, * *55P60, 55N34, 11F25. The author was partially supported by the National Science Foundation. cO0000 (copyright hold* *er) 1 2 CHARLES REZK 1.2. Power operations. A spectrum R which admits the structure of a commutative ring up to homotopy gives rise to a cohomology theory X 7! R*(X) taking values in graded commutative rings. The structure of commutative S-algebra on R is mu* *ch stronger than this; it provides not just a ring structure on homotopy groups, b* *ut also power operations, which encode "higher commutativity". Let m denote the symmet- ric group on m letters, and let B m denote its classifying space. If R is a com* *mutative S-algebra, there are natural maps Pm :R0(X) ! R0(B m x X), with the property that the composite of Pm with restriction along an inclusion * *{*}xX ! B m x X is the mth power map ff 7! ffm on R0(X). In other words, for a commutat* *ive S-algebra, the mth power map is just one of a family of maps parameterized (in * *some sense) by the space B m . An exposition of power operations and their propertie* *s can be found in [BMMS86 ]. For suitable R, one can construct natural homomorphisms of the form ae: R0(B m x X) ! D R0 R0(X), where D is an R0-algebra, and so get a cohomology operation of the form op :R0(X) Pm--!R0(B m x X) ae-!D R0 R0(X). Such functions op are what are usually called power operations. For instance, the Eilenberg-MacLane spectrum HR associated to an ordinary co* *m- mutative ring R is the spectrum which represents ordinary cohomology; when R = * *Fp, power operations are the Steenrod operations. Topological K-theory spectra, bot* *h real and complex, admit a number of power operations, including the exterior power o* *pera- tions ~k and the Adams operations _k. Other theories of interest include some e* *lliptic cohomology theories, including the spectrum of topological modular forms [Hop02* * ]; bor- dism theories, including the spectrum MU of complex bordism. 1.3. Formal groups and isogenies. Recall that a multiplicative cohomology theory is complex orientable if R*(CP 1) is the ring of functions on a one-dimensional* * com- mutative formal group. (In this paper, all formal groups are commutative and o* *ne- dimensional.) If R is a commutative S-algebra whose associated cohomology theory is complex orientable with formal group G, and op is a power operation on R which is a ring homomorphism as above, then R0(CP 1) ! D R0 R0(CP 1) is an homomorphism i*G ! G of formal groups; here i: R0 ! D is a map of rings, * *and i*G is the formal group obtained by extension of scalars along i. For example, * *complex K-theory is complex orientable, and K0(CP 1) is the ring of functions on the fo* *rmal multiplicative group ^Gm; the Adams operation _k corresponds to the k-th power * *map G^m ! ^Gm. The philosophy is that power operations (of degree m) on a complex orientabl* *e com- mutative S-algebra R should be parameterized by a suitable family of isogenies * *(of degree m) to the associated formal group. This philosophy is best understood in the c* *ase of Morava E-theories, which we now turn to. LOGARITHMIC COHOMOLOGY OPERATION 3 1.4. Power operations on Morava E-theory. Fix 1 n < 1 and a prime p. Let k be a perfect field of characteristic p, and 0 a height n formal group over k* *. Such a formal group admits a Lubin-Tate universal deformation [LT66 ], which is a form* *al group defined over a ring O Wk[[u1, . .,.un-1]]; W k is the ring of p-typical Witt vectors on k. Morava E-theory is a 2-periodic complex orientable cohomology theory with as its associated formal group; thu* *s ss*E O[u, u-1] with O in degree 0 and u in degree 2. The Hopkins-Miller theorem (see* * [GH ], [RR04 ]) states that Morava E-theories admit a canonical structure of commutati* *ve S- algebra. Power operations for Morava E-theories were constructed by Ando [And95 ]; se* *e also [AHS04 ]. These operations are parameterized by level structures on the associa* *ted Lubin- Tate universal deformation. To each finite subgroup A of the infinite torsion * *group * (Qp=Zp)n is associated a natural ring homomorphism _A :E0X ! D E0 E0X, where D is the E0-algebra representing a full Drinfel'd level structure f : * !* * i* ; the ring D was introduced into homotopy theory in [HKR00 ]. The associated isog* *eny i*G ! G has as its kernel the subgroup generated by the divisor of the image of* * f|A in i*G. An expression in the _A's which is invariant under the action of the automor* *phism group of * descends to a maprE0X ! E0X (see x1.12 below). When n = 1 and E is p-adic K-theory, then _A = _p for A Z=pr * Qp=Zp. A precise definition * *of the operations _A is given in x11.7. 1.5. Units of a commutative ring spectrum. To a commutative S-algebra R is associated a spectrum gl1(R), which is analogous to the units of a commutative * *ring (see x2). The 0-space of the spectrum gl1(R) is denoted GL1(R), and it is equivalent* * up to weak equivalence with subspace of 1 R. Write Hq(X; E) def=Eq(X) for a space X and a spectrum E. Then gl1(R) gives a generalized cohomology theory which, in degree q = 0 is given by H0(X; gl1(R)) (R0(X))x . The higher homotopy groups for the spectrum gl1(R) are given by ssqgl1(R) = eH0(Sq; gl1(R)) (1 + eR0(Sq))x (R0(Sq))x , q > 0. In particular, there is an isomorphism of groups ssqgl1(R) ssqR for q > 0, de* *fined by "1 + x 7! x". This isomorphism of homotopy groups is induced by the inclusi* *on GL1(R) ! 1 R of spaces, but not in general by a map of spectra. The main interest in the cohomology theory based on gl1(R) is that the group H1(X; gl1(R)) contains the obstruction to the R-orientability of vector bundles* * over X, according to the theory of [May77 ]. The present paper was motivated by one * *partic- ular application: the construction of a MO<8>-orientation for the topological m* *odular forms spectrum. This application will appear in joint work with Matt Ando and M* *ike Hopkins. 1.6. K(n) localization. Let F be a homology theory. Bousfield F -localization consists of a functor LF on the homotopy category of spectra, and a natural map 'X :X ! LF X for each spectrum X, such that 'X is the initial example of a map of spectra out of X which is an F*-homology isomorphism [Bou79 ]. Distinct homo* *logy 4 CHARLES REZK theories may give rise to isomorphic Bousfield localizations, in which case the* *y are called Bousfield equivalent. Given a Morava E-theory spectrum, there is a an associated "residue field" F* * , a spectrum formed by killing the sequence of generators of the ideal m = (p, u1, * *. .,.un-1) in ss0E, so that ss*F k[u, u-1]. The spectrum F is not a commutative S-algeb* *ra, although it is a ring spectrum up to homotopy. The Bousfield class of F depends only on the prime p and the height n of the* * formal group of E, and this is the same as the Bousfield class of the closely related * *Morava K-theory spectrum K(n). (The spectrum F is isomorphic to a finite direct sum of suspensions of K(n).) As is standard, we will write LK(n) for the localization * *functor associated to any of these Bousfield equivalent theories. In many respects, K(n)-localization behaves like completion with respect to * *the ideal m O. In particular, K(n)-localization allows us to define a modification of t* *he homol- ogy functor E* associated to a Morava-E theory, called the completed E-homology E^*and defined by E^*(X) def=ss*(LK(n)(X ^ E)). This functor takes values in complete E*-modules; if E*X is a finitely generate* *d E*- module, then E^*(X) (E*(X))^m. See [HS99 , x8] for a discussion of K(n)-local* *ization and completed homology. 1.7. The logarithmic cohomology operation. For each commutative S-algebra R, there is a natural family of "logarithm" maps from gl1(R) to various "completio* *ns" of R. For each prime p and n 1, there exists a natural map `n,p:gl1(R) ! LK(n)R. This map is defined using the construction due to Bousfield and Kuhn [Bou87 ], * *[Kuh89 ], which is a functor n from spaces to spectra, with the property that n 1 (X) LK(n)X for any spectrum X. If R is a commutative S-algebra, the spaces 1 gl1(* *R) and 1 R have weakly equivalent basepoint components, and so the Bousfield-Kuhn construction gives an equivalence LK(n)gl1(R) LK(n)R of spectra. The map `n,p* *is the composite 'gl1(R) gl1(R) ----! LK(n)gl1(R) LK(n)R. The construction of `n,pis described in detail in x3. The map `n,pgives a natural transformation of cohomology theories, and thus * *for any space X a group homomorphism `n,p:(R0X)x ! (LK(n)R)0(X), natural as X varies over spaces and R varies over commutative S-algebras. The purpose of this paper is the computation of this "logarithmic" map in te* *rms of power operations, when R is a reasonable K(n)-local commutative S-algebra. * *It is convenient to consider the cases n = 1 and n > 1 separately, though the proof f* *or n = 1 is really a corollary of the general case. 1.8. The logarithm for K(1)-local spectra. Let p be a prime, and let R be a K(1* *)- local commutative S-algebra, satisfying the following technical condition: the * *kernel of ss0LK(1)S ! ss0R contains the torsion subgroup of ss0LK(1)S. This condition is * *always satisfied if p > 2 (since ss0LK(1)S is torsion free for odd p), and is satisfie* *d at all primes when R is the p-completion of the periodic complex or real K-theory spectra, or* * if R is the K(1)-localization of the spectrum of topological modular forms. LOGARITHMIC COHOMOLOGY OPERATION 5 Such a ring R admits canonical cohomology operations _ and `, such that (in * *partic- ular) _ is a ring homomorphism, and for x 2 R0X, _(x) = xp + p`(x). (See x13.) When R is the p-completion of real or complex K-theory, then _ is t* *he classical pth Adams operation _p. 1.9. Theorem. Let p be any prime, and let R be a K(1)-local commutative S- algebra, satisfying the technical condition above. For a finite complex X, the * *logarithm `1,p:(R0X)x ! R0X is given by the infinite series 1X pk-1 ``(x)' k `1,p(x) = (-1)k____ ____p , k=1 k x which converges p-adically for any invertible x, and so is a well-defined expre* *ssion. Note that the series can be formally rewritten as ` ' ` ' p `1,p(x) = 1_plog_____1_____1 +=p1`(x)=xp_plogx__(x), and that this new expression is still meaningful, up to p-torsion. Since xp=_(* *x) 1 mod p for invertible x, this can be written `1,p(x) = (id-1_p_)(log(x)), which is meaningful (up to p-torsion) when x - 1 is nilpotent. If x = 1 + ffl w* *ith ffl2 = 0, then the formula of (1.9)becomes `1,p(1 + ffl) = ffl - `(ffl) = ffl - 1_p_(ffl). The proof of (1.9)is given in x13, as a corollary of (1.11)below. 1.10. The logarithm for Morava E-theory. For general n 1, we give a result for Morava E-theory, in terms of power operations. We have 1.11. Theorem. Let p be any prime, n 1, and let E be a Morava E-theory associ* *ated to a height n formal group law over a perfect field of characteristic p. Then t* *he logarithm `n,p:(E0X)x ! E0X is given by 1X pk-1 1 `n,p(x) = (-1)k-1____M(x)k = _ log(1 + p . M(x)), k=1 k p where M :E0X ! E0X is the unique cohomology operation such that Yn` Y ' (-1)jp(j-1)(j-2)=2 1 + p . M(x) = _A(x) . j=0 A *[p] |A|=pj Here *[p] * denotes the kernel of multiplication by p on *. In the case when n = 1, then 1 + p . M(x) = xp=_Z=p(x), and thus we recover * *the result for K-theory. 6 CHARLES REZK 1.12. Interpretation in terms of Hecke operators. Define formal expressions Tj,p for 0 j n and T (pk) for k 0 by X 1 X Tj,pdef=1_pj _A and T (pk) = __k _A; A *[p] p A * |A|=pj |A|=pk these give rise to well defined additive operations E0X ! p-1E0X, which we call* * Hecke operators. The T (pk) operators can be expressed as polynomials in the Tj,poper* *ators, and there is an action of the commutative ring Hn,p= Z[T1,p, . .,.Tn,p] on p-1E* *0X. (An account of this action is sketched in x14.) Formally, we can rewrite the expression of (1.11)as `n,p(x) = F1(logx) (using the fact that logtakes products to sums, and that the _A's are ring homo* *mor- phisms), where Xn FX = (-1)jpj(j-1)=2Tj,p. Xj 2 Hn,p[X]. j=0 In particular, if x = 1+ffl 2 (E0X)x with ffl2 = 0, then `n,p(1+ffl) = F1(ffl) * *(up to torsion). The formal operator inverse of FX is 1X FX-1= T (pk) . Xk 2 Hn,p[[X]], k=0 and both these expressions appear in the theory of automorphic forms. Namely, i* *f f is an eigenvector of the action of the algebraPHn,pon a space of automorphic forms* *, then Fp-1-sf = Lp(s; f)f, where Lp(s; f) = akp-ks is the p-th Euler factor of the * *Hecke L-function of f, and Fp-sf = (1=Lp(s; f))f, so that 1=Lp(s; f) is a polynomial * *of degree n in p-s. (See [Shi71, 3.21].) It is notable that this expression from the theory of L-functions arises nat* *urally from a purely topological construction; it came as a suprise to the author, and he s* *till has no good explanation for it. It is also significant for the application to elliptic* * cohomology; in the presence of an elliptic curve, these Hecke operators coincide with the clas* *sical action of Hecke operators on modular forms. 1.13. Structure of the proof. The proof of (1.11)falls naturally into two parts. First part. The logarithm is equal to a certain power operation in E-cohomo* *logy, corresponding to a particular element v 2 E^0 1 S, the completed E-homology of * *the 0-th space of the sphere spectrum (5.8). Furthermore, the element v (the "logar* *ithmic element") is completely characterized by certain algebraic properties (9.3). Th* *e proof of the first part comprises xx4-9. Second part. An element v 2 E^0 1 Sis constructed, and shown to be a logarit* *hmic element (12.3). The explicit form of v gives the formula of (1.11). The proof* * of the second part comprises xx10-12. The proof of (1.11)is completed in x12.4. 1.14. Conventions on spaces and spectra. We write Spaces for a category of "spaces" (such as topological spaces or simplicial sets), and Spaces*for based * *spaces. We write Spectrafor any suitable category of spectra. Most of this paper takes * *place in suitable homotopy categories of spaces or spectra; therefore, we will usually n* *ot specify LOGARITHMIC COHOMOLOGY OPERATION 7 a particular model. There are exceptions, namely x2 (where we refer to the mod* *el of [LMSM86 ]) and x6 (where we use the model of [BF78 ]). By commutative S-algebra, we mean any suitable category of commutitive ring * *objects in spectra (e.g., the model of [EKMM97 ], or any equivalent model, see [MM02 * *]). In x2, we will use the particular model of algebras over the linear isometries operad,* * in the sense of [LMSM86 ]. There are pairs of adjoint functors with units and counits (- )+ :Ho SpacesAE HoSpaces*:(- )- , pK :K ! (K+ )- , qK :(K- )+ ! K, where K+ K q pt, and K- K with the basepoint forgotten, 1 : HoSpaces*AE HoSpectra: 1 , jK :K ! 1 1 K, fflX : 1 1 X ! X, and 1+: HoSpacesAE HoSpectra: 1 , j+K:K ! 1 1+K, ffl+X: 1+ 1 X ! X, so that 1+K 1 (K+ ) and 1 X ( 1 X)- . Also, note that 1 (X ^ Y ) 1 X* * ^ 1 Y , while 1+(X x Y ) 1+X ^ 1+Y ; we will use this identifications often. 1.15. Conventions on localization. When n 1 and the prime p are fixed, we wri* *te L = LK(n) for the Bousfield localization functor, and write 'X :X ! LX for its * *coaug- mentation. We make the following convention for the sake of legibility: in general we * *do not specify the augmentation 'X . Thus, if f :X ! Y is a map of spectra, the nota* *tion "Lf :X ! LY " is understood to denote the composite of Lf :LX ! LY with the coaugmentation 'X :X ! LX. Note that little information is lost, since Lf :LX !* * LY is in fact the unique factorization of X ! LY through 'X up to homotopy. Likewise, if f :X ^ Y ! Z is a map, the notation Lf :X ^ LY ! LZ denotes the unique extension of X ^ Y ! LZ along the map X ^ Y ! X ^ LY (which is a K(n)-homology equivalence). If R is a K(n)-local spectrum, we write R^qX def=ssqL(X ^ R), RqX def=[X, qR]. Both functors Rq and R^qtake K(n)-homology isomorphisms to isomorphisms. 1.16. Acknowledgements. This work began as a joint project with Paul Goerss and Mike Hopkins. In particular, the first proof of (1.9)when R is p-completed K-th* *eory was proved jointly with them. The original proof of this was somewhat different* * than the one offered here; it involved an explicit analysis of the Bousfield-Kuhn fu* *nctor in the K(1)-local case. I would like to acknowledge both Paul and Mike for their assistance at vario* *us points in this project. I would also like to thank Matt Ando for many tutorials on po* *wer operations and level structures. I would also like to thank Nick Kuhn and Nora * *Ganter for various comments which improved the paper. 2.The units of a commutative ring spectrum In this section, we describe the units spectrum of a structured commutative * *ring spec- trum. The notion of the units of a commutative S-algebra has a long history, pa* *ralleling the long history of constructions of structured ring spectra. The notion seems * *to have arisen from work of Segal (as in [Seg75]) and Waldhausen. Our discussion of the* * units spectrum is based on the construction of [May77 ], as corrected in [May82 ]. T* *here is 8 CHARLES REZK another approach for constructing the units spectrum due to Woolfson [Woo79 ], * *based on Segal's theory of -spaces. 2.1. Definition of the units spectrum. Let R be a commutative ring spectrum in * *the sense of [LMSM86 ], i.e., a spectrum defined on a universe, and equipped with * *an action of the linear isometries operad L. Then R(0) = 1 R, the 0-space of the spectru* *m R, is itself an algebra over L. Let GL1(R) 1 R denote the subspace of 1 R defined* * by the pullback square GL1(R) _j__//_ 1 R | | | | fflffl| fflffl| (ss0R)x_____//ss0R We write ` :GL1(R) ! 1 R for the inclusion. Then GL1(R) is a grouplike E1 -spa* *ce, and so by infinite loop space theory is the 0-space of a (-1)-connective spectr* *um, which we denote gl1R. Note that the identity element of GL1(R) is not the usual basep* *oint of 1 R. The construction which associates R 7! gl1R defines a functor HoS-alg! HoSpe* *ctra. (It can be lifted to an honest zig-zag of functors between underlying model cat* *egories; we don't need this here.) 2.2. Example. Let R = S. Then GL1(S) G, the monoid of stable self-homotopy equivalences of the sphere. 2.3. Example. Let R = HA, the Eilenberg-Mac Lane spectrum associated to a commu- tative ring A. Then gl1HA HAx , the Eilenberg-Mac Lane spectrum on the group * *of units in A. The spectrum gl1R defines a cohomology theory on spaces; it is convenient to* * write X 7! Hq(X; gl1R) for the group represented by homotopy classes of stable maps f* *rom 1+X to qgl1R. In general, there seems to be no convenient description of thes* *e groups in terms of the cohomology theory R, except when q = 0. 2.4. Proposition. There is a natural isomorphism of groups H0(X; gl1R) (R0X)x . Furthermore, if X is a pointed and connected space, then this isomorphism ident* *ifies eH0(X; gl1R) (1 + eR0X)x . In particular, if we take X = Sk for k 1, we obtain isomorphisms of groups ssk(gl1R) eH0(Sk; gl1R) (1 + eR0Sk)x eR0(Sk) sskR; this uses the isomorphism (1 + eR0Sk)x eR0Sk defined by 1 + ffl 7! ffl, and i* *s realized by a map of spaces GL1(R) ! 1 R. A main motivation for studying the units is their role in the obstruction to* * orientations. For instance, if V ! X is a spherical fibration, the obstruction to the existen* *ce of an orientation class in the R-cohomology of the Thom space is a certain class w(V * *) 2 H1(X; gl1(R)); see [May77 , IV,x3]. LOGARITHMIC COHOMOLOGY OPERATION 9 2.5. A rational logarithm. In x3 we will construct a "logarithm" in the K(n)-lo* *cal setting, for n 1. This construction does not extend to the case of n = 0, whe* *re "K(0)- local" means "rational". For completeness, notice an ad hoc logarithm in the ra* *tional setting; it will not be used elsewhere in the paper. Let R be a commutative S-algebra, and let RQ denote the rationalization of R* *; its homotopy groups are ssnRQ (ssnR) Q. Let (gl1R)1 denote the 0-connected cove* *r of the spectrum gl1R. The group H0(X; (gl1R)1) is equal to the subgroup of (R0X)x consisting of cl* *asses ff which restrict to 1 2 R0({x}) for each point x 2 X. 2.6. Proposition. There exists a map `0: (gl1R)1 ! RQ of spectra, unique up to * *ho- motopy, which when evaluated at a space X is a map H0(X; (gl1R)1) ! H0(X; RQ) given by the formula 1X (ff - 1)k ff 7! log(ff) = (-1)k-1_______ . k=1 k Proof.The indicated formula is in fact well-defined; convergence of the series* * fol- lows because H0(X, RQ) limH0(X(k), RQ), where X(k)is the k-skeleton of a CW - approximation to X, and because ff - 1 2 H0(X, R) is nilpotent when restricted * *to any finite dimensional complex. Therefore, the indicated formula gives rise to a natural transformation of f* *unctors to abelian groups, and therefore is represented by a map 1 (gl1R)1 ! 1 RQ of H-s* *paces. Since RQ is a rational spectrum, it is straightforward to show that this map de* *loops to a map of spectra, unique up to homotopy. When X = Sk, k 1, and ff 2 eR0Sk, this gives `0(1 + ff) = ff; that is, `0 * *is the "identity" on homotopy groups in dimensions 1. 3. The Bousfield-Kuhn functor and the construction of the logarithm Fix a prime p and an integer n 1. Write L = LK(n)for localization of spect* *ra with respect to the nth Morava K-theory, as in x1.15. 3.1. The Bousfield-Kuhn functor. 3.2. Proposition (Bousfield [Bou87 ], Kuhn [Kuh89 ], Bousfield [Bou01 ]). There* * exists a functor = n: Spaces*! Spectraand a natural weak equivalence of functors o : * * O 1 ~-!L. Furthermore, (f) is a weak equivalence whenever f :X ! Y induces an isomorphism on ssn for all sufficiently large n. That is, L: Spectra! Spectrafactors through 1 :Spectra! Spaces*, up to homo- topy. 3.3. Remark. In fact, a stronger result applies. There is a functor fnand a n* *atural equivalence LfK(n)= fnO 1 , where LfK(n)is Bousfield localization with respec* *t to a vn-telescope of a type n finite complex. In fact, the functor constructed in [B* *ou01 ] is fn, in which case n = LK(n) Fn. In fact, everywhere in this paper where LK(n) and n appear, they may be rep* *laced by LfK(n)and fn, including in the key results (5.8)and (9.3). 10 CHARLES REZK 3.4. The basepoint shift. Let (K, k0) be a pointed space and X spectrum, and let f :K ! 1 X be an unbased map. Write j(f): K ! 1 X for the based map defined by k 7! (jf)(k) = ~(f(k), '(f(k0))), where ~: 1 Xx 1 X ! 1 X and ': 1 X ! 1 X are the addition and inverse maps associated to the infinite loop space structure. Colloquially, (jf)(k) = f(k) -* * f(k0). In terms of the cohomology theory represented by X, this induces the evident proje* *ction X0(K) ! eX0(K) X0K to reduced cohomology summand. 3.5. Construction of the logarithm. For a commutative S-algebra R, we write ` :GL1(R) ! 1 R for the standard inclusion, as in x2. It is not a basepoint pr* *eserving map; however, j` is. 3.6. Definition. Define ` = `n,p:gl1(R) ! LR to be the composite gl1(R) ! L gl1(R) o-!~ (GL1(R)) -(j`)--! ( 1 R) LR. Note that the map (j`) is an equivalence, since j` is a weak equivalence on* * basepoint components. This construction (for fixed n and p) gives rise for each space K and each c* *ommutative S-algebra R a map `: (R0K)x ! (LR)0K, which is natural in the variables K and R. It factors through the composite (R0K)x ! ((LR)0K)x -`!(LLR)0K (LR)0K; thus, when attempting to calculate the effect of ` on R, it will suffice to ass* *ume that R is already L-local. 4.A formula for the Bousfield-Kuhn idempotent operator The Bousfield-Kuhn functor produces an idempotent operator which turns unsta* *ble maps between infinite loop spaces into infinite loop maps; the logarithm of x3 * *is the result of applying the Bousfield-Kuhn idempotent to the inclusion ` :GL1(R) ! * *1 R. To derive a formula for the logarithm, we will first give a formula for the Bou* *sfield-Kuhn idempotent. In this section, we do this for a version of the Bousfield-Kuhn op* *erator 'ewhich acts on basepoint preserving maps. In the next section, we extend this * *to an operator ' which acts on arbitrary maps; that form will apply to the logarithm. For based spaces K, L, we write [K, L]* for the set of basepoint preserving * *maps up to homotopy. For spectra X, Y , we write {X, Y } for the the set of maps in the* * stable homotopy category. In what follows, we assume that X and Y are spectra, and that Y is an L-lo* *cal spectrum, so that 'Y :Y ! LY is a weak equivalence. 4.1. The Bousfield-Kuhn operator. We define an operator 'e:[ 1 X, 1 Y ]* ! [ 1 X, 1 Y ]* which sends f : 1 X ! 1 Y to the map obtained by applying 1 to the composite X 'X--!LX 1 X -f-! 1 Y LY Y. The operator e'has the following properties, where f, f0 2 [ 1 X, 1 Y ]*: LOGARITHMIC COHOMOLOGY OPERATION 11 (a)e'is a natural with respect to maps of spectra g :X ! X0, and to maps of* * L-local spectra h: Y ! Y 0, in the sense that 'e( 1 h O f O 1 g) = 1 h O e'f O 1 g. (b) e'is additive: e'(f + f0) = e'f + e'f0, where addition is defined using * *the infinite loop structure of 1 Y . (c)If f = 1 g, then e'f = f. (d) e'f is an infinite loop map. In particular, e'2= e', and thus the group of stable maps {X, Y } can be identi* *fied with a summand of the group of unstable maps [ 1 X, 1 Y ]*. Our approach to this operation relies on two facts. First, the fact that all* * unstable maps f : 1 X ! 1 Y between infinite loop spaces factor as f = 1 1 f O j 1 X, where 1 1 f is an infinite loop map and j 1 X is the unit of the 1 - 1 adjun* *ction. Therefore, we really only need to understand the effect of the Bousfield-Kuhn f* *unctor on the "universal example" of an unstable map out of an infinite loop space, wh* *ich is j 1 X. Second, if 1 X 1 1 K where K is a based space, then we can translate the problem of understanding the effect of the Bousfield-Kuhn functor on j 1 1* * Kto that of understanding its effect on j 1 S; this is a consequence of the fact th* *at the Bousfield-Kuhn functor can be modelled as a simpliciial functor. 4.2. The natural transformation ~. Let ~X :X ! L 1 1 X be the map defined by X 'X--!LX 1 X -j-1-X--! 1 1 1 X L 1 1 X; ~ is a natural transformation between functors on the homotopy category of spec* *tra. It is the same natural transformation considered in [Kuh ]. 4.3. Proposition. Let f : 1 X ! 1 Y be a based map, and let ef: 1 1 X ! Y denote its stable adjoint. There is a commutative diagram ~X 1 1 X _______//_L X ppp fO'X|| ppppp fflffl|Lfewwppp LY Y so that e'f = 1 (LfeO ~X ). Proof.Observe that f is equal to the composite 1 ef 1 X j-1-X--! 1 1 1 X ---! 1 Y. Apply to this diagram. 4.4. Corollary. The transformation ~ is a section of Lffl. That is, LfflX O ~X * *= 'X :X ! LX. Proof.Set f = 1 'X in (4.3). 12 CHARLES REZK 4.5. Formula for e'. Let K denote an arbitrary based space. A map f 2 [ 1 X, 1* * Y ]* gives rise to a cohomology operation f*: eX0(K) ! eY(0K) byf*(ff) = f O ff. We * *remind the reader that f* is not necessarily a homomorphism of abelian groups, althoug* *h it is the case that f*(0) = 0. Our goal is to calculate the cohomology operation indu* *ced by 'ef in terms of that induced by f. The formula we give is in the form of a comp* *osite Xe0(K) Pe-!eX0( 1 S ^ K) f*-!eY(0 1 S ^ K) Qe-!eY(0K), where ePand eQare certain functions which we define now. Given ff 2 eX0(K), represented by a map a: 1 K ! X, write ePff 2 eX0( 1 S ^* * K) for the class represented by 1 ( 1 S ^ K) 1 1 S ^ 1 K fflS^a---!S ^ X X. If Y is an L-local spectrum, then we define a natural map eQ:eY(0 1 S ^ K) ! eY(0K) as follows: represent a class ff 2 eY(0 1 S ^ K) by a map a: 1 ( 1 S ^ K) ! Y * *, and let eQ(ff) be the class represented by 1 K S ^ 1 K ~S^1---!L 1 1 S ^ 1 K ! L 1 ( 1 S ^ K) La--!LY Y. The main result of this section is. 4.6. Proposition. We have an identity ('ef)* = eQO f* O ePof operations Xe0(K) ! Ye0(K). 4.7. The natural transformation ffi. For a based space K and a spectrum Z, let ffi = ffiZ,K: 1 Z ^ K ! 1 (Z ^ 1 K) be the map adjoint to 1 ( 1 Z ^ K) 1 1 Z ^ 1 K fflZ^1-1-K----!Z ^ 1 K. 4.8. Lemma. Given ff 2 eX0(K) represented by a stable map a: 1 K ! X, the elem* *ent Pe(ff) is represented by the stable map 1 a O ffiS,K: 1 S ^ K ! 1 X. Proof.The composite 1 a 1 S ^ K ffi-! 1 (S ^ 1 K) ---! 1 X is adjoint to 1 ( 1 S ^ K) 1 1 S ^ 1 K fflS^1-1-K----!S ^ 1 K a-!X, which represents eP(ff). 4.9. The proof of (4.6). Proof of (4.6).Let ff 2 Xe0(K), represented by a spectrum map a: 1 K ! X. Let b: 1 S ^ K ! 1 X be the map of based spaces which represents ePff 2 eX0( 1 S * *^ K); it is adjoit to the stable map fflS^a: 1 1 S^ 1 K ! S^X X. Let ef: 1 1 X !* * Y LOGARITHMIC COHOMOLOGY OPERATION 13 be the map adjoint to f : 1 X ! 1 Y . We will refer to the following diagram. ~S^' 1_K____L010(_1_S_^_K)______________________________* *_____________________________________________________________________________ _________________________________________________________* *_____________________________________________________________________________* *________________________________________________________ _______________________|L|1_ffi_____________________________* *_____________________________________________________________________________* *_________________________ ________________________fflffl|______________________________* *__________________ S ^ 1 K __________//_L 1 1 (S ^ 1 K) _L_1_b__________________* *___________________ ~S^ 1 K __________________________* *___________________ | | 1 1 ____________________________* *_________________ a| |L ______________________________* *_______________a fflffl| fflftt_______________________________* *_________________________________fl| X _______~X________//_Lh1 1 X hhhh fO'X|| hhhhhhhhh fflffl|tthhhLfeh LY Y We are going to prove that this diagram commutes. Given this, the proposition i* *s derived as follows. Note that the composite f O'X Oa is precisely the map representing* * the class ('ef)*(ff). We claim that the long composite S ^ 1 K ! Y around the outer edge * *of the diagram is a map representing (QeOf*OPe)(ff). The composite efO 1 b: 1 ( 1 S ^* *K) ! Y is adjoint to f O b, which represents f*(Peff), and so it is clear from the d* *efinition of eQ that the long composite in fact the desired class. To show that the diagram commutes, we need to check the commutativity of eac* *h of four subdiagrams. The central square commutes because ~ defines a natural trans* *for- mation 1 ! L 1 1 . The bottom triangle commutes by (4.3). The commutativity of the right-hand triangle of the diagram follows from (4.* *8). We defer the proof of the commutativity of the upper-left triangle(6.1)to x6. 5.An unbased Bousfield-Kuhn operator In this section, we define a Bousfield-Kuhn operator on unbased maps, using * *the stable basepoint splitting, and derive a formula for it similar to (4.6); from this we* * will produce the formuala (9.3)for the logarithm. For unbased spaces K and L, we write [K, L] for the set of unbased homotopy * *classes of (not necessarily basepoint preserving) maps. 5.1. The stable basepoint splitting. For a spectrum Y and a based space K, we consider functions [K, 1 Y ]* i-![K, 1 Y ] j-![K, 1 Y ]*. The function i is the evident inclusion, while j is the basepoint shift operato* *r of x3.4. These operations give rise to the direct sum decomposition [K, 1 Y ] [K, 1 Y ]* ss0Y. In particular, ji = id. Note this direct sum decomposition arises from the stab* *le splitting 1 K+ 1 K _ S, which is realized by maps _1+zK//_ _1_qK//_ 1+(pt)oo___ 1+K oo___ 1 K , 1+ssK flK where zK :pt ! K, ssK :K ! pt, and qK :K+ ! K are the evident maps of spaces, and flK = 1 1+K is the stable map with 1 qK O flK = 1 1 K and 1+ssK O flK = 0* *. With this notation, the operators i and j are induced by qK and flK respectively. We* * record the relation between j and fl in the following 14 CHARLES REZK 5.2. Lemma. Let K be a based space, and f :K ! 1 Y an unbased map. Then the fe based map jf is adjoint to 1 K flK--! 1+K -! Y , where efis adjoint to f. In p* *articular, ffl+YO fl 1 Y = fflY , since ffl+Yis adjoint to the identity map of 1 Y , whic* *h preserves the basepoint. Parameterized Version: Let K be a based space and L be an unbased space, and* * let f :K x L ! 1 Y be an unbased map, with ef: 1+K ^ 1+L ! Y its adjoint. Then feO (flK ^ 1 1+L) is adjoint to f - f O (zK ssK x 1L) 2 [K x L, 1 Y ]. In particular, if f is such that f O (zK x 1): L ! 1 Y is homotopic to the * *null map, then efO (flK ^ 1 1+L) is adjoint to jf. 5.3. The natural transformation ~+ . Let ~+X:X ! L 1+ 1 X be the map defined by X ~X--!L 1 1 X Lfl-1-X---!L 1+ 1 X. 5.4. The operator '. Now suppose that X and Y are spectra, and that Y is L-loca* *l. We define an operator ': [ 1 X, 1 Y ] ! [ 1 X, 1 Y ] on the set of unbased ma* *ps, by 'f def=(i O e'O j)f. The operator ' is idempotent, and has as its image the* * set of infinite-loop maps; it coincides with e'on the summand of basepoint-preserving * *maps. We are going to prove a formula for ' analogous to the one proved for e'. For an unbased space K we define natural functions P :X0(K) ! X0( 1 S x K) and Q: Y 0( 1 S x K) ! Y 0(K), as follows. Given ff 2 X0(K), represented by a map a: 1+K ! X, write P ff 2 X0( 1 S x K) for the class represented by ffl+S^a 1+( 1 S x K) 1+ 1 S ^ 1+K ---! S ^ X X. If Y is an L-local spectrum, then we define a natural map Q: Y 0( 1 S x K) ! Y 0(K) as follows: represent a class ff 2 Y 0( 1 S x K) by a map a: 1+( 1 S x K) ! Y * *, and let eQ(ff) be the class represented by ~+S^1 1 1 1 1 1 La 1+K S ^ 1+K ---! L + S ^ + K ! L + ( S x K) --!LY Y. 5.5. Proposition. We have an identity ('f)* = Q O f* O P of operations X0(K) ! Y 0(K). Proof.Let K be an unbased space, and let ff 2 X0(K) eX0(K+ ). We will show th* *at Q(f*(P ff)) = eQ((jf)*(Pe(ff))) = ('ejf)*(ff); the result follows when we note * *that there is an identity of cohomology operations (ig)* = g* when g 2 [ 1 X, 1 Y ]*, so * *that ('ejf)* = (i'ejf)* = ('f)*. We have a direct sum decomposition X0( 1 S x K) X0(K) eX0( 1 S ^ K+ ) X0(K) X0( 1 S x K, ptx K), which is produced by smashing K+ with the maps (pt)+ -z!( 1 S)+ -q! 1 S and ss :( 1 S)+ ! (pt)+ . LOGARITHMIC COHOMOLOGY OPERATION 15 The remaining projection of the splitting comes from smashing 1+K with the sta* *ble map fl = fl 1 S: 1 1 S ! 1 ( 1 S)+ . We claim that with respect to this split* *ting, the maps in Xe0(K+ ) P-!eX0(K+ ) eX0( 1 S ^ K+ ) f*-!eY(0K+ ) eY(0 1 S ^ K+ ) Q-!eY(0* *K+ ) satisfy P (ff) = (0, eP(ff)), f*(0, fi) = (f*(0), (jf)*(fi)), Q(ff, fi) = eQ* *(fi). The proposition will follow immediately. The formula for P follows, using a standard adjunction argument, from the fa* *cts that ffl+SO fl = fflS, by (5.2), and ffl+SO 1 z = 0. The formula for Q follows from the facts that L 1 q O ~+S= L 1 q O Lfl O ~S * *= ~S and L 1 ss O ~+S= L 1 ss O Lfl O ~S = 0. To prove the formula f*, let b: 1 S x K ! 1 X be the map representing fi 2 Xe0( 1 S ^ K+ ) X0( 1 S x K, ptx K); in particular, b O (z x 1): ptx K ! 1 X is the null map, and b can be taken to be basepoint preserving. Set f*(0, fi) =* * (x, y). It is clear that x is represented by f O b O (z x 1) = f O 0: ptx K ! 1 Y , an* *d so x = f*(0) 2 X0(K) Xe0(K+ ). We have that y is represented by the stable map feO 1+b O fl ^ 1, which by the parameterized version of (5.2)is adjoint to j(f* * O b). Since b preserves basepoints, j(f O b) = (jf) O b, which represents (jf)*(fi). 5.6. Application to ring spectra. We now introduce the additional hypothesis th* *at Y is a commutative ring spectrum; here we only need that Y be a commutative rin* *g up to homotopy, not a structured ring spectrum. We write Y0^(X) def=ss0L(X ^ Y ) f* *or the completed homology of a spectrum X with respect to Y . If K is a based space, w* *e write Y0^(K) for Y0^( 1+K). Let v 2 Y0^( 1 S) denote the Hurewicz image of ~+S2 ss0L 1 S; i.e., the homo* *logy class represented by ~+S^1Y 1 1 1 1 S 1S^~---!S ^ Y ----! L + S ^ Y ! L( + S ^ Y ), where ~: S ! Y represents the unit of the ring spectrum. Since Y is a ring spectrum, there is a slant product operation ff oe 7! ff=oe :Y 0(X1 ^ X2) Y0^(X2) ! Y 0(X1) defined by X1 1^oe--!X1 ^ L(X2 ^ Y ) ! L(X1 ^ X2 ^ Y ) L(ff^1)-----!L(Y ^ Y ) ! LY * * Y. 5.7. Proposition. For an L-local ring spectrum Y , and ff 2 Y 0(K), we have Q(ff) = ff=v. In particular, for ff 2 X0(K), we have ('f)*(ff) = f*(P ff)=v. Proof.The second statement is immediate from the first. The first statement is * *straight- forward from the definitions. Now suppose that Y = R is an K(n)-local commutative S-algebra, and that X = gl1(R). Let ` : 1 gl1(R) ! 1 R be the standard inclusion; it corresponds to t* *he "cohomology operation" R0(K)x ! R0(K) which is the inclusion of the units into * *the ring. The logarithm is ` = ('`)*, and thus we have proved 16 CHARLES REZK 5.8. Theorem. Let v 2 R^0 1+ 1 S denote the Hurewicz image of ~+S2 ss0L 1+ 1 S. Then `(ff) = `*(P ff)=v. We will often abuse notation by taking `* to be understood, so that we write* * `(ff) = (P ff)=v. 6.Simplicial functors For a spectrum X and a space K, recall (from x4.7) that ffiX,K : 1 X ^K+ ! * *1 (X ^ 1+K) is the map adjoint to fflX ^ id: 1 1 X ^ 1+K ! X ^ 1+K, where ffl is t* *he counit of the adjunction ( 1 , 1 ). The purpose of this section is to prove th* *e following. 6.1. Proposition. For every spectrum X and space K, the diagram LX ^ 1+K ___~X^id___//_L( 1 1 X) ^ 1+K | | 1 | |L( ffiX,K) fflffl| fflffl| L(X ^ 1+K) __~_______//_L 1 1 (X ^ 1+K) X^ 1+K commutes in the homotopy category of spectra; the vertical maps involve the loc* *alization augmentation, as described in x1.15. The idea of the proof is easy to describe: the functors in question come fro* *m simplicial functors on an underlying simplicial model category of spectra, and the natural* * transfor- mation ~ on the homotopy category comes from a natural transformation between t* *hese simplicial functors; the vertical maps in the square are certain natural transf* *ormations associated a simplicial functor. To carry out the proof, we will choose explici* *t models for these functors. 6.2. Simplicial functors. Recall that if C is a simplicial model category, then* * for any X, Y 2 C and K 2 sSetthere are objects mapC(X, Y ) 2 sSet, X K 2 C, Y K 2 C, which come with isomorphisms map C(X K, Y ) mapC (X, Y K) mapsSet(K, mapC(X, Y )). We will usually write TK X def=X K and MK Y def=Y K; these are functors TK , MK* * :C ! C. It is standard that these give rise to derived functors LTK , RMK :Ho C ! Ho* * C, and that furthermore LTK and RMK are adjoint to each other. See [GJ99 , IX] for* * a full treatment. A simplicial functor F :C ! D is a functor which is enriched over simplicial* * sets, in the sense that for every pair of objects there is an induced map map C(X, Y * *) ! map D(F X, F Y ) which on 0-simplicies coincides with F . For any such functor,* * there is a natural transformation aeFX,K:F (X) K ! F (X K), which is adjoint to the map K ! mapC (X pt, X K) ! mapD (F (X pt), F (X K)). LOGARITHMIC COHOMOLOGY OPERATION 17 A simplicial natural transformation is a natural transformation ff: F ! G su* *ch that the two evident maps map C(X, Y ) ! map D(F X, GY ) sending f to ffY O F f* * and Gf O ffX coincide. Such transformations give rise to commutative square aeFX,K F X K ____//_F (X K) ffX 1K|| |ffX|K fflffl| fflffl| GX K aeG_//_G(X K) X,K 6.3. Bousfield-Friedlander spectra. Let BFSpectradenote the Bousfield-Friedland* *er model category of spectra [BF78 ], [HSS00 ], [GJ99 , X.4]. This category has as* * objects X = {Xn 2 sSet*, fXn:S1 ^ Xn ! Xn+1}n 0, where where S1 = [1]=@ [1], with morphisms g :X ! Y being sequences {gn: Xn ! Yn}n 0 commuting with the structure maps f. It simplicial model category. We note the following functors; all of them are simplicial functors. (1) 1 :sSet*! BFSpectra, defined on objects by ( 1 K)n = (S1)^n^ K, with the evident structure maps. (2) 1 :BFSpectra, defined on objects by 1 X = X0. (3)F :BFSpectra! BFSpectra, defined on objects by (FX)n = colimm m Sing|Xm+n |, where Sing:sSet*AE Top* :|- | are geometric realization and singular com* *plex, and K = map*(S1, K). (The functors | |, Sing, and are simplicial func* *tors, and thus so is F.) (4) : sSet*! BFSpectra, the functor defined in [Bou01 ]; there it is constr* *ucted as a simplicial functor. (5)TK :sSet*! sSet, where K is an unpointed simplicial set, defined on obje* *cts by TK (L) = L ^ K+ . (6)TK :BFSpectra ! BFSpectra, where K is an unpointed simplicial set, defin* *ed on objects by TK (X)n = Xn ^ K+ . A spectrum X 2 BFSpectrais cofibrant if and only if each structure map fXn:X* *n ! S1 ^ Xn+1 is an inclusion of simplicial sets. For all K 2 sSet*, the spectrum * *1 K is cofibrant. A map f :X ! Y 2 BFSpectrabetween cofibrant objects is a weak equivalence if* * and only Ff is a weak equivalence of simplicial sets in each degree. The functor F * *comes equipped with a natural transformation 1 ! F, which on degree n of a spectrum X* * is the map adjoint to the evident map |Xn| ! colimm 1 |Xm+n |. The functors 1 , , and both versions of TK are homotopy functors: they pre* *serve weak equivalences. The functor F is a homotopy functor on the full subcategory* * of cofibrant objects. The functor 1 is not a homotopy functors; however, the com* *posite 1 F is a homotopy functor on the full subcategory of cofibrant spectra. There is a simplicial natural transformation j :id! 1 F 1 of functors sSet* **! sSet*; it is the evident map X ! colimm m Sing|Sm ^ X|. Define ~: 1 F = id 1 F ! 1 F 1 1 F to be the natural simplicial transformation j 1 F. 18 CHARLES REZK By the above remarks, both functors are homotopy functors on cofibrant spect* *ra. Both functors are simplicial functors. Therefore, the square TK 1 FX TK~X//_TK 1 F 1 1 FX ae|| ae0|| fflffl| fflffl| 1 FTK X ~T__//_ 1 F 1 1 FTK X KX commutes for all X. Applied to cofibrant X, it gives rise to a commutative diag* *ram in the homotopy category of spectra. It is clear that the horizontal arrows are p* *recisely those of (6.1). It remains to identify the arrows labelled ae and ae0. They can* * be factored 1 F 0 1 1 1 F 1 1 1 1 F 1 1 ae = ae and ae = ( F)( )ae O( F)ae ( F)Oae ( )( F* *). 6.4. Proposition. (a)The natural transformation ae 1 F: TK 1 F ! 1 FTK , on the homotopy category of spectra, induces the map LX ^ 1+K ! L(X ^ 1+K), which is the unique map up to homotopy compatible with the augmentations 'X ^ id:X ^ 1+K ! LX ^ 1+K and1': X ^ 1+K ! L(X ^ 1+K). (b) The natural transformation ae :TK 1 ! 1 TK , on the homotopy catego* *ry of spectra, induces the canonical equivalence 1 X ^ 1+K ~-! 1 (X ^ 1+K). (c)The natural transformation ae 1 F:TK 1 F ! 1 FTK , on the homotopy cat* *e- gory of spectra, induces the map ffiX,K : 1 X ^ 1+K ! 1 (X ^ 1+K) adjoint to fflX ^ id: 1 1 X ^ 1+K ! X ^ 1+K. Proof. (a) The composite 1 F is a model for the Bousfield localization func* *tor on spectra.1 (b) The map ae 1 is easily seen to be an isomorphism on objects in BFSpectra. (c)The map ae F is adjoint to the arrow ff in the square TK 1 1 F_TKfflF//_TK F ae 1 1~F|| ~ aeF|| fflffl| fflffl| 1 TK 1 F__ff_//FTK which is seen to commute by appeal to the definitions. The vertical arro* *w on the left is an isomorphism, and the vertical arrow on the right is a weak eq* *uivalence when applied to cofibrant spectra. This gives the desired result. LOGARITHMIC COHOMOLOGY OPERATION 19 7.Power operations and the units spectrum Power operations on a representable functor on the homotopy category of spac* *es are defined via the action of some E1 -operad on the representing space. We need to* * consider two flavors of E1 -spaces, and their associated power operations: the "additive* *" E1 - structure on the 0-space of any spectrum (power operations here amount to the t* *heory of transfers), and the "multiplicative" E1 -structure on the 0-space of any com* *mutative S-algebra. The purpose of this section is to show that both flavors are united * *in the E1 - space GL1(R), which can be viewed as "multiplicative" by the inclusion GL1(R) * * 1 R, or as "additive" by GL1(R) 1 gl1(R). There are some standard references for some of this material, notably [BMMS8* *6 ], who frame their power operations in terms of extended power of spectra. (Their form* *ulation works naturaly not just for commutative S-algebras (a.k.a., E1 -ring spectra), * *but with the weaker notion of an H1 -ring spectra, and much of what we say in this secti* *on applies in that context, too.) 7.1. Power operations associated to E1 -spaces. Let Z be an E1 -space, as in [May72 ]. That is, for a suitable E1 -operad E, there are structure maps Em x m* * Zm ! Z, satisfying certain axioms. Each space Em of the operad is a free contractible * *m -space; we will not refer explicitly to the operad E, but rather to a structure map E m x * *m Zm ! Z. Given a map f :K ! Z, we let bPm(f): E m x m Km ! Z denote the composite E m x m Km ! E m x m Zm ! Z. This gives rise to functions bPm and Pm on homotopy classes of maps Pbm [K, Z]____//[E m x m Km , Z] PPP PPPP | *m Pm PPPP''Pfflffl|| [B m x K, Z] Here m :B m x K E m x m K ! E m x m Km denotes the diagonal map, and Pm def= *mO bPm. The set [K, Z] is a commutative monoid via the E1 -structure of Z. With res* *pect to this, bPmand Pm are homomorphisms of commutative monoids. Furthemore, these operations are natural with respect to maps of spaces in the K-variable, and ma* *ps of E1 -spaces in the Z-variable. 7.2. Power operations associated to a commutative S-algebra. Let R be a com- mutative S-algebra, and Z = 1 R, viewed as an E1 -space using the "multiplicat* *ive" structure (which, we emphasize, is not the same as the "additive" structure com* *ing from Z being an infinite loop space). Then the constructions of the previous section* * specialize to natural power maps bPmand Pm : bPm 0 m R0K _____//OOOR (E m x m K ) OOOO |* Pm OOOO''O fmflffl|| R0(B m x K) These operations are multiplicative: bPm(fffi) = Pbm(ff)Pbm(fi) and Pbm(1) = 1* *, and similarly for Pm . Hence if ff 2 R0K is multiplicatively invertible, then Pbm(* *ff) and 20 CHARLES REZK Pm (ff) are also invertible, and so these operations restrict to functions (R0K* *)x ! (R0E m x m Kn)x ! (R0B m x K)x . These operations are not in general additive. Instead, we have 7.3. Proposition. We have X bPm(ff + fi) = eTij[Pi(ff)Pj(ff)] i+j=m and X Pm (ff + fi) = Tij[Pi(ff)Pj(fi)], i+j=m where eTijand Tijdenote the transfers associated to the covering maps E m x ix j Km ! E m x m Km and B ix B j ! B m respectively. Proof.This is [BMMS86 , Lemma 2.1]. 7.4. Power operations associated to an infinite loop space, and transfers. Let Y be a spectrum, and let Z = 1 Y , viewed as an E1 -space using the "additive" structure. Then we again have natural power maps, which we denote bPm+and Pm+: bP+m Y 0K_____//OY 0(E m x m Km ) OOO | OOOO |*m P+m OOO''O fflffl| Y 0(B m x K) These operations are additive: Pb+m(ff + fi) = bPm+(ff) + bPm+(fi) and bPm+(0) * *= 0, and so likewise for Pm+. We recall the well-known relation between such "power operations" and the th* *eory of transfers. 7.5. Proposition. The map Pb+m(j+K): E m x m Km ! 1 1+K is adjoint to the composite 1+(proj)1 1+(E m x m Km ) transfer-----! 1+(E m x m (m_x Km )) ------! + K, where m_ is a fixed set of size m permuted by m . In particular, Pb+m(j+*): B* * m ! 1 1+S is adjoint to the composite 1+(proj)1 1+B m -transfer----! 1+B m-1 ------! + (*) S, (take B -1 = ?). 7.6. The natural transformation ffi+ . Let Y be a spectrum and K an unbased spa* *ce. Let ffi+Y,K: 1 Y x K ! 1 (Y ^ 1+K) denote the map adjoint to ffl+Y^ id: 1+ 1 Y ^ 1+K ! Y ^ 1+K. This is a variant of the map ffi defined in x4.7. LOGARITHMIC COHOMOLOGY OPERATION 21 7.7. A total power operation for infinite loop spaces. Let K be a space and Y a spectrum. We define operations bP:Y 0K ! Y 0 1 1+K, and P :Y 0K ! Y 0( 1 1+Sx K), as follows. Given ff: K ! 1 Y , let bP(ff) def= 1 eff, where eff: 1+K ! Y* * is the adjoint to ff. Let P (ff) def=bP(ff) O ffi+S,K. Since 1 Y is an infinite loop space, it admits "additive" power operations * *of the type described in x7.4. The following lemma says that the two kinds of operations co* *incide, via the standard maps eimdef=bPm+(j+K): E m x m Km ! 1 1+K and im def=Pm+(j+*): B m ! 1 S. 7.8. Lemma. Let ff: K ! 1 Y be a map. The diagram Pb+m(ff) B m x K _____//E m x m Km _____//818Yp ppp im xidK|| eim|| ppppp fflffl| 1 fflffl|1bP(ff)pp 1 S x K __ffi+_// + K S,K commutes up to homotopy. Proof.The adjoint pair ( 1+, 1 ) gives a commutative diagram K ____ff__//919Ys sss j+K|| ssss1 fflffl|ss "ff 1 1+K The two maps ff and j+K from K give rise to maps Pb+m(ff) and Pb+m(j+K) out of E m x m Km , and the resulting triangle which appears in the statement of the p* *ropo- sition commutes, by the naturality of bPm+with respect to E1 -maps. The commutativity of the left-hand square comes from the fact that the map Pb+m(j+X): E m x m Xm ! 1 1+X is a natural transformation of functors, and ca* *n in fact be realized as a toplological natural transformation, by taking E m to be * *the m-th space of the little cubes operad. The example we are interested in is Y = gl1(R). In this case, "classical" * *power operations are just the standard power operations in R-cohomology, restricted t* *o units. 7.9. The universal example of the "infinite-loop" total power operation. The "infinite-loop" power operation bP:[K, 1 Y ] ! [ 1 1+K, 1 Y ] of the previou* *s sec- tion is completely determined by its restriction to a "canonical" class, namely* * the map j+K:K ! 1 1+K. This leads to the following. 7.10. Proposition. Given a map ff: K ! 1 Y , we have that ffl+ 1* *+K (a)bP(ff): 1 1+K ! 1 Y is adjoint to the composite 1+ 1 1+K ----! 1+K eff-!Y . (b) P (ff): 1 S x K ! 1 Y is adjoint to the composite ffl+S^id 1+K eff 1+ 1 S ^ 1+K -------! S ^ 1+K 1+K -! Y. 22 CHARLES REZK Here effdenotes the adjoint to ff. In particular, the operator P coincides with* * the one called P in x5.4. Proof.Consider the diagram 1+ffi+S,K 1 1 1 1+ 1 eff 1 1 1+ 1 S ^ 1+K __________//_ + + K_________//_ + Y UUUUU | + UUUUU |ffl 1+K ffl+Y| ffl+S^idU1+KUU**UUUUUfflffl| fflffl|| 1+K _______eff______//_Y It is clear that the square commutes (up to homotopy), because ffl+ is a natura* *l trans- formation, while the triangle commutes because ffi+S,Kis adjoint to ffl+S^ 1 1+* *K. bP(ff) = 1 effis adjoint to ffl+YO 1+ 1 eff, which equals effO ffl+ 1 by * *commutativity + K of the diagram. Thus P (ff) = bP(ff) O ffi+S,Kis adjoint to effO ffl+ 1+KO 1+f* *fi+S,K, which the diagram shows is equal to effO ffl+S^ id 1+K. 7.11. Power operations associated to homology classes. We now assume that R is a K(n)-local commutative S-algebra. From now on, we will be interested only* * in two related kinds of power operations: the operations Pbnand Pn associated to * *the mulplicative struture on 1 R (x7.2), and the total operations bPand P associat* *ed to the infinite loop space GL1(R) (x7.4). Given u 2 R^0B m , define opu:R0K ! R0K by opu(x) def=Pm (x)=u, using the sl* *ant product map -=u: R0(B m x K) ! R0(K). Similarly, for a class u 2 R^0 1 S we define opu:R0(K)x ! R0(K) by opu(ff) d* *ef= P (ff)=u, where we implicitly regard P (ff) as an element of R0( 1 S x K) by th* *e usual inclusion R0( 1 S xK)x R0( 1 S xK). Such operations defined using 1 S coinci* *de with those defined using B m , via the map in = Pbm(j+*) used in (7.8). That i* *s, if u 2 R0B n, and u0= im*(u) 2 R0 1 S, then opu= opu0. With these definitions, (5.8)becomes `(ff) = opv(ff). More generally, if D is a flat extension of R^0(pt), then we obtain operat* *ions opu: R0K ! D R R0K parameterized by elements u 2 D R R^0B m (respectively, opu: (R0K)x ! D R R0K parameterized by elements u 2 D R R^0 1 S). 8.The structure of the spaces parameterizing power operations We summarize here some structure which is relevant to power operations, and * *which is used in xx9-12. Most of what we say in this section is well-known; the summa* *ry is provided mainly to fix notation. For another summary of this sort of structure* *, in a similar context, the reader is directed to [ST97 ]. Let R be a K(n)-local commutative S-algebra, and D a flat extention of R^0(p* *t). Set h(X) def=D R^0(pt)R^0(X); this is a multiplicative, homological functor. Set h* *0(X) def= D R^0(pt)R0X. We write h for h(pt). The functor h admits a K"unneth map x: h(X)* * h h(Y ) ! h(X x Y ). Let the symbol M denote either B or 1 S. We give below a combined list of structure maps (numbered items) involving t* *he group h(M), and a list of properties they satisfy (lettered items). After the l* *ist, we will give the definitions of each of the structure maps, and sketch the proofs of ea* *ch of the LOGARITHMIC COHOMOLOGY OPERATION 23 properties. We will also show that in every case in which the structure maps de* *fined for both M = B and M = 1 S, they commute with the standard map B ! 1 S. It may be helpful to point out here that structures (1) through (7) make h( 1 S) i* *nto a Hopf ring. 8.1. Summary list of structure maps for B and 1 S. The letters u, v denote elements of h(M), while ff denotes an element of R0(K), for an arbitrary space * *K. (1)h(M) is a module over h = h(pt), such that (a)opu+v(ff) = opu(ff) + opv(ff), and opcu(ff) = c . opu(ff), for c 2 h* *(pt). (2)There is a distinguished class 1 2 h(M), and (3)a product u v 7! u . v :h(M) h h(M) ! h(M), such that (b) "." is associative and commutative, with unit 1, and (c)op1(ff) = 1 and opu.v(ff) = opu(ff)opv(ff). Structures (1), (2), and (3) can be summarized: h(M) is a commutative h-algebra* *, and for ff 2 h0(K), the map u 7! opu(ff): h0(M) ! h0(K) is a map of h-algebras. (4)There is a distinguished class [1] 2 h(M), and (5)a product u v 7! u O v :h(M) h h(M) ! h(M), such that (d) "O" is associative and commutative, with unit [1], and (e)op[1]= id. (6)There is an h-module map ss :h(M) ! h, such that (g)ss(1) = 1 and ss(u . v) = ss(u)ss(v), and (h) ss(u) = opu(1). (7)There is an h-module map x :h(M) ! h(M x M). Say that an element u 2 h(M) is grouplike if (i) ss(u) = 1, and (ii) * *x (u) = u x u. We write h(M)grpfor the set of grouplike elements. (i)If u 2 h(M) is grouplike, then opu is multiplicative: opu(1) = 1 and opu(fffi) = opu(ff)opu(fi). (j)If u is grouplike, then we have the identities u O 1 = 1 and u O (v * *. w) = (u O v) . (u O w). That is, x 7! u O x is a ring homomorphism if u i* *s grouplike. (k) 1 2 h(M)grp, and u, v 2 h(M)grpimplies u.v 2 h(M)grpand uOv 2 h(M)gr* *p. Thus, the set of grouplike elements h(M)grpforms a commutative semi-* *ring, in which "addition" is given by u . v, "multiplication" is given by * *u O v, the "zero" element is 1, and the "one" element is [1]. Furthermore, the * *grouplike elements of h( 1 S) admit "additive" inverses, so that h( 1 S)grpis * *not just a semi-ring but a ring. (8)There is an h-module map o :h(M) ! h, such that (l)o([1]) = 1 and o(u O v) = o(u)o(v), and (m) o(1) = 0 and o(u . v) = o(u)ss(v) + ss(u)o(v). In particular, o defines a semi-ring homomorphism h(M)grp! h. The following structures are only defined for M = B . (9)There is an h-module map i :h(B ) ! h(pt), such that (n) i(1) = 1 and i(u . v) = i(u)i(v), (o)i([1]) = 0 and i(u O v) = i(u)ss(v) + ss(u)i(v), and (p) opu(0) = i(u). (10)There is an h-module map + :h(B ) ! h(B x B ). Say that u 2 h(B ) is primitive if (i) i(u) = 0, and (ii) + (u) = ux1* *+1xu. (q) If u 2 h(B ) is primitive, then opuis additive: opu(0) = 0 and opu(f* *f+fi) = opu(ff) + opu(fi). 24 CHARLES REZK 8.2. Definition of the structure maps. Recall that 1+B DS, the free commu- tative S-algebra on the 0-sphere. (2)The class 1 2 h(M) is the image of the canonical class under the maps pt B 0 B i-! 1 S. Equivalently, it is induced by the unit map S 1-!DS of* * the ring DS. (3)The product "." is induced on B and 1 S by maps Bq: B x B ! B and Hq : 1 S x 1 S ! 1 S. Here q: x ! is the coproduct functor on finite sets; 1+Bq corresp* *onds to the ring product DS ^ DS ! DS. Hq is the "additive" H-space product on 1 S, obtained by applying 1 to the fold map S x S S _ S ! S. (4)The class [1] 2 h(M) is the image of the canonical class under the maps * *pt B 1 B -i! 1 S. Equivalently, it is induced by the "cannonical" map S ! DS. (5)The product "O" induced on B and 1 S by maps Bx: B x B ! B and Hx :B x B ! B . Here x: x ! is the cartesian product functor on finite sets, and Hx * *is the "multiplicative" H-space product on 1 S, and is adjoint to ffl+ ^ ffl+ * *: 1+ 1 S ^ 1+ 1 S ! S ^ S S. (6)The map ss is induced by the projection map M ! pt. (7)The map x is induced by the diagonal map M ! M x M. (8)The map o is induced on 1+B by the composites 1+(proj)1 1+B m -transfer----! 1+B m-1 ------! + (pt) = S, (let B -1 = ?). On 1+ 1 S the map o is induced by the counit map ffl+ : 1+ 1 S ! S. (9)The map i is induced on 1+B by maps jk: 1+B n ! 1+B 0 S, which is the identity if k = 0, and null homotopic if k > 0. Alternately, i is* * induced by the ring map DS ! S free on the null map S ! * ! S. (10)The map + is induced on 1+B by the transfer maps 1+B k ! 1+(B ix B j), i + j = k, associated to the inclusion ix j n. Alternately, + is induced by th* *e ring map DS ! D(S _ S) DS ^ DS which is free on the pinch map S ! S _ S. 8.3. Compatibility of the structure maps. We need to know that in each of the c* *ases (1) through (8), the two structure maps are compatable with respect to B ! 1 * *S. For (1), (2), (4), (6), and (7), compatibility is clear. For (3) and (5), compatibility amounts to saying that B ! 1 S is a map of * *"semi- ring spaces", which is well-known. For (8), this is (7.5). 8.4. Proof of properties. (a)The slant product h0(M x K) h h(M) ! h0(K) is h-linear. (b) Bq (resp. Hq) make M into a commutative and associative H-space. (c)For M = B , this follows from the fact that r*ijbPm(ff) = bPi(ff) x bPj(* *ff), where i + j = m and rij:(E ix i Ki) x (E j x j Kj) E m x ix j Km ! E m x n Km . Restricting along the diagonal maps iand j gives the resu* *lt. LOGARITHMIC COHOMOLOGY OPERATION 25 For M = 1 S, this follows from the fact that H*qbP(ff) = bP(ff)xPb(ff* *), viewed as a class in h0( 1 1+K x 1 1+K)x . (d) Bx (resp. Hx ) make M into a commutative and associative H-space. (e)P1:h0(K) ! h0(E 1 x 1 K) h0(K) is the identity map. (g)M ! ptis a map of H-spaces. (h) P :h0(pt) ! h0(M x pt) sends 1 to 1, and ss(u) equals the slant product * *of 1 with u. (i)If u is grouplike, then the slant product map -=u: h0(M x K) ! h0K is a * *map of rings. Since P :h0(K) ! h0(M x K) is mulitplicative, the result follo* *ws. (j)M is a (semi-)ring space. (k) The two products are defined via space-level maps M x M ! M, and so are compatible with diagonal. (l)For M = B , this is a consequence of the "double-coset formula" applied * *to the homotopy pullback square B k-1 x B `-1 _____//B k`-1 | | | | fflffl| Bx fflffl| B k x B ` _______//_B k`. For M = 1 S, it follows from the fact that the composite 1+Hx 1 1 ffl+ 1+ 1 S ^ 1+ 1 S ----! + S -! S, which induces u v 7! o(u O v), equals ffl+ ^ ffl+ . (m) For M = B , this is a consequence of the "double-coset formula" applied * *to the homotopy pullback square (B k-1 x B `) q (B k x B `-1)_____//B k+`-1 | | | | fflffl| Bq fflffl| B k x B ` ______________//_B k+`. To prove it for M = 1 S, note that for any spectrum X there is a diagram ( 1+ 1 ss1, 1+ 1 ss2)1 1 1 1 ffl+_ffl+ 1+ 1 (X x X) ________________//_ + X _ + X___//_X _ X WWWWW | WWWWWWW |fold |fold 1+ 1 (fold)WWWWWW++WWWWWWfflffl|| fflffl|| 1+ 1 X ___________//_X ffl+ where ssi:XxX ! X denote the projection maps. This diagram is commutative up to homotopy. Evaluating at X = S gives the desired result. (n) This is clear from the observation that i arises from a ring map DS ! S. (o)Bx maps B k x B ` to BQ0 only if either k or ` equals 0. (p) P :h0(pt) ! h0(B ) kh0(B k) sends 0 to (1, 0, 0, . .).. (q) This follows from (7.3). 9. The logarithmic element In this section, we characterize the element v 2 R^0( 1 S) which appears in * *the state- ment of (5.8). The main result is (9.3), which states that v is satisfies certa* *in algebraic identities, which characterize it uniquely; i.e., it is the unique logarithmic * *element. These 26 CHARLES REZK algebraic identities in some sense encode the properties of the idempotent oper* *ator ': namely, that ' is the identity on infinite loop maps, and is idempotent. We note that all the results in this section still hold if we take L to be t* *he telescopic localization functor LfK(n), rather than K(n)-localization; see (3.3). 9.1. Logarithmic elements. Recall from x8.1(8) the homomorphism o :R^0 1 S ! R^0S induced by the counit map ffl+S: 1+ 1 S ! 1+(pt) = S, and from x8.1(5) the product O: R^0 1 S R^0(pt)R^0 1 S ! R^0 1 S. A logarithmic element for R is an element v 2 R^0 1 S with the following two properties: (La) o(v) = 1 in R^0S. (Lb) x O v = o(x)v for all x 2 R^0 1 S. 9.2. Proposition. There is at most one logarithmic element in R^0 1 S. A map R * *! R0 of K(n)-local commutative S-algebras carries the logarithmic element for R (if * *it exists) to the logarithmic element for R0. Proof.The second statement is a consequence of the uniqueness of the logarithmi* *c ele- ment. If both v and v0are logarithmic elements in R^0 1 S, then v = 1 . v = o(v0) . v = v0O v = v O v0= o(v) . v0= 1 . v0= v0, using x8.1(b) and (d). 9.3. Theorem. The element v of (5.8)is a logarithmic element in R^0 1 S, and he* *nce is the unique logarithmic element. In particular, there is a logarithmic element for R = LS, the K(n)-localizat* *ion of the sphere. By (9.2)all logarithmic elements for K(n)-local commutative S-algeb* *ras are determined by the logarithmic element for the case R = LS. To give a proof of (9.3), we need some results involving the natural transfo* *rmations ~+ (x5.3) and ffi+ (x7.6). Recall the structure map fl of x5.1. 9.4. Proposition. The diagram fl 1 Y^1 1+K 1 1 Y ^ 1+K ____________// 1+ 1 Y ^ 1+K 1 ffiY,K|| ||1+ffi+Y,K fflffl| fflffl| 1 1 (Y ^ 1+K) fl_1_(Y_^_1//_ 1+ 1 (Y ^ 1+K) + K) commutes in the homotopy category of spectra. Proof.This is immediate from (9.5), and the stable basepoint splitting of x5.1. 9.5. Lemma. For every unbased space K and spectrum Y , the diagram 1+ss^1 1 q^1 1 1 1 1+(pt) ^ 1+Koo___ 1+ 1 Y ^ 1+K ______//_ Y ^ + K 1+ss|| 1+ffi+Y,K|| ||1 ffiY,K fflffl| fflffl| fflffl| 1+(pt)oo_1_____1+ 1 (Y ^ 1+K) __1_//_ 1 1 (Y ^ 1+K) + ss q commutes in the homomotpy category of spectra. (The map q was defined in x5.1.) LOGARITHMIC COHOMOLOGY OPERATION 27 Proof.The commutativity of the left-hand square follows from the fact that ffi+* * is a natural transformation, and so commutes with the map induced by the projection * *Y ! pt. The right-hand square is equal to 1 applied to the square ( 1 Y x K)+ ( 1 Y )+ ^ K+____q^1____// 1 Y ^ K+ (ffi+Y,K)+|| ffiY,K|| fflffl| fflffl| ( 1 (Y ^ 1+K))+_______q________// 1 (Y ^ 1+K) so it suffices to show that this square commutes in the homotopy category of po* *inted spaces. In fact, formal properties of adjunction show that q O (ffi+Y,K)+ = ffi+Y,K= ffiY,K O (q ^ 1). 9.6. Proposition. Let K be a space and Y a spectrum. The diagram ~+Y^id 1 1 1 LY ^ 1+K ____________//L + Y ^ + K | | 1 + | + |L( + ffiY,K) fflffl| ~Y ^ 1+K fflffl| L(Y ^ 1+K) __________//_L 1+ 1 (Y ^ 1+K) commutes. (We use the conventions for localization described in x1.15.) In particular, taking Y = S gives ~+ 1+K= L( 1+ffiS,K) O (~+S^ id). Proof.This square breaks up into two squares: LY ^ 1+K ___~Y^id___//_L 1 1 Y ^ 1+K_Lfl_1_Y^id//_L 1+ 1 Y ^ 1+K | 1 | | 1 + | L ffiY,K| |L + ffiY,K fflffl| fflffl| fflffl| L(Y ^ 1+K)__~________//L 1 1 (Y ^ 1+K)_________//_L 1+ 1 (Y ^ 1+K) Y ^ 1+K Lfl 1 (Y ^ 1+K) The commutativity of the left-hand square is (6.1), while commutativity of the * *right-hand square is proved by applying L to the square of (9.4). 9.7. Lemma. The composite ffi+X, 1 Y 1 (idX^ffl+Y) 1 X x 1 Y -----! 1 (X ^ 1+ 1 Y ) ---------! 1 (X ^ Y ) is the K"unneth map, i.e., the map representing the external product map X0K x * *Y 0L ! (X ^Y )0(K xL) in generalized cohomology. In particular, for X = Y = S, the com* *posite ffi+S, 1 S 1 ffl+S 1 S x 1 S -----! 1 1+ 1 S ----! 1 S is precisely the K"unneth product for S. Proof.The K"unneth map 1 X x 1 Y ! 1 (X ^ Y ) is characterized as the adjoint to ffl+X^ ffl+Y: 1+ 1 X ^ 1+ 1 Y ! X ^ Y. The result follows by factoring ffl+X^ ffl+Y= (1 ^ ffl+Y) O (ffl+X^ 1) and taki* *ng adjoints. 28 CHARLES REZK As a consequence, we have 9.8. Lemma. The diagram ~+S^id 1 1 1 1 LS ^ 1+ 1 S __________//_L + S ^ + S id^ffl+S|| |L|1+m fflffl| fflffl|11 LS __________~+________//L + S S commutes, where m denotes the K"unneth product map 1 S x 1 S ! 1 S. (Recall the conventions described in x1.15 for localization.) Proof.Consider the diagram ~+S^id 1 1 1 1 LS ^ 1+ 1 S ____//_L + S ^ + S | | 1 + | + L(|+ ffiS, 1 S) fflffl|~ 1+ 1 S fflffl| L( 1+ 1 S) ______//_L 1+ 1 ( 1+ 1 S) Lffl+S|| L|1+|1 ffl+S fflffl| ~+S 1fflffl|1 LS ______________//L + S The top square is just (9.6)specialized to K = 1 S and Y = S, and so commutes.* * The bottom square commutes because ~+ :L ! L 1+ 1 is a natural transformation. The composite of the right-hand vertical maps is L 1+m, by (9.7), and therefore the* * outer rectangle is the desired square. 9.9. Proof of (5.8). We must verify for v the identities (La) and (Lb) of the d* *efinition of logarithmic element. Proof of (9.3), property (La).We have that ffl+ = ffl O q, ~+ = Lfl O ~, and q * *O fl = id; thus Lffl+SO ~+S= idS. Proof of (9.3), property (Lb).Apply R-homology to the commutative square of (9.* *8)to get R^0( 1 S)_vx__//R^0( 1 S x 1 S) o|| O|| fflffl| fflffl| R^0(pt)___v.___//_R^0( 1 S) On the bottom and left, the composite is x 7! o(x) . v. On the top and right, * *the composite is x 7! x O v. 10.Level structures and the cocharacter map 10.1. Universal deformations. In this section, we fix a prime p and a height n * * 1, a perfect field k of characteristic p, and a height n-formal group 0 over k. W* *e let E denote the Morava E-theory associated to the universal deformation of 0. LOGARITHMIC COHOMOLOGY OPERATION 29 10.2. Level structures. For a profinite abelian p-group M, we write M* def=homcts(M, Qp=Zp), where Qp=Zp is given the discrete topology, and we write M[pr] for the subgroup* * of pr-torsion elements of M. If F is a formal group over a complete local ring R with maximal ideal mR, t* *hen F (mR) denotes the additive group with underlying set mR and group law given by* * F . For a discrete abelian group A, a homomorphism f :A ! F means a homomorphism A ! F (mR) of abelian groups. The set of such homomorphisms is denoted hom(A, F* * ). 10.3. Proposition. Let A be a finite abelian group. (a)The O-module E0BA* is free and finitely generated over O. There are natu* *ral isomorphisms E0(BA* x X) E0BA* O E0X and R O E0BA* homctsO-mod(E0BA*, R). (b) Let i: O ! R be a local homomorphism to a complete local ring R, classif* *ying a deformation F . Then there is a natural isomorphism hom ctsO-alg(E0BA*, R) hom(A, F ). Proof.Part (a) is [HKR00 , 5.10 and 5.11], and part (b) is [HKR00 , 5.12]. We set O(hom (A, )) def=E0BA*; it carries the universal homomorphism from A* * to a deformation of 0. A homomorphism f :A ! F is called a level structure if on the formal scheme F over R one has the inequality of divisiors X [f(a)] F [p], a2A[p] where the left-hand side is over the elements of the p-torsion subgroup of A, a* *nd the right- hand side denotes the divisor of the p-torsion subgroup of F . In terms of a co* *ordinate T on F , this amounts to the condition that Y (T + T (f(a))) divides [p]F (T ) in R[[T ]]. a2A[p] Write level(A, F ) for the set of level structures; note that by definition it * *is a subset of hom (A, F ). 10.4. Proposition. Fix a deformation F of 0, classified by the homomorphism i:* * O ! R. (a)For each finite abelian group A, there exists a complete local ring O(le* *vel(A, )) over O and natural bijections homO-alg(O(level(A, )), R) level(A, F ). The ring O(level(A, )) is a quotient of O(hom (A, )). (b) If f :A ! B is an inclusion of finite abelian groups, then there is an e* *vident natural transformation level(B, F ) ! level(A, F ). The map O(level(A, * * )) ! O(level(B, )) classifying the universal example of this transformation * *is finite and flat. 30 CHARLES REZK (c)The invariant subring of the evident action of the ring Aut ((Z=prZ)n) on O(level((Z=prN)n), ) is exactly O. Proof.Part (a) is [Str97, Proposition 22] or [AHS04 , 10.14]. The existence of * *the trans- formation of part (b) is clear. That the map classifying it is finite and flat* * is [Str97, Theorem 34(ii)], while (c) is [Str97, Theorem 34(iii)]. Level structures enter topology in the statement of the character theorem of* * [HKR00 ], though this point of view is not made explicit there. The most useful reference* *s for level structures in the context of algebraic topology are [Str97, x7] and [AHS04 , x1* *0]. 10.5. The cocharacter map. Fix Znp, so that * (Qp=Zp)n. Write Dr def= S O(level( *[pr], )), and let D def=Dr. The group GL( ) acts in a natural way on* * each Dr on the left, through the finite quotient GL( =pr ), in such a way that DGL(r* *) O. Let G denote a finite group (not necessarily abelian). If M is any profinite* * abelian p-group, we let GM def=homcts(M, G)=G, where G acts by conjugation. In the spe* *cial case M = it is called the set of of generalized p-conjugacy classes. There is* * an evident right action of GL( ) on G . In [HKR00 ], the authors define a character map, which is a ring homomorphism OG :E0BG ! map(G , D)GL( ). Their theorem [HKR00 , Thm. C] states that this becomes an isomorphism after in* *verting p. It is more convenient for us to use a dual construction, which we call a coc* *haracter map. The cocharacter map !G :G ! D O E0BG is defined as follows: An element x 2 G is represented by some homomorphism f : =pr ! G for sufficiently large r. T* *here is a homomorphism E0B =pr i Dr D classifying the underlying homomorphism of the universal =pr -level structure. Write !r 2 D O E^0B =pr for the homol* *ogy class corresponding to this homomorphism, by (10.3). Then we define !G (x) def=f*(!r) 2 D O E^0BG. One checks that this definition does not depend on the choice of r or f, and th* *at the HKR character map is derived from the cocharacter map: the evaluation of OG at a given x 2 G is given by Kronecker pairing with !G (x). Recall that an element u 2 D O E^0X is grouplike if ss*(u) = 1 where ss :X * *! *, and if *(u) = u x u, where "x" denotes the external product, and : X ! X x X * *is the diagonal. 10.6. Proposition. (a)The image of !G is contained in the grouplike elements of D O E^0BG. (b) Under the evident bijection (GxH) G xH , we have !GxH (x, y) = !G (x)x !H (y). (c)The cocharacter map !G is equivariant with respect to the actions of GL(* * ) on G and D. (d) If H is a subgroup of G, and T :E^0BG ! E^0BH denotes transfer, we have X T (!G (x)) = !H (xg), gH2(G=H)x( ) where x: ! G is a fixed representative of the generalized conjugacy cl* *ass, xg(~) def=g-1x(~)g, and (G=H)x( )is the subset of G=H fixed by the image x( ) G. LOGARITHMIC COHOMOLOGY OPERATION 31 Proof.For (a), it suffices to see that !r 2 D E^0B =pr is grouplike, which i* *s a conse- quence of (10.3)(a) and the fact that it is dual to a ring homomorphism E0B =pr* * ! D. The proofs of (b) and (c) are straightforward. The equation of part (d) is proved in the same way as the transfer formula f* *or charac- ters [HKR00 , Theorem D]. (The statement of the HKR transfer formula directly i* *mplies (d) modulo torsion, which is enough for our purposes.) 10.7. A congruence formula. Let i: O ! O(level(A, )) be the standard inclu- sion, and let f :A ! i* be the tautological level structure. Let I(level(A, * *)) O(level(A, )) denote the ideal generated by the elements {T (f(a))}a2A, where * *T is any coordinate for ; the ideal does not depend on the choice of T . 10.8. Proposition. Suppose A is a finite abelian p-group of rank 1 r n. The* *n there is an isomorphism O(level(A, ))=I(level(A, )) O=(p, . .,.ur-1), and so I(level(A, )) \ O (p, . .,.ur-1). Proof.By construction, the O-algebra O(level(A, ))=I(level(A, )) is universal* * for level structures which are trivial homomorphisms. A deformationrF of 0 admits at mos* *t one such level structure, and one exists if and only if T p divides [p]F (T ). Thus* * F admits such a trivial level structure if and only if i(uk) = 0 for k = 0, . .,.r - 1, * *and we conclude that O(level(A, ))=I(level(A, )) O=(p, . .,.ur-1). 10.9. Remark. As a special case of (10.8), we see that any deformation of 0 to* * a ring with p = 0 admits a unique Z=p-level structure. I am indebted to Mike Hopkins * *for pointing out this fact to me, which led to the proof of the congruence (12.2). Consider a finite subgroup V *. Let S denote the kernel of the projec* *tion dual to this inclusion: 0 ! S ! ! V *! 0. Thus S is an open subgroup of . The inclusion V * determines a homomorphism O(level(V, )) ! D, classifying the restriction of level structures. 10.10. Proposition. Let x, y 2 G , such that x|S = y|S in GS. Then !(x) !(y) mod I(level(V, )) . D O E^0BG. Proof.Let I def=I(level(V, )) . D. Let r be chosen sufficiently large such tha* *t pr S, and such that x and y are represented by maps f, g : =pr ! G. It suffices to s* *how, in j D O E^0B =pr __ss//_D=I O E^0B =pr oo___D=I O E^0BS=pr , jjjj f*||g*|| jjjjjjjj fflffl|fflffl|ttjjjj D=I E^0BG that ss(!r) is in the image of j; given this, the result follows, because x|S =* * y|S implies that f|S=pr and g|S=pr are conjugate by an element of G, and so induce identica* *l maps E^0BS=pr ! E^0BG. Dualizing, we are asking that a dotted arrow exist making the 32 CHARLES REZK following a commutative square of rings: !r E0B =pr ______//D | | | | fflffl| fflffl| E0BS=pr ` ` `//D=I The existence of such a dotted arrow is a tautology; it amounts to a factorizat* *ion 0_____//V_______//__*_____//_S*___//_v0 | v | v v fflffl|zzv (mD=I) of a diagram of abelian groups. 11.Burnside ring elements 11.1. The Morava E-theory of B and 1 S. 11.2. Proposition. For all k, E^0B k is a finitely generated, free O-module. Proof.This is [Str98, Theorem 3.3]. 11.3. Proposition. E^0 1 S is the completion of an infinitely generated free O-* *module. It is flat over O, and thus in particular is p-torsion free. The union of the i* *mages of the maps E^0{k} x B ` ! E^0 1 S are dense. Proof.Let f :B ! B denote the map given by fk: B k-1 ! B k. It is well known that 1 S is stably equivalent to hocolim(B f-!B f-!. .).. Furthermore, fk ad* *mits a stable retraction 1+B k ! 1+B k-1 (the stable homotopy version of a theorem* * of Dold [Dol62]). From this and (11.2), it is clear that ssk(E ^ f-1 B ) is a free* * O-module for even k, and 0 for odd k. Thus E^0 1 S is the m-adic completion of this free* * module. The flatness result follows from (11.4)below. 11.4. Lemma. Let A be a Noetherian commutative ring, I A an ideal. Then the I-adic completion of any free A-module is flat over A. Proof.For a free module on one generator, this is well-known; the usual proof (* *e.g., [AM69 , Prop. 10.14]) using the Artin-Rees lemma generalizes to give the lemma,* * as we show below. L Let S be a set, and define a functor on A-modules by F (M) def= s2SM ^I. * *We claim that (i)F is exact on the full subcategory of finitely generated A-modules, and (ii)the evident map F (A) A M ! F (M) is an isomorphism when M is finitely generated. Then F (A) A - is exact on the full subcategory of finitely generated modules,* * and thus F (A) is flat. Recall the Artin-Rees lemma [AM69 , Thm. 10.11]: given a finitely generated * *module M and a submodule M0, there exists c 0 such that for all k 0,LIk+cM0 LIkM * *\M0 Ik-cM0. Therefore the same is true when M and M0 are replaced by sM and sM0. This implies (i), by [AM69 , Cor. 10.3]. To prove (ii), note that the map is an* * isomorphism if M is free and finitely generated, and therefore surjective for all finitely * *generated LOGARITHMIC COHOMOLOGY OPERATION 33 M, using the exactness result (i). Since A is Noetherian, (ii) follows by a 5-* *lemma argument. 11.5. The cocharacter map for B and 1 S. As in x10, define the profinite abel* *ian group def=Znp. Let A+k( ) denote the set of isomorphism classes of discrete c* *ontinuous -sets which have exactly k elements. We identify A+k( ) ( k) : to a general* *ized conjugacy class x: ! k associate X = k= k-1, regarded as a -set via x. ` + Let A+ ( ) def= kAk ( ). The set A+ ( ) admits the structure of a semi-ring,* * with addition and multiplication corresponding to coproduct and product of sets. Let* * A( ) be the ring obtained from A+ ( ) by adjoining additive inverses; it is the Burn* *side ring of , isomorphic to the direct limit A( =S) where S ranges over open subgroups * *of . For the following proposition we need the notation introduced in x8.1. 11.6. Proposition. The cocharacter maps (10.5)for symmetric groups fit together* * to give a map !+ :A+ ( ) ! D O E^0B . It is a homomorphism into the semi-ring of grouplike elements. That is, for x, * *y 2 A+ ( ), !+ (0) = 1, !+ (1) = [1], !+ (x + y) = !+ (x) . !+ (y), !+ (xy) = !+ (x* *) O !+ (y). If X is a transitive -set, then !+ ([X]) is also primitive. Furthermore, this map extends uniquely to a map ! :A( ) ! D O E^0 1 S which is a homomorphism into the ring of grouplike elements in D O E^0 1 S. We have that for x 2 A( ), o(!(x)) = d(x), where d: A( ) ! Z is the ring homomorphism defined by d([X]) = #(X ). The maps !+ and ! are GL( )-equivariant, and so ! induces a map A( )GL( )! E^0 1 S. Proof.That !+ lands in the grouplike elements follows from (10.6)(a). That it * *is a homomomorphism of semi-rings follows from the fact that the operations of sum a* *nd product on A+ ( ) are derived, using (10.6)(b), from the maps ( k) x ( `) ( k x `) q-!( k+`) and ( k) x ( `) ( k x `) x-!( k`) , which are also the origin of the product maps "." and "O", as defined in x8.1. * *Similarly, the additive and multiplicative units of A+ ( ) arise as the unique elements of* * ( 0) and ( 1) , respectively. The primitivity of !+ ([X]) when X is transitive follows from [ST97 , 4.3]. The map o is induced by the stable map B k transfer-----!B k-1 proj--!pt. Th* *e transfer formula (10.6)(d) gives X transfer(! k(x)) = ! k-1(xg). x2( k= k-1)x( ) The element !G (y) is always a grouplike element (10.6)(a), and so goes to 1 un* *der the projection BG ! pt. Under this projection, the element on the right-hand side o* *f the equation becomes an integer, equal to the size of ( k= k-1)x( )= X . 34 CHARLES REZK To extend !+ to !, note that the grouplike elements of D E^0 1 S are inver* *tible in the "." product, so we may set !([X] - [Y ]) def=!([X]) . !([Y ])-1. The equivariance property follows from (10.6)(c). 11.7. Power operations. Let A * be a finite subgroup of order pr. Dualizing gives a surjective homomorphism f : ! A*. Using this map, we can regard A* as a set with a transitive -action, and hence an element in the Burnside ring, de* *noted s(A) 2 A+pr( ). We define _A :E0(X) ! D O E0(X) by _A = op!+(s(A)). According to the remarks of the previous section, s(A) is both grouplike and primitive, a* *nd thus _A is a ring homomorphism (though not an O-algebra homomorphism). These operations coincide with the ones constructed by Ando [And95 ], though the construction is* * not identical, since Ando did not have available to him the fact that the Morava E-* *theories are commutative S-algebras. Some dicussion of these operations is given in [AHS* *04 ]. 12. Construction of the logarithmic element 12.1. The element defined. We define a certain element e 2 A( )GL( )as follows: nX e = p (-1)jpj(j-1)=2ej, j=0 where X ej = 1_pj [ =S]. p S =S (Z=p)j This element e really lives in A( ) and not just A( ) Q, since 1 + j(j - 1)=2* * - j = (j - 1)(j - 2)=2 0 when j 0. In this section, we will prove the following. 12.2. Proposition. The element !(e) is congruent to 1 modulo p in E^0 1 S. 12.3. Proposition. Let m 2 E^0 1 S such that 1 + p . m = !(e). The resulting el* *ement X1 pk-1 1 v def= (-1)k-1____mk = _ log!(e) k=1 k p is the logarithmic element for E. 12.4. Proof of the main theorem. We can now complete the proof of (1.11). By (5* *.8) and (9.3), we have that `(ff) = opv(ff), where v is the logarithmic element for* * E. From x8.1(a) and (c) we have that opu+u0(ff) = opu(ff) + opu0(ff) and opuu0(ff) = opu(ff)opu0(ff), and so 1 X pk-1 `(ff) = (-1)k-1____opm (ff)k. k=1 k The operation M of (1.11)is simply opm. By construction, the operation M satisfies the formula for 1 + pM given in t* *he state- ment of (1.11). We claim it is the unique such operation. It is clear that the * *formula characterizes M up to p-torsion. Any operation E0 ! D O E0 corresponds to an e* *le- ment of D O E0 1 E. By, e.g., [BH04 , Thm. 1.4], E^0 1 E is a free E*-module i* *n even LOGARITHMIC COHOMOLOGY OPERATION 35 degress, whence D O E0 1 E is torsion free, and thus M must be the unique oper* *ation with this property. 12.5. Congruence for !(e). We use the notation of x11.7. In these terms, using * *(11.6), we have Yn` Y ' (-1)jp(j-1)(j-2)=2 !(e) = !(s(U)) . j=0 U *[p] |U|=pj Recall that *[p] = (Z=p)n (Q=Z(p))n = *. 12.6. Proposition. In D E E^0 1 S we have the congruence !(e) 1 mod I(level(V, )) . D O E^0 1 S, where V *[p] is any subgroup which is isomorphic to Z=p. Reduction of (12.2)to (12.6).Let I = I(level(V, )). By (10.8)we have an inclus* *ion of short exact sequences 0 ____//_pO____//fflfflO//_fflfflO=pO//_fflffl0 | | | | | | fflffl| fflffl| fflffl| 0 ____//_ID____//D____//_D=ID____//_0. Tensoring with the flat module E^0 1 S (11.3)preserves exact sequences and mono* *mor- phisms. The element 1 - !(e) 2 D O E^0 1 S lives in E^0 1 S by (11.6), and liv* *es in ID O E^0 1 S by (12.6), and so must be an element of pE^0 1 S. 12.7. Lemma. Consider a decomposition V V ? __ *[p] where V Z=p; let T = Ker( ! V *). Given a subgroup U *[p], let U V ? *[p] denote the image * *of the projection of U to V ?. Then in A+ (T ), ( __ s(U)|T = s(U )|T_ if V 6 U, p . s(U )|Tif V U. As a consequence, we obtain the congruences ( __ !(s(U)) !(s(U ))__ if V 6 U, !(p . s(U ))if V U, modulo the ideal I(level(V, )) . D O E^0 1 S. Proof.Let i: U ! * denote the given_inclusion,_and j :U ! * denote the map factoring through the projection to U. By definiton, i j mod V , and so both* * i and j define the same composite U ! * ! T *. Dualizing, we see that i*|T = j*|T, vie* *wed as maps T ! ! U*. __ If V 6 U, then j*|T is surjective, whence s(U)|T_= s(U )|T. If V U, then * *j*|T has cokernel isomorphic to Z=p, whence s(U)|T = p . s(U )|T. The congruences follow immediately from (10.10). 36 CHARLES REZK Proof of (12.6).Choose any decomposition V V ? *[p] with V Z=p, as in the lemma. Let d(j) = (-1)jp(j-1)(j-2)=2; note that d(j + 1) = -d(j)pj-1. We have Yn Y n-1YY Yn Y !(e) = !(s(U))d(j)= !(s(U))d(j). !(s(U))d(j) j=0U *[p] j=0 V 6 U j=1V U |U|=pj |U|=pj |U|=pj which by (12.7)is congruent mod I to n-1YY __ Yn Y __ I !(s(U ))d(j). !(p . s(U ))d(j) j=0 V 6 U j=1V U |U|=pj |U|=pj which we reindex according to subgroups of V ?, to get n-1YY j Yn Y = !(s(W ))d(j)p. !(s(W ))d(j)p j=0W V ? j=1W V ? |W|=pj |W|=pj-1 n-1YY j = !(s(W ))d(j)p +d(j+1)p. j=0W V ? |W|=pj Since the exponents are always 0, the expression reduces to 1. 12.8. M"obius functions and the logarithmic element property. It remains to show that the element v of (12.6)is in fact a logarithmic element. To do this, * *we first show that e=p 2 A( ) Q is the idempotent associated to the augmentation d: A( )* * ! Z sending d([X]) = #X . We recall the theory of idempotents in a Burnside ring [Glu81], in the speci* *al case when the group is finite abelian. Thus, let G be a finite abelian group and A(* *G) its Burnside ring. The M"obius function of G is the unique function ~G defined on * *pairs C B of subgroups of G, characterized by the property that ( X 1 if A = B, ~G (C, B) = A C B 0 if A 6= B, where the sum is over all subgroups C contained in B and containing A. Then the elements X eA def= ~G_(B,_A)_[G=B] 2 A(G) Q B A #(G=B) as A ranges over the subgroups of G are the primitive idempotents of A(G) Q [* *Glu81, p. 65], and furthermore, yeA = dA(y)eA, where dA :A(G) ! Z is given by dA([X]) = #(XA ). Set ~G = ~G (0, G). Since the value of ~G (C, B) really only depends on the* * poset of subgroups of G between C and B, we see that ~G (C, B) = ~B=C , and that ~G o* *nly depends on the isomorphism class of G. The following lemma calculates ~A for al* *l abelian A. 12.9. Lemma. Let A be a finite abelian group. LOGARITHMIC COHOMOLOGY OPERATION 37 Q Q (1)If A Ap where the Ap are p-groups for distinct primes, then ~A = ~* *Ap. (2)If A is elementary p-abelian of rank j 0, then ~A = (-1)jpj(j-1)=2. (3)If A is a p-group but not elementary p-abelian, then ~A = 0. Proof.Part (1) is straightforward. To prove part (2), note that it amounts to the identity [Shi71, Lemma 3.23] Xn ~n ~ (1 if n = 0, (-1)jpj(j-1)=2 = j=0 j p 0 if n > 0, n Q j-1(pn-pi) where jp = i=0______(pj-pi)is the Gaussian binomial coefficient, which is th* *e number of elementary abelian subgroups of rank j inside (Z=p)n. PWe prove part (3) by induction on the size of A. We have for A nontrivial ~* *A = - A)B ~B , the sum taken over proper subgroups of A. A proper subgroup B ( A is one of two types: (a) it is an elementary abelian p-group, or (b) it isn't. * * For (a), suchPB are exactly the subgroups of A[p] ( A, the subgroup of p-torsion element* *s, and A[p] B~B = 0, since A[p] 6= 0. For (b), we have ~B = 0 by induction. For r 1, the elements er def=e =pr 2 A( =pr ) Q are idempotents, and the homomorphisms A( =pr ) Q ! A( =pr+1 ) Q carry er to er+1, as can be seen from the explicit formula for these elements together with (12.9). Thus the limiting* * element e1 2 A( ) Q of this sequence is an idempotent in this ring, with [X]e1 = d([X* *])e1 , where d([X]) = #(X ). By (12.9), we see that the element e defined in x12.1 is* * equal to pe1 , and thus we obtain 12.10. Proposition. In A( ) we have (a)d(e) = p, and (b) for all y 2 A( ), ye = d(y)e. 12.11. Proof of the logarithmic element property. 12.12. Lemma. Let ff 2 D O E^0 1 S be an element of the form ff = 1 + pfi, and* * let P pk log(ff) def=k 1(-1)k-1__kfik-1. (a)We have that o(log(ff)) = o(ff)_ss(ff). (b) If w 2 D O E^0 1 S is a grouplike element, then w O log(ff) = log(w O ff). Proof.First, note that the operations o, ss, and O on D O E^0(- ) are continuo* *us with respect to the maximal ideal topology, since they are induced by maps of spectr* *a. To prove (a), recall that o is a derivation (8.1)(m) with respect to the "."* * product, so that o(fik) = ko(fi)ss(fi)k-1. Thus 0 1 X pk X pk o(log(ff))= o @ (-1)k-1__fikA = (-1)k-1__o(fik) k 1 k k 1 k X = (-1)k-1pkss(fi)k-1o(fi) = o(fi)(1 + pfi)-1. k 1 38 CHARLES REZK To prove (b), recall that if w is grouplike, then w O (- ) is a homomorphism* * of D- algebras (8.1)(j). Thus, 0 1 X pk w O log(ff)= w O @ (-1)k-1__fikA k 1 k X pk = (-1)k-1__(w O fi)k = log(1 + p(w O fi)) = log(w O ff). k 1 k Proof of (12.3).We are going to prove that v = (1=p) log!(e) is a logarithmic e* *lement. We have that o(1_plog!(e)) = 1_po(!(e))_ss(!(e))= 1_pd(e)_1= p_p= 1, using (12.12)(a), (11.6), and (12.10)(a). This proves (La) of the logarithmic * *element property, that o(v) = 1. Now we need to prove (Lb); that x O v = o(x)v for all x 2 E^0 1 S. The union* * of the images of E^0{k}xB ` ! E^0 1 S is dense, with respect to the maximal ideal topo* *logy, by (11.3). Thus it suffices to prove (Lb) for those x which are in the image of* * one of these maps. It is enough to do this after faithfully flat base change to D. Now* *, E^0 1 S is p-torsion free, and by the HKR theorem, p-1D E^0B m is spanned by elements* * in the image of the cocharacter map. Thus, it suffices to check (Lb) when x = !(y)* * for any y 2 A( ). So let y 2 A( ). We have that !(y) O 1_plog!(e)= 1_plog(!(y) O !(e)) by (12.12)(b), = 1_plog!(ye) = 1_plog!(d(y)e) by (12.10)(b), = 1_plog!(e)d(y)= d(y)_plog!(e) = o(!(y))1_plog!(e) by (11.6). Thus !(y) O v = o(!(y))v, as desired. 13.The logarithm for K(1)-local ring spectra In this section, we describe the structure of ss0LK(1) 1+ 1 S (completely at* * an odd prime, and modulo torsion at the prime 2), outline its relation to power operat* *ions on K(1)-local commutative ring spectra, and give a proof of (1.9). 13.1. The p-adic K-theory of some spaces. We recall results on the p-completed K-homology of 1 S, due to [Hod72 ]. We have that K^0(B m ; Zp) homcts(R m , Zp), where R m is the complex representation ring, topologized with respect to the i* *deal of representation of virtual dimension 0. Furthermore K^0(B ; Zp) (Z[ 0, 1, 2, . .].)^p. LOGARITHMIC COHOMOLOGY OPERATION 39 The elements k, k 0, are characterized implicitly by Witt polynomials X pj Wk = pi i , i+j=k where Wk 2 K0(B pk; Zp) is the element corresponding to the continuous homomor- phism R pk! Zp defined by evaluation of characters on an element g 2 pk which * *is a cycle of length pk. Thus K^0( 1 S; Zp) (Z[ 0 , 1, 2, . .].)^p. k Since Wk W0p mod pK^0B , the elements Wk become invertible in K^0 1 S. The cocharacter map !+ :A(Zp) ! K^0(B ; Zp)grpsends [Zp=pk] to Wk. The opera- tion corresponding to Wk is the Adams operation _pk. According to (12.3), the logarithmic element for K^pis 1_log!(p[*] - [Z=p]) = 1_logW0p_= 1_log____1_____= X (-1)k-1pk-1__k1_. p p W1 p 1 + p 1= p0 k 1 k pk0 13.2. The K(1)-local homotopy of B and 1 S. Recall that if X is a spectrum, then there are cofibration sequences KO ^ X ! KO ^ X ! K ^ X and ~ LX ! L(K ^ X) (_--1)^id------!L(K ^ X) if p > 2, and ~ LX ! L(KO ^ X) (_--1)^id------!L(KO ^ X) if p = 2, where ~ 2 Zxp(respectively, ~ 2 Zx2={ 1}) is a topological generator. In partic* *ular, we have that ( ss0LK(1)S = Zp if p > 2, Z2[x]=(x2, 2x)if p = 2, where x comes from the non-trivial element of ss1KO Z=2. 13.3. Proposition. Let f denote either of the Hurewicz maps ss0L 1+B k ! K^0B k or ss0L 1+ 1 S ! K^0 1 S. The map f is an isomorphism if p is odd, while if p =* * 2 it is surjective, and its kernel is the ideal generated by x 2 ss0LS. Proof.This follows from the cofibration sequences mentioned above, together wit* *h the fact that the Adams operations _~, for ~ 2 Zxp, act as the identity map on K0(B* * k; Zp). 13.4. The proof of (1.9). Let R be any K(1)-local commutative S-algebra satisfy* *ing the technical condition described in x1.8. By the above proposition, there is a* * natural factorization i: K^0 1 S ! R^0 1 S of the map ss0L 1+ 1 S ! R^0 1 S. Define pow* *er operations in R-theory by _ def=opi(W1)= opi( p0+p 1)and ` def=opi(.1) There is an identity _(x) = xp + p`(x), and _ is a ring homomorphism. The derivation of (1.9)is now straightforward, since the logarithmic element* * for R must be the image under i of the logarithmic element for K, since both are the * *image of the logarithmic element for LS. 40 CHARLES REZK 13.5. Exponential maps for K-theory and KO-theory. The logarithm maps gl1(K^p) ! K^pand gl1(KO^p) ! KO^pare seen to be weak equivalences on 3-connect* *ed covers in the first case, and 1-connected covers in the second. In other words,* * the loga- rithm admits inverse "exponential" maps e: KSU(X; Zp) ! (1 + KSU(X; Zp))x and e: KSO(X; Zp) ! (1 + KSO(X; Zp))x , where KSU(- ; Zp) and KSO(- ; Zp) denote the cohomology theories defined by the* *se connective covers. Let `k: K0(X; Zp) ! K0(X; Zp) denote the operation correspon* *ding to the element k 2 K^0B pk described above, so that we have k X i pj _p (ff) = p `i(ff) . i+j=k 13.6. Proposition. The exponential maps in K and KO theory are both given by the formula 0 1 1Y X1 ` (ff)pj e(ff) = exp@ _i____jA, i=0 j=0 p ` P pk ' which converges p-adically. Formally this equals exp 1k=0___(ff)_pk. Proof.We give the proof for K-theory; at the end, we indicate the changes neede* *d for KO-theory. P 1 Tpj Let f(T ) def=exp( j=0 ___pj) 2 Z(p)[[T ]] be the Artin-Hasse exponential. * *We will define Q1 a map e by e(ff) def=i=0f(`i(ff)). We will show below that this expression conv* *erges for ff 2 KSU(X; Zp) when X is a finite complex. It is then straightforward to c* *heck that `(e(ff)) = ff for any ff 2 KSU(X; Zp) where X is a finite complex. The represe* *nting space for KSU(- ; Zp) is BSU^p; the set KSU(BSU^p; Zp) = limKSU(X; Zp) as X ranges over finite subcomplexes, and thus we can verify the identity ` O e = id* *on the universal example, which proves that e is the desired inverse. S def Suppose given a CW-model Xk = X. Let Ik = ker[K(X; Zp) ! K(Xk-1; Zp)]; the Ik's give a filtration of K(X; Zp) by ideals such that IkIk0 Ik+k0. We hav* *e that KSU(X; Zp) = I4. Since the `i are natural operations, and `i(0) = 0, they prese* *rve the ideals Ik; in particular, `j(I4) I4. Since X is finite, Ik = 0 for k sufficie* *ntly large, and so for ff 2 I4 each expression f(`i(ff)) is actually a finite sum, contained in* * 1 + I4. We now show that `i(ff) ! 0 as i ! 1 in the p-adic topology. Since I4 has a * *finite filtration by the Ik's, it suffices to do this one filtration quotient at a tim* *e. By the Atiyah- Hirzebruch spectral sequence, this amounts to a calculation on the reduced K-th* *eory of spheres. Thus, Ik=Ik-1 = 0 if k is odd, and `i(ff) = pi(k=2-1)ff for ff 2 Ik=2=* *Ik=2-1if k is even and k > 0. In particular, for ff 2 Ik, k 4, we see that the sequenc* *e `i(ff) mod Ik+1 approaches 0 p-adically. It follows that f(`i(ff)) ! 1 as i ! 1, and therefore the infinite product c* *onverges. The argument for KO-theory is almost the same, except for the calculations o* *n fil- tration quotients. Here the additional observation is that `i(ff) = 0 for ff 2 * *Ik=Ik-1, for all k such that k 2 and k 1, 2 mod 8 (but not when k = 1). 13.7. Exponential operations of Atiyah-Segal. In [AS71 ], the authors construct explicit exponential maps on K-theory and KO-theory completed at some prime p. * *Their construction startsPwith the observation that on any ~-ring R, the operator t:* *R ! R[[t]] given by t(x) = i 0~i(x)ti is exponential. By setting t to particular values* * ff 2 Zp, LOGARITHMIC COHOMOLOGY OPERATION 41 one can sometimes obtain series which converge p-adically, on some subsets of s* *uitable p- adic ~-rings R. In this way, Atiyah and Segal can piece together exponential op* *erations, and construct an exponential isomorphism for KO-theory (though not for K-theory* *). Their construction involves arbitrary choices, and leads to an operation which * *is not infinite-loop. To compare our construction with theirs, we note thatPin a ~-ring we can set St(x) = ( -t(x))-1; the operators si defined by St(x) = i 0si(x)ti correspond* * to taking symmetric powers of bundles. Adams operations are related to St by the e* *qua- tion 2 3 X _m (x) St(x) = exp4 ______tm 5. m 1 m Thus, our exponential operator is a kind of "p-typicalization" of the symmetric* * powers, evaluated at t = 1. 14.The action of Hecke operators on Morava E-theory We give a quick and dirty exposition of a fact which does not seem to be pro* *ved in the literature, but should be well-known; namely, that the Morava E-theory of a* * space carries an action by an algebra of Hecke operators. 14.1. Hecke operators. Let be a monoid, and a subgroup. Define H = homZ[ ](Z[ = ], Z[ = ]) where Z[ ] denotes the monoid ring of , and Z[ = ] is the left-Z[ ]-module spa* *nned by cosets. If M is a left Z[ ]-module, then the -invariants M are naturally * *a left H-module. Consider two examples: (a)The algebra H = Hn, associated to = End(Zn) \ GLn(Q) and = GLn(Z). (b) The algebra H = Hn,p, associated to = End(Znp)\GLn(Qp) and = GLn(Zp). In either case, H has a basis which is in one-to-one correspondence with double* * cosets \ = ; a double cosetP x corresponds to the unique endomorphism "T x of Z[ = ] which sends 1 7! [y ], where y ranges over representatives of the finite set * * x = . In these terms, Hn is the same as the Hecke ring for GLn as described for insta* *nce in [Shi71, Ch. 3]. One sees also that O Hn,p Z[T"1,p, . .,."Tn,p] and Hn Hn,p, p where "Tj,pcorresponds to the double coset of the diagonal matrix which has p i* *n j entries and 1 in the other n - j entries. 14.2. Morava E-theory is a module over Hn,p. We want to show that the algebra Hn,pacts on the functor X 7! E0(X), where E is a Morava E-theory of height n. (Warning: this only agrees up to scalar with the action described in x1.12; see* * x14.5 below.) Let = Znp. The right cosets = are in one-to-one correspondence with* * open subgroups of , by x 7! x . The sum X !+ ([ =y ]) 2 D E^0B , 42 CHARLES REZK where y ranges over representatives of x = , is invariant under the action of * *GL( ) on A+ ( ), and so lives in E^0B , by (11.6). We define _ x :E0X ! E0X to be the operationPassociated to this class. In terms of the notation used in* * x11.7, we have i O _ x = _A, where i: E0X ! D O E0X is the evident inclusion (D is faithfully flat over O), and the sum is over all finite subgroups A * suc* *h that ker( ! A*) = y for some y 2 x . 14.3. Proposition. The assignment "T x7! _ x makes E0X into an Hn,p-module. This is a statement about compositions of the additive cohomology operations* * _ x . We will prove it by reducing to results about the composition of certain ring o* *perations, proved in [AHS04 , App. B]. 14.4. Lemma. For each x 2 there is a ring homomorphism _x: D O E0X ! D O E0X natural in X, with the following properties: (a)If x 2 = GL( ), then _x acts on D O E0X purely through the D-factor, * *via the the action of GL( ) on D described in x10.5. (b) Under the inclusion i: E0X ! D O E0X, we have _x O i = _A, where A * is the kernel of the adjoint x*: * ! * to x. (c)If X is a finite product of copies of CP 1, then x 7! _x gives an action* * of the monoid on D O E0X. Proof.Given x 2 , consider the following diagram of formal groups and level st* *ructures over D: * A _____// *_x___//_ * `|| |`0| fflffl| fflffl| i* __f_//_j* Here is the universal deformation formal group over O, i: O ! D is the usual * *inclusion, A = kerx*, f is an isogeny with kernel `(A), such that modulo the maximal ideal* * of D, f reduces to a power of frobenius. Therefore the codomain of f is a deformation* * of 0 to D, classified by a map j :O ! D. There is a commutative diagram D MMoo____O = E0(pt)_________//_E0(X) MM MMM _| _| O`0MMM&&MMAfflffl|| fAflffl|| D = D O E0(pt) _____//D O E0X where _A is the operation associated to A as in x11.7. On the cohomology of a p* *oint, the map _A :O ! D equals j. The map labelled O`0is the map classifying the pair consisting of the formal group j* over D, and the level structure `0. We define _x: D O E0X ! D O E0X by x y 7! O`0(x)_A(y). Properties (a) and (b) are immediate. Property (c) is proved by the arguments o* *f [AHS04 , App. B] when X = ptor X = CP1 ; since the _x act as ring homomorphisms, they are compatible with K"unneth isomorphisms, and so property (c) holds for finite pro* *ducts of projective spaces. LOGARITHMIC COHOMOLOGY OPERATION 43 Proof of (14.3).We first show that the cohomology operations _ x make E0X into* * a Hn,p-module when X is a finite product of complex projective spaces, and hence * *when X is a finite product of CP1 's. 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