On the coalgebraic ring and Bousfield-Kan spectral sequence for a Landweber exact spectrum Martin Bendersky*& John R. Huntonyz 5 December 2001 Abstract We construct a Bousfield-Kan (unstable Adams) spectral sequence based on an arbitrary (and not necessarily connective) ring spectrum E with unit and which is related to the homotopy groups of a certain unsta- ble E completion X^Eof a space X. For E an S-algebra this completion agrees with that of the first author and R. Thompson [7]. We also es- tablish in detail the Hopf algebra structure of the unstable cooperations ______________________________ *Hunter College and the Graduate Centre, CUNY, New York, USA. Email: mben- ders@shiva.hunter.cuny.edu yDepartment of Mathematics and Computer Science, University of Leicester, Un* *iversity Road, Leicester, LE1 7RH, England. Email: J.Hunton@mcs.le.ac.uk zCorresponding author. Tel.: ++44-116 252 5354; fax: ++44-116 252 3915. 1MSC: 55P60, 55Q51, 55S25, 55T15. Secondary: 55P47. 2Keywords: Bousfield-Kan spectral sequence, unstable Adams spectral sequence* *, Hopf rings, coalgebraic algebra, Landweber exact, unstable cooperations, unstable co* *mpletions 1 (the coalgebraic module) E*(E_*) for an arbitrary Landweber exact spec- trum E, extending work of the second author with M. Hopkins [15] and with P. Turner [20] and giving basis-free descriptions of the modules of primitives and indecomposables. Taken together, these results enable us to give a simple description of the E2-page of the E-theory Bousfield- Kan spectral sequence when E is any Landweber exact ring spectrum with unit. This extends work of the first author and others and gives a tractable unstable Adams spectral sequence based on a vn-periodic theory for all n. Introduction An unstable Adams spectral sequence computes homotopy theoretic information for a space X from homological information. More specifi- cally, such a spectral sequence based on a homology theory E*(-) seeks, under certain hypotheses, to compute the homotopy of an appropriate E-completion of X from an Ext group (in a suitable category) involv- ing E*(X). This paper identifies, for E a general ring spectrum with unit, an unstable E-completion X^Eof X and an associated E-theory Bousfield-Kan spectral sequence with E2-term the homology of a cer- tain unstable cobar complex. When E is an arbitrary Landweber exact spectrum [22] we obtain a more tractable description of the E2-term, and, when E additionally has the structure of an S-algebra in the sense of [13], our completion X^Eand spectral sequence agree with those of [7]. In order to obtain the description of the E2-term we prove a number of results on the generalised homology of the spaces in the -spectrum for 2 a Landweber exact theory E, that is, on the coalgebraic ring F*(E_*). These results are of independent interest; see, for example, [14]. The first example of an unstable Adams spectral sequence based on a theory E other than ordinary homology was that of the first author with E. Curtis and H. Miller [5] which considered the case of a connective theory E and concentrated in particular on the case of BP -theory. This provided a sequence that converged to the p-localisation of the unstable homotopy of an odd dimensional sphere and identified the E2-term as an Ext group in a non-abelian category of unstable BP*(BP )-coalgebras. Using results of Wilson [25], the E2-term was given a simpler, and more computationally practical, interpretation as the homology of an unstable cobar complex and which could be further considered [4] as the homology of a certain subcomplex of the stable cobar complex. This spectral sequence and subsequent variations were generalisations of that of Bousfield and Kan [9] and we refer throughout to all these models as Bousfield-Kan spectral sequences. A more mysterious gadget however has been that of a Bousfield-Kan spectral sequence based on a periodic theory E. Theories E one would naturally wish to consider include complex K-theory, the Johnson- Wilson theories E(n) and the Morava E- and K-theories; for techni- cal reasons one is probably going to make easiest headway with those theories E which are also Landweber exact, as in the connective ex- ample BP successfully dealt with by [5] and subsequent papers. With R. Thompson, the first author has developed a framework [7] to define and study sequences based on periodic theories. The requirements on E to set such a sequence up, to identify the E2-term in a practical and 3 computable manner, and to prove convergence to an identifiable object are however significant. In brief, convergence is proved, in appropriate cases, to a certain `unstable E-completion' of the underlying space X, where this completion is defined as Tot of a certain cosimplicial space, and is defined only when E is represented by an S-algebra in the sense of [13]. Of the example theories E listed above, to present knowledge this rules out all but complex K-theory and the Morava E-theories. The understanding of the E2-page of any of these Bousfield-Kan spectral sequences involves in large part having good and well under- stood structure in the un stable E-theory cooperation algebras, that is, in the coalgebraic ring [19] or Hopf ring [24] E*(E_*) where the E_r denote the spaces in the -spectrum for E-theory. In [7] the E2-term was identified in a practical manner under the hypotheses that each E*(E_r) was free as an E*-module and that the submodule of primitives P E*(E_r) inject under infinite stabilisation in the stable cooperation ring E*(E). Work of the second author and M. Hopkins [15] showed that these hypotheses were satisfied for a Landweber exact theory E whose coefficients were `not too large': this incuded the cases of K-theory and the Johnson-Wilson theories E(n), but not the S-algebra examples of the Morava E-theories. Between these two sets of requirements on E - for convergence and for the computation of the E2-term - fully satisfac- tory results in [7] were obtainable only when E was taken as complex K-theory. The main results of this paper fall into three sections. In section 1 we study in depth the homology and generalised homology of the spaces E_r in the -spectrum for an arbitrary Landweber exact theory E, assuming 4 only that the coefficients E* are concentrated in even dimensions (an assumption which fails only in rather artifical examples). These results extend those of [15], removing the size restrictions on the coefficients, proving for example (Theorem 1.4) that the algebras F*(E_r) for a wide class of homology theories F*(-) are polynomial or exterior for r even or odd respectively. However, they go further than the type of results in [15], giving also basis-free descriptions (Theorem 1.10) of the modules of primitives and indecomposables associated to the F*(E_r). Unusually for results on the coalgebraic ring F*(E_*) for theories F and E, these results give explicit descriptions of the individual Hopf algebras F*(E_r), rather than just implicit descriptions in terms of the global object F*(E_*). We also relate (Corollary 1.11) the modules P F*(E_r) and QF*(E_r) to the primitives and indecomposables of the universal example MU*(MU__*). These results are of independent interest in the study of the homology of -spectra, having applications, for example, to group cohomology [14] and [18]. In the case of certain completed spectra, such as the Morava E-theories and the Baker-Würgler completions [E(n)[3], these results have parallels with those of [16] where the homological effects of completion on -spectra are examined using rather different methods. In section 2 we suppose E merely to be a ring spectrum with a unit. For a space X we define a notion (Definition 2.2) of E-completion of X, denoted X^E. If E is an S-algebra then the space X^E turns out to be homotopy equivalent to the E-completion of X as defined in [7]. The E-theory Bousfield-Kan spectral sequence related to the homotopy groups of this space X^Eis introduced and we identify (Theorem 2.8) the E2-page as the homology of an unstable cobar complex. 5 The results of section 2 are very general but as they stand offer small hope for specific computation. In section 3 we build on them in the special case of a Landweber exact ring spectrum (with unit), using the work of section 1 on the coalgebraic ring for such a spectrum. The main result here is a `change of rings' theorem that identifies (Theo- rem 3.1) the E2-page of the E-theory Bousfield-Kan spectral sequence of section 2 as an Ext group in a convenient, moreover abelian, cate- gory. This applies to spaces X such as torsion free H-spaces and odd dimensional spheres. We note also (Remark 3.11) that a similar result holds for spaces such as S2n+1, though care is needed for such exam- ples as, by the work of section 1, the relevant Hopf algebras E*(E_2r) in the computation are not primitively generated. Taken together, the results of this article allow for the construction and description of an unstable Adams spectral sequence based on a vn-periodic theory for any positive integer n, extending the framework of [7] which established the v1-periodic case. Notation The convention we use for denoting spaces, spectra, etc. re- lated to a theory E is as follows. For a theory E we write E*(-) and E*(-) for the generalised E-homology and cohomology, E for the asso- ciated spectrum when we wish to consider it as an explicit object in the stable category, and E_rand E_*for the spaces in the -spectrum and for the -spectrum itself. Thus the space E_rrepresents the cohomological functor Er(-) in the sense that Er(X) = [X, E_r] for any space X. The E_rare related by equivalences E_r+1' E_r. 6 1 The coalgebraic module F*(E_ *) Throughout this section we shall assume that E_* is an -spectrum representing a Landweber exact cohomology theory [22]. Such theo- ries include the examples of complex cobordism MU and the Brown- Peterson theories BP [1], the Johnson-Wilson theories E(n) [21] and their In-adic completions [E(n)[3] as well as Morava E-theory, complex K-theory, various forms of elliptic cohomology [23] and their comple- tions. For simplicity in the statement of our results, we assume the coefficients E* are concentrated in even degrees; this is satisfied by all standard examples including all those just mentioned. As E is neces- sarily a module spectrum over MU , the mod p homology H*(E_*; Fp) will be a coalgebraic module over both H*(MU__*; Fp) and Fp[MU*] in the sense of [19]; if, as will in fact generally be the case, E is a ring spectrum, H*(E_*; Fp) will be a coalgebraic ring (Hopf ring) and a coal- gebraic algebra over these objects as well. We assume the notation and results on H*(MU__*; Fp) to be found in [24] and the notions of coal- gebraic algebra as in [19, 24]. If E is a p-local spectrum then similar statements hold on replacement of MU by BP . __ The work of [19] establishes in particular a tensor product in the category of Fp[MU*] coalgebraic modules; note that this is quite distinct from the tensor product of the underlying Fp coalgebras. The main theorem of [20] tells us __ * Theorem 1.1 H*(E_*; Fp) ~=H*(MU__*; Fp) Fp[E ]. Fp[MU*] Corollary 1.2 H*(E_2r+1; Fp) is an exterior algebra. 7 Proof Consider the indecomposable quotient QH*(E_2r+1; Fp). Un- __ winding the definition of , elements in this quotient are represented __ by sums of O products of elements of the form q x where q represents an indecomposable in an odd MU space and x 2 Fp[E*] = H0(E_*; Fp) (this follows from the fact that E* is concentrated in even dimensions). As QH*(MU__s; Fp) lies in odd homological dimensions if s is odd, [24], we conclude that QH*(E_2r+1; Fp) lies in odd homological dimension. Thus any finite dimensional sub-Hopf algebra of H*(E_2r+1; Fp) lies in a finite dimensional sub-Hopf algebra generated by odd dimensional elements, and so is an exterior algebra. As H*(E_2r+1; Fp) is the colimit of its finite dimensional subalgebras, the result follows. Corollary 1.3 H*(E_2r; Fp) is a polynomial algebra and homology sus- pension induces an isomorphism QH*(E_2r; Fp) ~=QH*(E_2r+1; Fp). Proof This is an immediate consequence of Corollary 1.2 and the ho- mology Eilenberg-Moore spectral sequence [12] Cotor H*(E_2r+1;Fp)(Fp, Fp) =) H*(E_2r; Fp). As H*(E_2r+1; Fp) is exterior, the E2-page is already polynomial and is concentrated in even degrees. The sequence thus collapses and the result follows. We require knowledge of F*(E_*) for more general theories F than just mod p homology; for example we need results with F = E for the unstable homotopy spectral sequences later, but other examples are of importance too. For the remainder of this article we shall assume that F is a p-local ring spectrum, with coefficients torsion free and concentrated 8 in even dimensions. We shall have occasion also to consider a version of a theory F with coefficients reduced mod p; such homology of a space X will be denoted F*(X; Fp). Theorem 1.4 Suppose E_*and F are as above. Then F*(E_s) is a free F* module, with algebra structure polynomial for s even and exterior for s odd. Proof Begin with the case F = HZ(p). As H*(E_2r; Fp) is polynomial, and in even dimensions, its generators lift to polynomial generators of the torsion free algebra H*(E_2r; Z(p)). From this we can deduce that the homology of the odd spaces H*(E_2r+1; Z(p)) are torsion free, exterior algebras, generated by the suspensions of generators of H*(E_2r; Z(p)). The result for general F follows by a collapsing Atiyah-Hirzebruch spec- tral sequence argument. Corollary 1.5 For E_* and F as above, F*(E_*) is a F*(MU__*) coal- gebraic module; if E is a ring spectrum, F*(E_*) is also a coalgebraic ring. Proof This result is a standard and formal argument (see, for example, [24]) and follows as soon as a Künneth theorem F*(E_rx E_s) ~=F*(E_r) F*(E_s) F* is established. This holds by the freeness result of (1.4). For E a ring spectrum, recall the algebraic model coalgebraic rings F*R(E_*) and F*Q(E_*) constructed in [24] and [17] respectively. The for- mer, F*R(E_*), is the free F*[E*] coalgebraic algebra generated by certain 9 classes arising from the complex orientation on E, modulo specific re- lations arising from the interaction of the E and F formal group laws; the latter, F*Q(E_*), can be constructed as a certain sub-coalgebraic ring of the rational object F Q*(EQ_*). There are natural maps F*(E_*) -fi F*R(E_*) -! F*Q(E_*). Corollary 1.6 For E and F as above, there are isomorphisms of coal- gebraic rings F*(E_*) ~=F*R(E_*) ~=F*Q(E_*). Proof That ø is an isomorphism follows from [20] and the correspond- ing result for MU, [24]. This, together with the fact that F*(E_*) is torsion free by Theorem 1.4, gives the second isomorphism using [17] corollary 6.3. We seek now to describe the modules of primitives P F*(E_s) and indecomposables QF*(E_s) for the Hopf algebras F*(E_s). One of the main ideas of [17] is that when F*(E_*) is torsion free (as here), simple descriptions of its algebra structure can be obtained by identifying its image in the rational coalgebraic ring F Q*(EQ_*). The following result allows analagous descriptions of P F*(E_s) and QF*(E_s) by embedding these modules in the stable object F*(E). Proposition 1.7 For E and F as above, homology suspension induces monomorphisms P F*(E_s)- ! F*-s(E) QF*(E_s) - ! F*-s(E). 10 Proof The proof is essentially given by the following commutative di- agram P F*(E_s) -'! QF*(E_s)? -ffs! F*-s(E)? ?? ? y Qæ* ?yæ* QF Q*(E_s) -ffs!F Q*-s(E). The map ' is the natural map from primitives to indecomposables; as F*(E_*) is torsion free, by (1.4), this is an inclusion. Infinite homology suspension from the sth space is denoted by oes and æ indicates the rationalisation map F ! F Q. Then Theorem 1.4 tells us that the left hand vertical map Qæ* is a monomorphism, and the analysis of rational coalgebraic rings in [17] shows that the suspension oes: QF Q*(E_s) ! F Q*-s(E) is also monic. This result allows us to give a basis free description of P F*(E_s) and QF*(E_s) as a submodule of the stable module F*(E). This generalizes the construction of [4], definition 2.13. First though we need to recall some standard notation for elements in coalgebraic rings; see [24] for further details. Recall there are classes bt 2 H2t(MU__2). When localized at a prime p (as here) it is customary to denote the class bpsby b(s)2 H2ps(MU__2). We shall use this notation throughout, reserving the names bt (without brackets) for elements of the stable module F*(E), as below. There is also the suspension element e 2 H1(MU__1) with the relation eOe = -b(0). For v a homogeneous element of MU*, say v 2 MU|v|= ß0(MU__-|v|), we have the element [v] 2 H0(MU__-|v|), its Hurewitz image; note that v 2 MU|v|= MU-|v|and |v| > 0. By [24], H*(MU__*; Z(p)) is generated as a coalgebraic ring by the classes [v], e and b(s). The algebraic models 11 F*R(E_*) are by definition generated by the analogous elements [v], e and b(s)with the v homogenous elements of E*, and by Corollary 1.6 we know the corresponding classes also generate F*(E_*). The suspension homomorphisms, oes: F*(E_s) ! F*-s(E) send b(i) to bi2 F2pi-2(E), kill * products and take O products to multiplication in F*(E); note in particular that oe2: b(0)7! 1. For v a homogeneous element of E*, we denote also by v the image of [v] under suspension in F|v|(E); this is additionally the image of v in F*(E) under the right unit F* ! F*(E). We shall denote the free E* module generated by a class 'n in dimen- sion n by Mn, or, when we need to indicate the spectrum considered, by MEn. This is useful for keeping track of the domain of the stabiliza- tion map, oes: F*(E_s) ! F*-s(E) and is accomplished by defining the range of oes to be F*-s(E) Ms. In this notation b(i)2 F2pi(E_2) maps E* to bi '2 while a class such as b(i)O[v] 2 F2pi(E_2-|v|) maps to bi v'2-|v|. Notice that the suspension homomorphisms now preserve dimension. For each finite sequence of non-negative integers I = (i1, i2, . .,.in) we write bI for the stable element bI = bi11bi22. .b.inn2 F*(E). The length of I is the integer l(I) = i1 + . .+.in. Write bOI for the unstable element bOi1(1)O . .O.bOin(n)2 F*(E_2l(I)). Of course oe2l(I):bOI 7! b* *I. More generally, any element of the form (bOr(0)O bOI+ decomposables) suspends to bI. Definition 1.8 Let M be a free, graded E*-module. Write UF (M) for the sub-F*-module of F*(E) M spanned by all elements of the form E* 12 bI m where 2l(I) < |m|. Definition 1.9 Let M be a free, graded E*-module. Write VF (M) for the sub-F*-module of F*(E) M spanned by all elements of the form E* bI m where 2l(I) 6 |m|. The special case of the next result for E = F = BP was proved in [4, 5]. Theorem 1.10 The image of the suspension homomorphism, oes: QF*(E_s) ! F*-s(E) Ms ~=F*(E) lies in V (Ms) and E* oes: QF*(E_s) ! V (Ms) is an isomorphism. Furthermore the image of oes|PF*(E_s)lies in U(Ms) and oes: P F*(E_s) ! U(Ms) is an isomorphism. Proof We start with the identification of the image of the indecompos- ables QF*(E_s) with VF (Ms). As oes on QF*(E_s) is monomorphic this will prove the first statement. We begin with the case where s is even. By Corollary 1.6 we know that any element of QF*(E_s) can be written as a F*-linear sum of elements of the form bOr(0)O bOIO [v] with r > 0. Such an element suspends to bI v's. The condition that an element bOr(0)O bOIO [v] lies in the F -homology of the sthspace E_sis that 2r + 2l(I) - |v| = s, thus 2l(I) 6 |v| + s = |v's| and so the image of oes lies in VF (Ms). 13 Conversely, if bI v's lies in VF (Ms) then 2l(I) - |v| 6 s and so bI v's = oes(bOr(0)O bOIO [v]) where 2r = s + |v| - 2l(I) > 0. Hence oes is onto VF (Ms) and the isomorphism for even spaces is shown. The result for odd spaces is very similar; note that circle multiplica- tion by e induces a one to one correspondence between QF*(E_2t) and QF*(E_2t+1). The result for primitives is again similar and follows immediately after making the observation that P F*(E_s) for even s is the F*-linear span of elements of the form bOr(0)O bOIO [v] with r > 0. Also, for odd s there is an isomorphism P F*(E_s) ~=QF*(E_s). The F*-module QF*(E_s) is not as it stands an E* module, but may be modified to be so. Looking at all the spaces together, the bigraded object QF*(E_*) is an E* module under the action x v 7! x O [v] for x 2 QF*(E_*) and v 2 E*. (Verification that x O [v + w] = x O [v] + x O [w] in QF*(E_*) is left as an exercise in coalgebraic modules: see the axioms listed in [24].) We may modify the construction of VF (Ms) so as also to carry the action of E* by considering the corresponding bigraded object VF (M*) equipped with the action (y v's) w 7! y vw's-|w|. Define a global suspension map oe :QF*(E_*) -! VF (M*) as oes on the compo- nent QF*(E_s). With these definitions and the previous result it may easily be checked that oe is a F*-E* bimodule isomorphism. Similar constructions may be made and results established for the objects of primitives P F*(E_*) and UF (M*). Our second description of the modules of primitives and indecom- posables for F*(E_s) can now be given in terms of a simple relation to 14 those of the universal theories. As in the underlying philosophy of [24], etc., this requires us to consider all spaces E_stogether. Corollary 1.11 Let E and F be as above. Then there are isomor- phisms QF*(E_*) = F* QMU*(MU__*) E* = QF*(MU__*) E* MU* MU* MU* P F*(E_*)= F* P MU*(MU__*) E* = P F*(MU__*) E* MU* MU* MU* where means tensor product of modules in the standard sense. As- suming E is p-local, analgous results hold on replacing MU by BP . Proof We prove the first line, concerning the indecomposable functor: the proof of the version involving the primitives is essentially identical. Note also that the equality F* QMU*(MU__*) E* = QF*(MU__*) E* MU* MU* MU* follows immediately since each MU*(MU__s) is a free (left) MU* algebra and hence F* QMU*(MU__*) = QF*(MU__*). MU* We show that QF*(E_*) = QF*(MU__*) E*. By Theorem 1.10 MU* it suffices to show that VF (ME*) = VF (MMU*) E*. Of these, the MU* left hand side is the (bigraded) sub-F*-module of F*(E) ME*spanned E* in grading s by elements bI m's satisfying 2l(I) |m's|. As E is Landweber exact, and, by definition, ME* and MMU* are free over E* and MU* respectively in each grading, F*(E) ME*= F*(MU) E* ME*= F*(MU) MMU* E*. E* MU* E* MU* MU* 15 Under this equivalence, the element bI m's 2 VF (ME*) F*(E) ME* E* is then identified with bI 's m 2 VF (MMU*) E* F*(MU) MMU* E*. MU* MU* MU* This is also onto VF (MMU*) E* as the map MU ! E induces a left MU* inverse. The constructions UF and VF may be extended to other E* modules M. For an arbitrary non-negatively graded left E*-module M let F1-f!F0 ! M ! 0 be exact with F0 and F1 free over E*. Then UF may be extended to M by defining UF (M) = coker(UF (f): UF (F1) ! UF (F0)). VF is similarly extended to such E*-modules. Proposition 1.12 VF (Ms Z=p) ~= QF*(E_s; Fp) UF (Ms Z=p) ~= Im(P F*(E_s; Fp) ! QF*(E_s; Fp)). Proof Since F*(E_s) is a free algebra, there is a diagram with rows short exact 0 ! QF*(E_s) xp-! QF*(E_s) ! QF*(E_s; Z=p) ! 0 k k 0 ! VF (Ms) xp-! VF (Ms) ! VF (Ms Z=p) ! 0. Hence there is an induced isomorphism QF*(E_s; Fp) ! VF (Ms Z=p). A similar proof gives the second isomorphism. 16 Remark 1.13 Since in practice our cohomology theories tend to be Z(p)local it can be advantageous to use BP generators. The generators hi = c(ti), where the ti are the standard generators for BP*(BP ) and c denotes the canonical anti-isomorphism, have proven to be useful for unstable calculations. Following ([5], 8.5) we may replace the generator bi with hi in Theorem 1.10. We conclude this section with an example to help clarify these defi- nitions. Example 1.14 Let F = E = BP . We claim that ph1 '1 defines a non-zero element in VBP (M1 Z=p) = QBP*(BP_1; Fp), but which suspends to zero in QBP*(BP_2; Fp). To see that ph1 '1 is indeed in VBP (M1 Z=p), note that the right action formula tells us that ph1 = v1 . 1 - 1 . v1. Thus ph1 '1 = v1 '1 - 1 v1 . '1, an element of VBP (M1). This element is not divisible by p in VBP (M1), and so is not zero in VBP (M1 Z=p). On the other hand, h1 '2 is an element of VBP (M1) and so ph1 '2 is p-divisible in VBP (M1) and thus is zero in VBP (M1 Z=p). In general, VF (Ms Z=p) is not a submodule of F*(E): when working mod p the unstable classes do not necessarily inject into the stable module. In a similar fashion the right action formula for ph1 can be used to show that phn1is a non-zero element in VBP (M2n-1 Z=p) but which suspends to zero in VBP (M2n Z=p). 17 Example 1.15 Consider the Araki generators wi2 BP2pi-2, as in [2], X pi pmn = mi(wn-i); w0 = p. 06j6n We prefer the Araki generators to the Hazewinkel generators because of the integral form of Ravenel's formulæ F*X i F*X i hpj. wi= wpj. hi. Here F c(fli) is the formal group sum, c is the canonical anti-isomorph- P F* ism and F*fli = c( F c(fli)). (i.e., looks like the usual formal group law, but the formal group coefficients act on the right). It is easy i * pi to check that the Ravenel formulæ imply that F*hpj. wi= F wj . hi hold also in VF (MEs) and VF (MEs Z=p) with s > 2. There are similar formulæ involving the Hazewinkel generators, but they are only true stably mod p. We do not know if the Hazewinkel generators satisfy similar, mod p formulæ, unstably. If E = E(1), Adams' summand of p-local K-theory, or equivalently the first Johnson-Wilson theory and we take F to be H, integral ho- mology, these formulæ reduce to _ F* ! X p hj . w1 's = 0 inVH*(Ms Z=p) ifs > 2. Using the grading, this implies that hpj. w1 's = 0 here. (Notice that hpj.w1 '2 = hpj w1.'2 and w1.'2 has degree 2p so this class is defined.) 18 2 The BKSS for E-theory Let S be the category of pointed CW complexes and suppose E is a ring spectrum with unit. Associated to E is a functor TE :S ! S given by sending X to 1 (E ^ 1 X). There are natural transformations OE: 1S ! TE and ~: TE2= TE O TE ! TE induced by the unit and the multiplication in E respectively and these make (TE , OE, ~) a triple up to homotopy. See, for example, [6, x2], [7, x4] and [8] for details of the notions of triple, cotriple, their associated categories and derived functors, as used in this and the next section. If E is an S-algebra in the sense of [13] (for example, K-theory), it is shown in [7] that (TE , OE, ~) is in fact a triple on the category S. Following [10] there is then a cosimplicial space, TE X, with coface maps and codegeneracies denoted di and sj respectively. The completion of X with respect to E is taken as X^E= Tot(TE X). The E2-page of the Bousfield-Kan spectral sequence associated to X^E is identified [7] with the homology of the unstable cobar complex, Es2(X) = ßsß*TE X = Hs(ß*TE X, @), where ß*TE X is considered as a cochain complex with coboundary map @ = (-1)iß*di. We wish however to be able to consider an `E-completion' of a space X and a corresponding E-theory Bousfield-Kan spectral sequence when- ever E is an arbitrary ring spectrum with a unit. In this section we use the results of [11] to construct (2.2) a space X^Efor any such E, and 19 prove it to be homotopic to the construction in [7] if E is an S-algebra. In Theorem 2.8 we identify the E2-term of the E-theory Bousfield-Kan spectral sequence as an unstable cobar complex. We recall the notion [11] of a restricted cosimplicial space, i.e., a öc simplicial space" without the codegeneracies. Definition 2.1 Suppose (T, OE) is an augmented functor on S, i.e., a functor T :S ! S equipped with a natural transformation OE: 1S ! T . Let X be a space in S. Define the restricted cosimplicial space bTX to be the restricted cosimplicial resolution with respect to T given by (bTX)k = T k+1X in codimension k, and coface maps given by i k k TiffiTk-ik+1 ((bTX)k-1-d! (bTX) ) = (T X ----! T X). We may describe a restricted cosimplicial space as a diagram in S as follows. Let restdenote the restricted simplicial category, that is the category whose objects are finite ordered sets [n] = {0, 1, . .,.n} (n > 0) and whose morphisms are strictly monotone maps. A restricted, unaugmented, cosimplicial space, Crestis equivalent to a functor Crest: rest! S. In particular bTEX 2 S rest. The full simplicial category, , is the category whose objects are the sets [n] and whose morphisms are all weakly monotone maps. Then a cosimplicial space is a functor C: ! S. 20 So C 2 S . Let J : rest! be the inclusion functor. Then there is a natural transformation J* :S ! S rest, essentially the forgetful functor from cosimplical spaces to restricted cosimplical spaces. Definition 2.2 For a general ring spectrum with unit E, define X^E, the E-completion of X, to be holim-bTEX. Strictly speaking, this definition only requires E to have a unit. How- ever, we shall need E to have a ring structure directly after the next definition, which introduces an object lying between a cosimplicial space and a restricted cosimplicial space. Definition 2.3 A modified cosimplical space is a restricted cosimpli- cial space with codegeneracies that satisfy cosimplicial-like identities djdi = didj-1 i < j sjdi ' disj-1 i < j ' id i = j, j + 1 ' di-1sj i > j + 1 sjsi ' si-1sj i > j where the first identity is the usual cosimplicial identity, but the rest are required to hold only up to homotopy. Remark 2.4 If E is a ring spectrum with unit, then, for X 2 S, the triple (TE , OE, ~) induces a modified cosimplicial space which we also 21 denote by TE X. Clearly any cosimplicial space C is also a modified cosimplicial space and so if X is an S-algebra the two objects denoted TE X agree. Remark 2.5 Corollary 3.9 of [11] proves that Tot(C) = holim-(Crest) when C = J*Crest. In particular, if E is an S-algebra, the completion X^Edefined in [7] agrees with that of Definition 2.2. Remark 2.6 It is not possible to apply Tot to modified cosimplicial spaces. However, after applying ß* we obtain a cosimplicial group ß*TE X which we view as a diagram ß*TE X 2 A , where A is the category of abelian groups. Applying ß* to bTEX gives an object in A restwhich is J*(ß*TE X). For a wide class of diagrams X_2 SI Bousfield and Kan [10], XI 7.1, define a spectral sequence related to the groups ß*holim-X_. Definition 2.7 For X 2 S and E a ring spectrum with unit, define E*,*r(X), the E-theory Bousfield-Kan spectral sequence of X, as the Bousfield-Kan spectral sequence for bTEX 2 S rest. Theorem 2.8 Es,*2(X) is isomorphic to the homology of the unstable cobar complex. That is to say Es,*2(X) = ßsß*TE X Remark 2.9 Recall the cohomotopy ßsA_of a cosimplicial abelian group A_is defined [10], X 7.1, as the cohomology Hs(ch(A_), @) where (ch(A_), @) P is the cochain complex given by ch(A_)n = A_nand @ = (-1)idi. Proof of (2.8) Let I be either or rest. For X_2 SI the E2-page is given by Es,t2= lim-sßtX_ 22 ([10] page 309). Since ß*TE X is a cosimplicial group, lim-sß*TE X = ßsß*TE X ([10], XI 7.3 (i)) and it suffices to show that lim-sß*bTEX = lim-sß*TE X. For any fixed n, denote by K_I(n) 2 SI the diagrams of Eilenberg- Mac Lane spaces K(A, n) which correspond to ß*TE X 2 A and ß*bTEX 2 A restfor the respective I (see [10], XI 7.2). Then for s 6 n (again from [10], XI 7.2) lim-sß*TE X = ßn-s holim-K_(n) lim-sß*bTEX = ßn-s holim-K_ rest(n) However, J : rest! is left cofinal ([11] page 193). Thus J* :holim-K_ (n) ! holim-K_ rest(n) is a homotopy equivalence. Since n was arbitrary, it follows that lim-sß*TE X = lim-sß*bTEX for all s. 3 The Unstable Cobar Complex for E- theory Section 2 identifies the E2-page of the Bousfield-Kan spectral sequence for a ring spectrum with unit E as the homology of the cochain complex ch(ß*TE X). However, for practical purposes, as in [5, 7], etc., it is important to be able to reinterpret this in terms of a more manageable object, in practice as the homology of a sub-complex of the stable cobar 23 complex, i.e., as an Ext group over a more convenient (in particular, abelian) category. We suppose for this section that E is a Landweber exact ring spec- trum with unit and (largely for convenience) that E is p-local with coefficients E* concentrated in even dimensions. Let M be the cate- gory of free, graded E*-modules. Drawing on the results of [5, 6, 7] and those of sections 1 and 2, we introduce a certain associated abelian category U. Our main theorem is the following. Theorem 3.1 Suppose E is a Landweber exact ring spectrum with unit. Suppose M 2 M has E*-module generators only in odd degrees and suppose X is a space with E*(X) ~= (M) as coalgebras, where (M) is the E*-Hopf algebra defined by letting M be the submodule of primitives, i.e., (M) is the exterior algebra on M. Then the E2-term of the E- theory Bousfield-Kan spectral sequence of X can be identified as Es,t2(X) ~=ExtsU(E*(St), M). Example 3.2 Spaces X satisfying the hypotheses of the theorem in- clude torsion free H-spaces and odd dimensional spheres. We begin by defining functors G and U :M ! M. Here and below we draw on a number of the results of section 1 with F = E, i.e., in this section we deal only with the coalgebraic ring E*(E_*). Definition 3.3 For a free E*-module M define (a) G(M) to be E*(EM__0), where EM denotes the spectrum realizing the homology theory E*(-) M. E* (b) U(M) to be P G(M), the primitive elements in G(M). 24 Both G and U are functorial; they take values in M, the category of free E* modules, by the results of section 2. Remark 3.4 (a) As M is a free E*-module, it is helpful to observe that EM__0= 1 EM is a product of spaces in the spectrum associated to E indexed by a set of generators of M. In particular, if {gi} are a set of E* generators of M with gi in dimension |gi|, _ ! ` Y EM__0= 1 -|gi|E = E_-|gi|. i i W Q Moreover, with this notation, M ~=ß* i -|gi|E = ß* iE_-|gi|. (b) Note that G is closely related to the functor TE :S ! S of section 2. For a space X 2 S with E*(X) 2 M, there is an isomorphism G(E*(X)) ~=E*(TE (X)). (c) Note also that U(M) is identical to the construction UE (M) of section 1. There is of course a similar functor V :M ! M based on the indecomposable quotient of G(M) and given by the construction VE (M) of section 1, but it will play no part in the proof of Theorem 3.1. Proposition 3.5 The unit and product in E respectively induce natural transformations ffiG :G ! G2 fflG :G ! I making (G, ffiG , fflG ) a cotriple on the category M. There are similar natural transformations ffiU :U ! U2, fflU :U ! I making (U, ffiU , fflU ) also a cotriple on M and a sub-cotriple of (G, ffiG , fflG ). 25 Proof The proof is essentially as in sections 6 and 7 of [5]; moreover, with the first observations of Remark 3.4 the maps ffiG and fflG , for example, may be written explicitly. Alternatively, for Landweber exact E, given the definition (1.8) and Theorem 1.10, the result on (U, ffiU , fflU ) also follows from the coaction formulæ for the bi. Remark 3.6 As usual the cotriples define categories G and U of G, respectively U, coalgebras: writing C for either G or U, recall that a C coalgebra in M is an object M 2 M with a map _ :M ! CM such that fflC _ = IdM : M ! M and ffiC _ = (C_)_ :M ! C2M (see [5, x5] for details). In particular, recall that if M 2 M then CM is naturally a C coalgebra with map _ on CM ! C2M given by ffiC . There are adjoint functors -C! M C - J where J denotes the forgetful functor. The adjunction gives natural isomorphisms Hom C(D, CM) ~=Hom M (D, M) for any D 2 C (where we identify D with its image under the forgetful functor). Strictly speaking, we shall abuse notation and write C not only for the functor M ! C above, but also for the functor JC :M ! M of the cotriple (C, ffiC , fflC ) on M and for the other composite, CJ :C ! C, the functor of the adjoint triple (C, ~C , jC ) on C, as in [5, x5]. 26 For C = G or U and objects W 2 C we recall the notions of cosimpli- cial resolution over C, as in [8, 2.5] and [6, 2.2] and the resulting derived functors ExtC(E*, W ). Definition 3.7 A cosimplicial resolution, N, over C, of W 2 C consists of objects Nn 2 C for n > -1 and, for every pair of integers (i, n) with 0 6 i 6 n, coface and codegeneracy maps (in C) di:Nn-1 ! Nn , si:Nn+1 ! Nn satisfying the usual cosimplicial identities (cf. 2.3) and such that (a)N-1 = W ; (b)for n > 0 there is an Mn 2 M with Nn = CMn; (c)Hn(JN) = 0 for n > -1. Here J :C ! M is the forgetful functor and the homology of JN is the homology of the cochain complex with groups JNn and boundary maps (-1)iJdi. The Ext groups ExtC(E*, W ) are then defined as the homology of chain complex associated to Hom C(E*, gJN), where gJNdenotes the unaug- mented complex 0 ! JN0 ! JN1 ! JN2 ! . ... These are the derived functors of Hom C(E*, -) by [8]. Example 3.8 The C cobar complex provides a standard example of a cosimplicial resolution. We illustrate it for C = U; the case of G is similar. 27 For W 2 U, consider the resolution with qth module Uq+1(W ). The maps in the U resolution are displayed in the diagram of E*-modules -d0! 0 d1 W -d! U(W ) -! . . . -s0 and are defined in terms of the triple (U, ~U , jU ) by di= UijU Un-i : Un (W ) ! Un+1 (W ), 0 6 i 6 n, si= Ui~U Un-i : Un+2 (W ) ! Un+1 (W ), 0 6 i 6 n. The U cobar complex is then the complex W -@! U(W ) -@!U2(W ) -@! . . . P n where @ = i=0(-1)ndi:Un (W ) ! Un+1 (W ). The embedding of the primitives in the stable cooperations, (1.7) and (1.10), shows that the acyclicity condition is satisfied since there is an extra codegeneracy s-1 :Uq+1(W ) ! Uq(W ) induced by the counit in E*(E): Uq+1(C) ! E*(E) Uq(C)-ffl!1Uq(C). In particular, again by (1.10), ExtU (E*, W ) is the homology of a sub-complex of the stable cobar complex. These constructions and the link between the functors G and TE of Remark 3.4(b) allow us to rewrite Theorem 2.8 as follows. Theorem 3.9 For E a ring spectrum with unit and X 2 S such that E*(X) 2 M, there is a natural isomorphism Es,t2(X) = ExtsG(E*(St), E*(X)). 28 Theorem 3.1 will now follow upon proving Theorem 3.10 Suppose E is a Landweber exact ring spectrum with unit. For M 2 M with generators in odd degree and (M) denoting the exterior algebra on M with M (M) the submodule of primitives, there is a natural isomorphism ExtsG(E*(St), (M)) ~=ExtsU(E*(St), M). Proof Let us write UM for the U cobar complex as in Example 3.8, i.e., with qth space Uq+1(M). Applying the functor (-) gives a complex UM : (M) ! (U(M)) ! (U2(M)) ! . ... Now let Y q= G(Uq(M)) for q > 0. Since M is concentrated in odd degrees the same is true for Uq(M). By the theorems (1.4) and (1.10) we have natural isomorphisms G(Uq(M)) ~= (Uq+1(M)) and we can identify the complex UM as a complex Y : (M) ! G(M) ! G(U(M)) ! G(U2(M)) ! . ... The maps in Y are coalgebra maps and E*(E)-comodule maps. By [5, 7.3] the maps are in G (note that [5, 7.3] does not require the assumption [5, 7.7] that the homology of the spaces in the -spectrum be cofree coalgebras - this is not satisfied in general). The extra codegeneracy in the U cobar complex passes via to an extra codegeneracy in Y, showing Y to be acyclic. Thus Y is a G-resolution of (M). 29 The Ext groups ExtsG(E*(St), (M)) can be obtained using the com- plex Y by computing the homology of the complex Hom G(E*(St), Y s) = Hom G(E*(St), G(Us(M))). However, by the adjunction isomorphism mentioned in Remark 3.6 (ap- plied twice), shows Hom G(E*(St), G(Us(M))) = Hom M (E*(St), Us(M)) = Hom U(E*(St), Us+1(M)). Thus ExtG(E*(St), (M)) is isomorphic to the homology of the U-cobar complex which by definition is precisely ExtU(E*(St), M). Remark 3.11 The results of section 1 on the algebra structure of E*(E_*) allow further results to follow. For example, suppose for M 2 M we write oe-1M for the isomorphic E*-module with degrees shifted downward by one, i.e., we let oe-1Mt = Mt+1. Then Theorem 1.4 and its proof shows oe-1U(M) = QG(oe-1M) . If we take M = E*(S2n+1) then E*( S2n+1) = oe-1M and an argument similar to that for BP -theory in [6], x6, shows that the complex Y used in the proof of Theorem 3.10 may also be used to compute the E2-page of the E-theory Bousfield-Kan spectral sequence for S2n+1: for any odd dimensional sphere S2n+1 there is an isomorphism Es,t-12( S2n+1) ~=Es,t2(S2n+1) . 30 Acknowledgements Both authors are pleased to thank the Japan-U.S. Mathematics Institute (JAMI), Johns Hopkins University, and its or- ganisers J.M. Boardman, D. Davis, J.-P. Meyer, J. Morava, G. Nishida, W.S. Wilson, and N. Yagita for support during Spring 2000 and at which this research was initiated. The first author thanks also Em- manuel Dror-Farjoun for helpful discussions on the material in section 2. The second author thanks the University of Leicester for sabbatical leave and the Leverhulme Foundation for a Research Fellowship during which most of this paper was written. References [1]J. F. 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