THE GENERALIZED HOMOLOGY OF PRODUCTS MARK HOVEY Abstract.We constructQa spectral sequence that computes the generalized homology E*( Xff) of a product of spectra. The E2-term of this spectral sequence consists of the right derived functors of product in the catego* *ry of E*E-comodules, and the spectral sequence always converges when E is the Johnson-Wilson theory E(n) and the Xffare Ln-local. We are able to prove some results about the E2-term of this spectral sequence; in particular,* * we show that the E(n)-homology of a product of E(n)-module spectra Xffis ju* *st the comodule product of the E(n)*Xff. This spectral sequence is relevant* * to the chromatic splitting conjecture. Introduction The basic tools of computation in algebraic topology are homology theories. Homology theories preserve coproducts, but can behave very badly on products. There are examples of homology theories E and setsQof spectra (generalized spac* *es) {Xff}, for which E*Xff= 0 for all i and yet E*( ffXff) 6= 0. Indeed, we can ta* *ke E = HQ, rational homology, where we have (HQ)*(HZ=pk) = 0 for all k, but Y Y (HQ)*( HZ=pk) = ( Z=pk) Q 6= 0 k k since, for example, the element (1, 1, 1, . .).is not torsion. Despite this counterexample,Qin this paper we build a spectral sequence that converges to E*( Xff) in good cases. The most important good case is when E = E(n), the Johnson-Wilson theory of great importance in stable homotopy theory. The E2 term of this spectral sequence is made up of the right derived functors of product applied to {E*Xff}. Of course, the product is exact in the category of E*-modules, so these derived functors are instead taken in the cate* *gory of E*E-comodules, where products remain mysterious. The usefulness of this spectral sequence will depend on our knowledge of its E2-term.Q At this point, the author knows very little about the derived functors s ff E(n)*E(n)M of product in the categoryQof E(n)*E(n)-comodules. The most important conjecture about them is that sE(n)*E(n)Mff= 0 for all s N for some N, so that the spectral sequence has a horizontal vanishing line at the E2 term (we show that the spectral sequence does have a horizontal vanishing line * *at some Er term). We expect that N is very close to n itself. ____________ Date: March 16, 2005. 1991 Mathematics Subject Classification. 55N45, 55P60, 55T99, 18G10, 16W30. 1 2 MARK HOVEY We do prove that derived functors of product can be computed using relatively injective resolutions, such as the cobar complex, rather that honest injective * *reso- lutions. It follows that Y Y E(n)*( Xff) ~= E(n)*Xff E(n)*E(n) for a family of E(n)-module spectra Xff. We also construct a spectral sequence relating derived functors of product in the category of E(n)*E(n)-comodules to derived functors of product in the category of BP*BP -comodules. The category of BP*BP -comodules is easier to cope with since BP*BP is connective and free over BP*. These results give the author hope that these derived functors will be understood at some point, though at the moment he does not even understand them in the simple case of E(1)*E(1)-comodules. The reason for the author's interest in this spectral sequence is the chromat* *ic splitting conjecture [5] of Mike Hopkins. Recall that the simplest form of the chromatic splitting conjecture is that K(n - 1)*LK(n)X is a direct sum of two copies of K(n - 1)*X, for X a finite p-complete spectrum. Also recall that LK(n* *)X is a homotopy inverse limit holimI(LnX ^ S=I) analogous to completion at the ideal (p, v1, . .,.vn-1). This result is due to Hopkins; a precise statement of* * it can be found in [8, Proposition 7.10]. Therefore, if one has a spectral sequence f* *or the E(n - 1)-homology of a homotopy inverse limit, one might be able to compute E(n - 1)*(LK(n)X) and therefore K(n - 1)*(LK(n)X). This approach to the chromatic splitting conjecture is due to Mike Hopkins, a* *nd is based on the work of Paul Goerss [4], who constructed a spectral sequence fo* *r the mod p homology of a homotopy inverse limit of spaces. Hopkins suggested this id* *ea to Hal Sadofsky and the author after a talk by Goerss. Sadofsky has constructed* * a spectral sequence for the E(n)-homology of a homotopy inverse limit, as envisio* *ned by Hopkins, and has proved some results about it that are relevant to the chrom* *atic splitting conjecture. Unfortunately, Sadofsky has not yet made a preprint of h* *is work available. The author decided instead to begin with the simpler case of products, though the methods used in this paper can also be used to construct a version of Sadof- sky's spectral sequence. To the author's knowledge, Sadofsky has not considered products. But the author acknowledges his heavy debt to the work of Sadofsky. He also would like to thank Mike Hopkins for his original suggestion, and Paul Goe* *rss for his paper [4], without which this paper would never have been written. 1.The modified Adams tower The first step in constructing a spectral sequence is to resolve the object o* *ne is considering. In our case, the resolution we need is called the modified Adams t* *ower and is due to Devinatz and Hopkins [3]. The idea is to mimic the usual construc* *tion of an injective resolution using E*-injectives, where E is a well-behaved homol* *ogy theory. We will have to assume that E is a commutative ring spectrum such that E*E is flat over E*; it is well-known [11, Proposition 2.2.8] that this implies* * that (E*, E*E) is a flat Hopf algebroid and that E*X is naturally a left E*E-comodule for a spectrum X. It also implies that E*E-comodules form an abelian category [* *11, Theorem A1.1.3] with enough injectives [11, Lemma A1.2.2]. The following definition is taken from [3]. THE GENERALIZED HOMOLOGY OF PRODUCTS 3 Definition 1.1. Let E be a commutative ring spectrum such that E*E is flat over E*. Define a functor D from injective E*E-comodules to the stable homotopy category S as follows. Given an injective E*E-comodule I, consider the functor * *DI from spectra to abelian groups defined by DI(X) = Hom E*E(E*X, I). Then DI is a cohomology functor, so there is a unique spectrum D(I) such that there is a natural isomorphism DI(X) ~=[X, D(I)]. The hypotheses we have given on E are sufficient to define D(I), but appar- ently insufficient to compute E*D(I). For this we need some form of the followi* *ng definition; this particular form comes from [6]. Definition 1.2. A ring spectrum E is called topologically flat if E is the mini* *mal weak colimit of a filtered diagram of finite spectra Xi such that E*Xi is a fin* *itely generated projective E*-module. Minimal weak colimits are discussed in [7, Section 2.2]. Adams [1, Section II* *I.13] proves that many standard spectra such as BP are topologically flat; in additio* *n, any Landweber exact commutative ring spectrum over BP or MU is topologically flat [6, Theorem 1.4.9]. Note that if E is topologically flat, then E*E is flat* * over E*, since it is the colimit of projective modules. The following theorem is a translation of Theorem 1.5 of [3] to this terminol* *ogy. Theorem 1.3. Suppose E is a topologically flat commutative ring spectrum, and I is an injective E*E-comodule. Then there is a natural isomorphism E*D(I) ~=I. We can now describe the modified Adams tower. Let E be a topologically flat commutative ring spectrum, and suppose we have a spectrum X. Let C = E*X, and choose an injective resolution 0 -!C j-!I0 o0-!I1 o1-!. . . of C in the category of E*E-comodules. Let js: Cs -!Is denote the kernel of os, so that j0 = j. As explained in [3, Section 1], we can use this resolution of C to build a to* *wer over X with good properties. More precisely, we have the following lemma, which is easily proved by induction on n. Lemma 1.4. Let E be a topologically flat commutative ring spectrum, let X be a spectrum, and choose an injective resolution of E*X as above. Then there is a tower g g g X = X0 ---0- X1 ---1- X2 ---2- . . . ?? ? yf0 ?yf1 K0 K1 over X satisfying the following properties. (a) Ks = -sD(Is). (b) Xs+1 is the fiber of fs. (c) E*Xs ~= -sCs. (d) The map fs is induced by the inclusion Cs -!Is. 4 MARK HOVEY (e) E*gs = 0, and the boundary map Ks -! Xs+1 induces the surjection -sIs -! -sCs+1 on E*-homology. We call this tower the modified Adams tower for X based on E-homology. Of course, it actually depends on the injective resolution as well. We obtain a sp* *ectral sequence by applying [Z, -] for any Z to get the modified Adams spectral sequen* *ce of Devinatz [3]; its E2-term is Ext**E*E(E*X, E*Y ), it is independent of the c* *hoice of resolution from the E2 page on, and in good cases it converges to [Z, LE X]*. 2.Products of comodules In order to understand the spectral sequence for products of spectra, we need to know a little about products of comodules. So suppose (A, ) is a flat Hopf algebroid. As mentioned above, basic facts about the category of -comodules can be found in [11, Appendix 1], though he does not discuss products. A more in-de* *pth look at the global structure of the category of -comodules, including products* *, can be found in [6]. The main point of interest here is that the forgetful functor to A-modules do* *es not preserve products. It is easiest to understand this when is freePover A. * *In this case, every element m in a -comodule M has a diagonal of the form fli mi, where fli runs through a basis of as a rightQA-module, and all but finitely m* *any of the mi are zero. In the A-module product ffMffof comodules Mff, there may well be elements whose diagonal would have toQbe infinitely long. In fact,Qwhen is projective over A, the comodule product Mffis the submodule of Mff consisting of those elements whose diagonal lands in Y Y A Mff ( A Mff). To construct the product when is only assumed to be flat over A, one first checks that Y Y ( A Nff) ~= A ( Nff) for A-modules Nff, where A P denotes the extendedQ -comodule, in which coacts only on the factor. One then constructs fff, where fffis an arbitra* *ry map of extended comodules. Finally, given arbitrary comodules Mff, we have exact sequences of comodules ff 0 -!Mff-_! A Mff-f-! A Nff, where Nffis the cokernel of _, and fffis the composite A Mff-! Nff-_! A Nff. Q Q It follows that Mff~=ker fff. Details can be found in [6]. This construction shows that the product of comodules is more complicated than one would want; in particular, it is not always exact (see the example bef* *ore Proposition 1.2.3 of [6]). As a right adjoint, of course, the product is left * *exact. Since thereQare enough injective -comodules, the product will have right deriv* *ed functors sMfffor s 0. Almost nothing is known about these right derived functors, but they are what will appear as the E2-term in our spectral sequence. For the construction of our spectral sequence, we need the following proposit* *ion. THE GENERALIZED HOMOLOGY OF PRODUCTS 5 Proposition 2.1. Suppose E is a topologically flat commutative ring spectrum, a* *nd {Iff} is a family of injective E*E-comodules. Then there is a natural isomorphi* *sm Y Y D( Iff) -! D(Iff). E*E Q Here the notation E*Edenotes the product in the category of E*E-comodules. Q Proof.Note that E*EIffis again an injective comodule. The functoriality of D guarantees the existence of this map. Now, if X is an arbitrary spectrum, we ha* *ve a chain of isomorphisms Y Y Y [X, D( Iff)] ~=Hom E*E(E*X, Iff) ~= Hom E*E(E*X, Iff) E*E E*E ~=Y [X, D(Iff)] ~=[X, Y D(Iff)]. This gives us the desired isomorphism. 3.Construction of the spectral sequence We can now use the modified Adams towers of Lemma 1.4 to construct our spectral sequence. Theorem 3.1. Let E be a topologically flat commutative ring spectrum, and let {Xff} be a family of spectra. There is a natural spectral sequence E***({Xff}) * *with ds,tr:Es,tr-!Es+r,t+r-1rand E2-term Q s Es,t2~=( E*E E*Xff)t. This is a spectral sequence of E*E-comodules, in the sense that each ds,*ris a * *map of E*E comodules of degree r - 1. Furthermore, every element in E0,t2in the ima* *ge of the natural map M Q E*Xff-! E*E E*Xff is a permanent cycle. Proof.We have modified Adams towers Xffsfor each Xff. Taking the product gives us the tower below. Q Q gff0Q Q gff1Q Q gff2 Xff ---- Xff1---- Xff2---- . . . ? ? (3.2) ?yQfff0 ?yQfff1 Q Q Kff0 Kff1 By applying E*-homology, we getQan associated exact coupleQand spectral sequenc* *e. That is, we let Ds,t1= Et-s( Xffs) andQEs,t1= Et-s( Kffs). We define i1: D -!D of bidegree (-1,Q-1) by is,t1= Et-s( gffs), we define j1: D -!E of bidegree (0* *, 0) by js,t1= Et-s( fffs), and weQdefine k1:QE -!D of bidegree (1, 0) in bidegree * *(s, t) to be Et-sof the boundary map Kffs-! Xffs+1. All of these maps are maps of comodules, and therefore the resulting spectral sequence will be a spectral seq* *uence of comodules, as claimed. By combining Proposition 2.1 with Theorem 1.3, we see that Y Y Y Es,t1~=Et-s( -sD(Iffs)) ~=EtD( Iffs) ~=( Iffs)t. E*E E*E 6 MARK HOVEY Q One can easily check that the first differential d1 is E*Eoffs, and therefore* * that the E2-term is as claimed. Nautrality now follows in the usual way; a collection of maps Xff-!Y ffinduces non-canonical maps of the injective resolutions in question, and hence the modi* *fied Adams towers. Taking products gives us a map of spectral sequences, which is canonical from E2 onwards. Finally, we can also construct a spectral sequence by taking the wedge of the modified Adams towers ofLthe Xffand applyingLE* homology. This gives a spectral sequence with Ds,t1= ( Cffs)tLand Es,t1= ( Iffs)t. The d1 differential is * *the obvious one, and so E0,*2~= E*Xffand Es,t2= 0 for s > 0. There is a map from the spectral sequence to the spectral sequence for the product of the Xff. Anyt* *hing in the image of this map of spectral sequences must be a permanent cycle. 4.Convergence of the spectral sequence We now discuss the convergence of our spectral sequence. This is a delicate question, in general, as the example given at the beginning of the paper shows. However, the spectral sequence always converges when E = E(n) and each Xffis E(n)-local. Theorem 4.1. Suppose E = E(n) and each Xffis Ln-local.Q Then the spectral sequence of Theorem 3.1 converges strongly to E(n)*( Xff). Furthermore, it has a horizontal vanishing line at some Er term. Proof.First note that each Xffsis Ln-local, since Kffs= -sD(Iffs) is clearly L* *n- local. Each map gffs:Xffs+1-!Xffshas E(n)*(gffs) = 0. It follows from [10, Theo- rem 5.10] that there is an N, depending on n but independentQof ff, suchQthat e* *ach N-fold composite Xffs+N-!Xffsis null. Hence each composite Xffs+N-! Xffsis null, giving us our desired horizontal vanishing line. Hence Y Y limsE(n)*( Xffs) = lim1sE(n)*( Xffs) = 0 Q so the spectral sequence converges conditionally to E(n)*( Xff) [2]. It is al* *so clear that lim1rEs,tr= 0, and so the spectral sequence converges strongly as we* *ll [2, Theorem 7.3]. 5. Relatively injective resolutions and an application Although we cannot prove very much about the derived functors of products, we can at least show that one can use relatively injective comodules to compute th* *em. This allows us to compute the E(n)-homology of products of E(n)-module spectra. Proposition 5.1. Let (A, ) be a flat Hopf algebroid,Qand suppose Mffis a rela- tively injective -comodule for all ff. Then sMff= 0 for s > 0. Proof.Since Mffis relativelyQinjective, it is a retract of A Mff. It theref* *ore suffices to show that s( A Mff) = 0 for all s > 0. To do so, choose an inje* *ctive resolution Iff*of Mffin the category of A-modules. Since is flat over A, A * *Iff*is a resolution of A Mffin the category of -comodules. Furthermore, each A If* *fs is an injective -comodule [11, Lemma A1.2.2]. Hence Qs ff ~ s Q ff ~ s Q ff ( A M ) = H ( ( A I* )) = H ( A ( I* )). Since products are exact on the category of A-modules, and since is flat, the* *se groups are 0 for s > 0. THE GENERALIZED HOMOLOGY OF PRODUCTS 7 This yields an immediate topological corollary. Corollary 5.2. Suppose Xffis an E(n)-module spectrum for all ff. Then Y Y E(n)*( Xff) ~= E(n)*(Xff). E(n)*E(n) In particular, Y Y E(n)*( E(n) ^ Xff) ~=E(n)*E(n) E(n)*( E(n)*Xff). Proof.Since Xffis an E(n)-module spectrum, it is Ln-local. Furthermore, E(n)*Xff is a retract of E(n)*(E(n) ^ Xff) ~=E(n)*E(n) E(n)*E(n)*Xff, so is relatively injective. Proposition 5.1 then implies that the E2-term of o* *ur spectral sequence is 0 except in bidegree (0, t). It therefore collapses, and w* *e get the desired isomorphism. It also follows, using standard homological algebra, that we can use relative* *ly injective resolutions to compute the derived functors of product. For example, * *we can use the cobar resolution C*(M) described in [11, Definition A1.2.10]. Corollary 5.3. Suppose (A, ) is a flat Hopf algebroid, and {Mff} is a set of -comodules. Let C*(Mff) denote the cobar resolution on Mff. Then Q s ff~ s Q * ff M = H C M . This corollary tellsQus, for example, that if JMff= 0 for some invariant idea* *l J and all ff, then J sMff= 0 for all s. 6.BP*BP-comodules and E(n)*E(n)-comodules In this section, we exploit the close relationship between BP*BP -comodules and E(n)*E(n)-comodules studied in [9] to get some partial understanding of the product of comodules. We begin with BP*BP -comodules, which are easier to handle because BP*BP is connective and projective over BP*. As mentioned inQSection 2, the product o* *f a family {Mff} of BP*BP -comodules is the submodule of Mffconsisting of those elements whose diagonal has finite length. Definition 6.1. A family of BP*BP -comodules {Mff} is uniformly bounded below if there is a d 2 Z such that Mffn= 0 for all n < d and all ff. The product and its derived functors are particularly simple for a uniformly bounded below family. Theorem 6.2. Suppose (A, ) = (BP*, BP*BP ), and {Mff} is a family of - comodules that is uniformly bounded below. Then Q ff~ Q ff Q s ff M = M and M = 0 for alls > 0. Q Proof.Since every element of Mffmust have finite diagonal, the first statement is clear. For the second statement, consider the cobar_resolution_C*Mffof Mff by relatively injective comodules. We have CsMff= A s A Mff, so, since 8 MARK HOVEY is connective, the family {CsMff} is uniformly bounded below for each s. We therefore have Q s ff~ s Q ff~ s Q ff M = H C*M = H C*M = 0 for s > 0, using Corollary 5.3 and the fact that products of modules are exact. To relate this to E(n)*E(n)-comodules, we recall from [9] and [10] the exact functor * from BP*BP -comodules to E(n)*E(n)-comodules defined by *M = E(n)* BP* M. The functor * has a fully faithful right adjoint *, the composite * * is naturally isomorphic to the identity, and the composite Ln = * * is the localization functor on the category of BP*BP -comodules with respect to the hereditary torsion theory of vn-torsion comodules. The functor Ln is left exact* *, but has right derived functors Lqnfor 0 q n, studied in [10]. As a left adjoint, we do not expect * to preserve products. We do, however, have the following result. Theorem 6.3. Suppose {Mff} is a family of BP*BP -comodules. Then there is a natural isomorphism Y Y *Mff-! *( LnMff). E(n)*E(n) BP*BP In fact, there is a convergent first quadrant spectral sequence Ep,qrof E(n)*E(* *n)- comodules with Q p Q p+q Ep,q2~= *( BP*BP (LqnMff)) ) E(n)*E(n) *Mff. Proof.Since * is a right adjoint, we have Y Y Y *Mff~= * *( *Mff) ~= * (LnMff), E(n)*E(n) E(n)*E(n) BP*BP as required. The spectral sequence is the Grothendieck spectral sequence for the derived functors of the composition, described in [12, Section 5.8]. Recall tha* *t this spectral sequence has Ep,q2= (RpF )(RqG)(-) and converges to Rp+q(F G)(-), un- der the assumption that (RpFQ)(GI) = 0 for all injectives I and p > 0. In the c* *ase at hand, the functor F is * BP*BP (-) and the functor G is Ln (applied object- wise to the product category). Since Ln preserves injectives [10, Corollary 2.* *4], the Grothendieck spectral sequence exists. Since * is exact and products of in* *jec- tives are injective,Qwe can use another Grothendieck spectral sequence argument* * to see that RpF = * pBP*BP(-). Similarly, since * is exact and preserves injec- tives [10, Corollary 2.5], another Grothendieck spectral sequence argument shows that Q Q Rp+q(F G)(-) = Rp+q( E(n)*E(n) *)(-) = p+qE(n)*E(n) *(-), completing the proof. This proposition allows us to compute some products of E(n)*E(n)-comodules. For example, we have Y Y ffE(n)* ~=E(n)* BP* ffBP*, E(n)*E(n) as long as the ff are bounded below. To see this, use the fact that BP* is Ln- local [10], Theorem 6.2, and Theorem 6.3. THE GENERALIZED HOMOLOGY OF PRODUCTS 9 In fact, we have Q s ff E(n)*E(n) E(n)* = 0 for 0 < s < n and Qs ff ~ Q s-n ff 1 1 1 E(n)*E(n) E(n)* = * BP*BP BP*=(p , v1 , . .,.vn ) for s n, again under the hypothesis that the ff are bounded below. This follo* *ws from the spectral sequence of Theorem 6.3 and the fact [10] that LqnBP* = 0 exc* *ept when q = 0 and q = n, where L0nBP* = BP* and LnnBP* = BP*=(p1 , v11, . .,.v1n). Q Note that we do not know whether BP*BP ffBP*=(p1 ,Qv11, . .,.v1n) is all vn- torsion or not, and therefore we do not know whether nE(n)*E(n) ffE(n)* is ze* *ro or not. References [1]J. F. Adams, Stable homotopy and generalised homology, Chicago Lectures in * *Mathematics (University of Chicago Press, Chicago, Ill., 1974). MR 53 #6534 [2]J. Michael Boardman, Conditionally convergent spectral sequences, Homotopy * *invariant al- gebraic structures (Baltimore, MD, 1998), Contemp. Math., vol. 239, Amer. Ma* *th. Soc., Providence, RI, 1999, pp. 49-84. MR 2000m:55024 [3]Ethan S. Devinatz, Morava modules and Brown-Comenetz duality, Amer. J. Math* *. 119 (1997), no. 4, 741-770. MR 98i:55008 [4]Paul G. Goerss, The homology of homotopy inverse limits, J. Pure Appl. Alge* *bra 111 (1996), no. 1-3, 83-122. [5]Mark Hovey, Bousfield localization functors and Hopkins' chromatic splittin* *g conjecture, The ~Cech centennial (Boston, MA, 1993), Contemp. Math., vol. 181, Amer. Mat* *h. 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A. Weibel, An introduction to homological algebra (Cambridge University * *Press, 1994). Department of Mathematics, Wesleyan University, Middletown, CT 06459 E-mail address: hovey@member.ams.org