LOCAL COHOMOLOGY OF BP*BP -COMODULES MARK HOVEY AND NEIL STRICKLAND Abstract.Given a spectrum X, we construct a spectral sequence of BP*BP- comodules that converges to BP*(LnX), where LnX is the Bousfield localiz* *a- tion of X with respect to the Johnson-Wilson theory E(n)*. The E2-term of this spectral sequence consists of the derived functors of an algebraic * *version of Ln. We show how to calculate these derived functors, which are closely related to local cohomology of BP*-modules with respect to the ideal In+* *1. Introduction The most common approach to understanding stable homotopy theory involves localization. One first localizes at a fixed prime p; after doing so there is a* * tower of localization functors . .L.n-!Ln-1 -!. .-.!L1 -!L0. called the chromatic tower [Rav92 , Section 7.5]. Each functor Ln retains a li* *ttle more information than the previous one Ln-1; the chromatic convergence theo- rem [Rav92 , Theorem 7.5.7] says that the homotopy inverse limit of the LnX is X itself for a finite p-local spectrum X. These localization functors come from * *the Brown-Peterson homology theory BP , where BP*(S0) ~=Z(p)[v1, v2, . .]. with |vi| = 2(pi- 1). There are different possible choices for the generators v* *n, but the ideals In = (p, v1, . .,.vn-1) for 0 n 1 are canonical. The functor Ln * *is Bousfield localization with respect to v-1nBP ; all the different choices for v* *n give the same localization functor. In previous work [HS03 ], the authors constructed an algebraic endofunctor Ln on the category of BP*BP -comodules, analogous to the chromatic localization Ln on spectra. This functor Ln is the localization obtained by inverting all maps * *of comodules whose kernel and cokernel are vn-torsion (or, equivalently, In+1-tors* *ion). The Ln-local comodules are equivalent to the category of E(n)*E(n)-comodules, or to the category of E*E-comodules for any Landweber exact commutative ring spectrum with E*=In+1 = 0 but E*=In 6= 0. In [HS03 ], our main interest was algebraic. In this paper, we compare our al* *ge- braic version of Ln with the topological one. As always, when one has a topolog* *ical version of an algebraic construction, one expects a spectral sequence converging to the topological construction whose E2-term involves the derived functors of * *the algebraic construction. Since the algebraic Ln is left exact, it has right der* *ived functors Lin. We prove the following theorem. ____________ Date: July 22, 2004. 1 2 MARK HOVEY AND NEIL STRICKLAND Theorem A. Let X be a spectrum. There is a natural spectral sequence E***(X) with dr: Es,tr-! Es+r,t+r-1rand E2-term Es,t2(X) ~= (LsnBP*X)t, converging strongly to BPt-s(LnX). This is a spectral sequence of BP*BP -comodules, in the sense that Es,*ris a graded BP*BP -comodule for all r 2 and dr: Es,*r-!Es+r,*r is a BP*BP -comodule map of degree r - 1. Furthermore, every element in E0,*2 that comes from BP*X is a permanent cycle. For this spectral sequence to be useful, we need to be able to compute the E2- term. The derived functors Linturn out to be closely related to local cohomolog* *y, which is well-known in commutative algebra and was introduced to algebraic topo* *l- ogy by Greenlees [Gre93]. Recall from [GM95 ] that, given an ideal I in a ring * *R, one can form the local cohomology H*I(-) and the Cech cohomology ~CH*I(-) of an R-module M. Although it is not phrased this way in [GM95 ], the functor ~CH0In+1 on the category of BP*-modules is the localization functor that inverts all maps of modules whose kernel and cokernel are vn-torsion. Thus ~CH0In+1is the analog of Ln in the category of BP*-modules, and hence Cech cohomology is simply the derived functors of this localization functor on the category of BP*-modules. The following theorem describes the behavior of Ln itself. Theorem B. Let M be a BP*BP -comodule. (1) LnM ~=~CH0In+1M. (2) If vj acts isomorphically on M for some 0 j n, then LnM = M. (3) 8 >>>BP* Q ifk = n = 0; 0; Ln(BP*=Ik) = > n * n >>:BP*=Ik ifk < n; 0 ifk > n. (4) Ln commutes with filtered colimits, arbitrary direct sums, and finite li* *mits. Part (1) of this theorem is proved in Theorem 4.5, part (2) in Proposition 1.* *6, part (3) in Proposition 1.1(c), Corollary 1.5, and Corollary 1.7, and part (4) * *in Proposition 1.8. The derived functors of Ln are described in the following theorem. Theorem C. Let M be a BP*BP -comodule. (1) We have LinM ~=~CHiIn+1M. (2) Lin(M) = 0 for i > n. (3) Lin(M) is In+1-torsion for all i > 0. (4) If vj acts isomorphically on M for some j, then LinM = 0 for i > 0. (5) If i > 0, then Lin(BP*=Ik) = 0 unless k < n and i = n - k, in which case we have Ln-kn(BP*=Ik) = BP*=(p, v1, . .,.vk-1, v1k, . .,.v1n). (6) For i > 0, Lincommutes with filtered colimits and arbitrary direct sums. Part (1) of Theorem C is proved in Theorem 4.5, part (2) in Theorem 3.7, part* * (3) in Proposition 3.4, part (4) in Theorem 3.5, part (5) in Corollary 3.6, and par* *t (6) in Theorem 3.8. Most of Theorem C, except part (6), would follow from part (1) of it, and kno* *wn facts about local cohomology. However, local cohomology is generally considered LOCAL COHOMOLOGY OF BP*BP-COMODULES 3 only for Noetherian rings, and BP* is not Noetherian. This turns out not to be a problem, but because there is no discussion of non-Noetherian local cohomology in the literature, and because it is not very hard, we offer direct proofs of t* *he remaining parts of Theorem C. In the light of Theorem C the reader may naturally wonder whether there is a connection between the local cohomology spectral sequence of [Gre93] and [GM95 ] and our spectral sequence. Recall that Greenlees and May begin with the category of modules over a strictly commutative ring spectrum; since BP is not known to be such, we must begin with MU. Combining Theorems 5.1 and 6.1 of [GM95 ], and applying them to the MU-module spectrum MU ^ X, then gives a spectral sequence converging to MU*(LnX) whose E2-term is ~CH-s,-tIn+1(MU*X). Our spec- tral sequence would have the same E2-term as this Greenlees-May spectral sequen* *ce if we used MU instead of BP (and we reindexed the spectral sequence). However, our construction allows us to conclude that we have a spectral sequence of como* *d- ules, which the Greenlees-May construction does not. This significantly restric* *ts the possible differentials and extensions that can occur in the spectral sequen* *ce. The authors would like to thank the Isaac Newton Institute for Mathematical Sciences for its support during our collaboration. We also thank John Greenlees for many helpful conversations on the subject matter of this paper. 1. The functor Ln In this section, we define our localization functor Ln and prove almost all of Theorem B, though we postpone Theorem B(1) to Section 4. In this section, and throughout the paper, n will be a fixed nonnegative inte* *ger, and : (BP*, BP*BP ) -!(E(n)*, E(n)*E(n)) will denote the evident map of Hopf algebroids. There is an induced exact funct* *or *: BP*BP -comod-! E(n)*E(n)-comod of the categories of graded comodules that takes M to E(n)* BP* M. As explained in [Hov04 , Proposition 1.2.3], * has a left exact right adjoint *: E(n)*E(n)-comod -!BP*BP -comod. On relatively injective comodules, * is defined by *(E(n)*E(n) E(n)*N) = BP*BP BP* N. Since every comodule is a kernel of a map between relatively injective comodule* *s, and * is left exact, this determines * in general. Explicitly, as pointed out* * by Mark Behrens, *(N) = (BP*BP BP* E(n)*) E(n)*E(n)N, where the symbol denotes the cotensor product (see [Rav86 , Lemma A1.1.8]). We prove in Section 2 of [HS03 ] that * is a fully faithful embedding with * *M ~=M, and in Section 4 of that paper that the composite functor * * is localization with respect to the hereditary torsion theory consisting of vn-torsion comodule* *s. It is this composite * * that we denote by Ln. For a quick review of the theory 4 MARK HOVEY AND NEIL STRICKLAND of localization with respect to hereditary torsion theories, see the discussion* * im- mediately following Corollary 4.3. Since the vn-torsion comodules are the small* *est hereditary torsion theory containing BP*=In+1 [HS03 ], Ln is localization away * *from BP*=In+1 so is analogous to Lfnon the category of spectra. On the other hand, t* *he vn-torsion comodules are precisely the kernel of * [HS03 ], so Ln is also anal* *ogous to the functor Ln on the category of spectra. Because the collection of vn-torsion comodules is a hereditary torsion theory* *, the submodule TnM of vn-torsion elements in a comodule M is in fact a subcomodule. This also follows from [JY80 , Corollary 2.4] or [Lan79, Corollary 2]. The most basic facts about Ln are contained in the following proposition. Proposition 1.1. (a)Ln is left exact and idempotent. (b) Given a map f of comodules, Lnf is an isomorphism if and only if the kernel and cokernel of f are vn-torsion. (c) For a comodule M, LnM = 0 if and only if M is vn-torsion. In particular, Ln(BP*=Ik) = 0 for k > n. (d) For any comodule M, there is an exact sequence of comodules 0 -!TnM -!M -!LnM -!T 0-!0 where T 0is a vn-torsion comodule. (e) A comodule M is Ln-local if and only if Hom *BP*(BP*=In+1, M) = Ext1,*BP*BP(BP*=In+1, M) = 0. Remark. In part (e), note that the Hom group is defined in the category of BP*- modules, whereas the Ext group is defined in the category of BP*BP -comodules. However, we know from [Lan79, Corollary 4] that Hom *BP*BP(BP*=In+1, M) = 0 iff Hom *BP*(BP*=In+1, M) = 0 iff M has no vn-torsion. We will see in Lemma 1.3 that the Ext condition can also be reformulated in the category of modules. Proof.Parts (a), (b), and (c) are immediate consequences of the fact that Ln is localization with respect to the vn-torsion comodules. Part (e) is proven in Co* *rol- lary 4.3 of [HS03 ]. Part (d) follows from the other parts; since Ln is idempot* *ent, the map ': M -!LnM is an Ln-equivalence, so its kernel and cokernel are vn-torsion. Part (e) implies that LnM has no vn-torsion, so the kernel of ' is TnM. We now identify some Ln-local comodules. Proposition 1.2. Suppose j < n and M is a vj-1-torsion comodule on which (vj, vj+1) is a regular sequence. Then M is Ln-local. For this proposition, we need the following lemma. We will prove the converse of this lemma in Corollary 4.6. Lemma 1.3. Suppose M is a BP*BP -comodule with no vn-torsion such that Ext1,*BP*(BP*=In+1, M) = 0. Then M is Ln-local. Proof.We must show that Ext1BP*BP(BP*=In+1, M) = 0. So suppose we have a short exact sequence 0 -!M -!X p-!BP*=In+1 -!0 LOCAL COHOMOLOGY OF BP*BP-COMODULES 5 L of comodules. We know that X ~=M BP*=In+1 as BP*-modules, and TnM = 0, so p induces an isomorphism TnX -! Tn(BP*=In+1) = BP*=In+1 (of comodules). The inverse of this isomorphism splits the sequence. Proof of Proposition 1.2.First note that M has no vj-torsion, so M has no vn- torsion either. In light of Lemma 1.3, we must show that Ext1,*BP*(BP*=In+1, M) = 0. For notational simplicity, we will assume that * = 0, but the proof works for a* *ny value of *. Suppose we have a short exact sequence (1.4) 0 -!M f-!X g-!BP*=In+1 -!0 of BP*-modules. As in Lemma 1.3, we observe that g induces an injective map Tn+1X -! BP*=In+1; if we can show that this is also surjective, then the inverse will split the sequence, as required. Choose an x 2 X such that g(x) = 1. Then g(vjx) = g(vj+1x) = 0, so there are elements y and z in M such that f(y) = vjx and f(z) = vj+1x. Then f(vj+1y) = f(vjz), so = vj+1y = vjz. Since (vj, vj+1) is a regular sequence on * *M, we conclude that y = vjw for some w. This means vjvj+1w = vjz, so, since vj is not a zero-divisor on M, z = vj+1w. Now consider the element x0= x - f(w). We claim that this element defines a splitting of the short exact sequence 1.4. Ce* *rtainly g(x0) = 1 and vjx0= vj+1x0= 0. Now suppose k n. We claim that vkx0= 0. Certainly g(vkx0) = 0, so vkx0= f(t) for some t. But then f(vjt) = 0, so vjt = * *0. This forces t to be 0, as required. Proposition 1.2 immediately gives us part of Theorem B(3). Corollary 1.5. Suppose k < n. Then BP*=Ik is Ln-local. Another class of local comodules is given by the following proposition, which* * is Theorem B(2). Proposition 1.6. If vj acts invertibly on a BP*BP -comodule M for some j n, then M is Ln-local. Proof.If vj acts invertibly on M and j < n, then (vj, vj+1) is a regular se- quence on M, so Proposition 1.2 implies M is Ln-local. If vn acts invertibly on M, then certainly M has no vn-torsion. By Lemma 1.3, it suffices to show that Ext1,*BP*(BP*=In+1, M) = 0. Since vn acts by 0 on BP*=In+1, it also acts by 0 on this Ext1,*group. On the other hand, since vn acts isomorphically on M, it acts isomorphically on this Ext1,*group as well. Hence Ext1,*(BP*=In+1, M) = 0 as required. The following corollary completes the proof of Theorem B(3). Corollary 1.7. Suppose M is a vn-1-torsion BP*BP -comodule. Then LnM ~= v-1nM. Note that this includes the case n = 0, where we interpret v0 = p and v-1 = 0. Proof.Note that v-1nM is a BP*BP -comodule by [JY80 , Proposition 2.9]. The map M -!v-1nM obviously has vn-torsion kernel and cokernel, so is an Ln-equivalence. Proposition 1.6 implies that v-1nM is Ln-local, so it must be LnM. 6 MARK HOVEY AND NEIL STRICKLAND We conclude this section with the proof of Theorem B(4). Proposition 1.8. The functors * and Ln commute with filtered colimits, arbitra* *ry direct sums, and finite limits. Proof.The functor * is a right adjoint, so preserves all limits, and Ln is left exact, so preserves finite limits. To complete the proof, we show that * prese* *rves filtered colimits. It will then follow that * preserves arbitrary direct sums,* * which are filtered colimits of finite direct sums, and that Ln = * * preserves filte* *red colimits and arbitrary direct sums, completing the proof. So suppose Xiis a filtered diagram of E(n)*E(n)-comodules. There is certainly a natural map ff: colim *Xi-! *(colimXi). We need to recall that a BP*BP -comodule (resp. E(n)*E(n)-comodule) P is called dualizable if it is finitely generated and projective over BP* (resp. E(* *n)*), and that the dualizable comodules generate the category of BP*BP -comodules (resp. E(n)*E(n)-comodules). See [Hov04 , Section 1.4]. Thus, to show ff is an isomorphism, it suffices to check that it is an isomorphism upon applying BP*BP -comod(P, -) for any dualizable BP*BP -comodule P , since the dualizable comodules generate. The main point is that * preserves dualizable comodules, and dualizable comodules are finitely presented. This implies Hom BP*BP(P, colim *Xi) ~=colimHom BP*BP(P, *Xi) ~=colimHomE(n)*E(n)( *P, Xi) ~=Hom E(n)*E(n)( *P, colimXi) ~=Hom BP*BP (P, *(colimXi)), so * preserves filtered colimits. 2.Injective BP*BP -comodules In order to construct the spectral sequence of Theorem A and in order to comp* *ute the right derived functors Linof Ln, we need to known something about injective objects in the category of BP*BP -comodules. Very little seems to have been wri* *tten about these absolute injectives; relative injectives are easier to understand a* *nd have been used much more often. The object of this section is to learn a little more* *; in particular, we prove that Ln, *, *, and Tn all preserve injectives. The most basic fact about injective BP*BP -comodules is the following well- known lemma. Recall that a Hopf algebroid (A, ) is said to be flat if is fla* *t as a left (or, equivalently, right) A-module. Lemma 2.1. Let (A, ) be a flat Hopf algebroid. (a) If I is an injective A-module, then the extended -comodule A I is an injective -comodule. (b) There are enough injective -comodules. (c) A -comodule is injective if and only if it is a comodule retract of * *A I for some injective A-module I. Proof.Because the extended comodule functor is right adjoint to the forgetful f* *unc- tor from -comodules to A-modules, we have Hom (-, A M) ~=Hom A(-, M) from which part (a) follows. LOCAL COHOMOLOGY OF BP*BP-COMODULES 7 Now, if M is an arbitrary -comodule, choose an injective A-module J so that there is an embedding M j-!J. The composite M _-! A M 1-j-! A J is a comodule embedding of M into an injective -comodule, proving part (b). If M is itself injective, this embedding must have a retraction, proving part (c). This lemma is of little practical assistance, since injective BP*-modules are* * ex- tremely complex. They must not only be vn-divisible for all n, but also x-divis* *ible for every nonzero homogeneous element x in BP*. This is the reason one generally uses relatively injective BP*BP -comodules, as they are much simpler. However, * *to compute right derived functors of Ln, we must use absolute injectives. Indeed, * *as explained following the proof of Corollary 3.6, Lin(BP*BP ) is not always zero * *for positive i. The first step is to understand the vn-torsion in an injective comodule. Proposition 2.2. Suppose M is a vn-torsion BP*BP -comodule and N is an es- sential extension of M in the category of BP*BP -comodules. Then N is vn-torsio* *n. In particular, the injective hull of M is vn-torsion. Proof.Suppose N is notpvn-torsion._ Let x be an element of N that is not vn- torsion, and let I = Ann x. Since x is not vn-torsion, vn is not in I. Theorem* * 1 of [Lan79] guarantees that I is an invariant ideal of BP*, so we must have I = * *Ik for some k n. Theorem 2 of [Lan79] tells us that there is some primitive y in N such that Ann (y) = Ik. Hence BP*=Ik is isomorphic to a subcomodule of N. This subcomodule has no vn-torsion, so cannot intersect M nontrivially. This contradicts our assumption that N is an essential extension of M. This proposition leads to the following useful theorem. Theorem 2.3. Suppose I is an injective BP*BP -comodule, and let TnI denote the vn-torsion in I. Then TnI and I=TnI are injective, and I ~=TnI I=TnI. Proof.The injective hull of TnI must be a subcomodule of I, since I is injectiv* *e, and it must be vn-torsion by Proposition 2.2. Hence it must be TnI itself. Corollary 2.4. Suppose I is an injective BP*BP -comodule. Then LnI = I=TnI. In particular, Ln preserves injectives. Proof.Certainly the map I -!I=TnI is an Ln-equivalence. But I=TnI is an injec- tive comodule by Theorem 2.3, and has no vn-torsion. It is therefore Ln-local, * *by Proposition 1.1(e). Corollary 2.5. The functor * preserves and reflects injectivity, and the funct* *or * preserves injectives. Proof.The functor * is right adjoint to the exact functor *, so preserves in- jectives. Conversely, suppose *I is injective, j :M j-!N is an inclusion of E(n)*E(n)-comodules, and f :M -! I is a map. Applying *, we find a map h: *N -! *I such that h O *j = *f. Since * is fully faithful, we conclude that h = *g for some extension g of f. Hence I is injective. Now Ln = * * preserves injectives by Corollary 2.4. Since * reflects injec- tives, we conclude that * must preserve injectives. 8 MARK HOVEY AND NEIL STRICKLAND Theorem 2.3 divides the study of injective BP*BP -comodules into those with no vn-torsion and those which are all vn-torsion. About all we know about injective comodules which are all vn-torsion is the following proposition. Proposition 2.6. Suppose I is an injective BP*BP -comodule that is all vn-torsi* *on. Then I is vn+1-divisible. Proof.Suppose x 2 I. Because every BP*BP -comodule is a filtered colimit of finitely presented BP*BP -comodules, there is a map P -! I from a comodule P that is a free finitely generated BP*-module, whose image contains x. Since I is vn-torsion, and therefore vi-torsion for all i n, this map factors through Q = P=JP -g!I for some invariant ideal J = (pi0, vi11, . .,.vin-1n-1). There is some k such t* *hat vkn+1 is invariant modulo J. Thus multiplication by vkn+1defines a monomorphism of comodules Q -! -tQ. Because I is injective, g must extend to a map -tQ -!I, showing that x is divisible by vn+1. We now turn our attention to injectives that have no vn-torsion. Theorem 2.7. Suppose M is a BP*BP -comodule with no vn-torsion. Then there is an embedding of M into an injective BP*BP -comodule with no vn-torsion, and this embedding can be chosen to be functorial on the category of BP*BP -comodul* *es with no vn-torsion. Proof.Since the category of E(n)*E(n)-comodules is a Grothendieck category (see Section 1.4 of [Hov04 ]), there is a functorial embedding of any E(n)*E(n)-como* *dule into an injective E(n)*E(n)-comodule. (Apply Quillen's small object argument to the set of subobjects of a generator). In particular, for M a BP*BP -comodule, * *we get a functorial embedding *M -!I. Applying * gives us a functorial embedding LnM -! *I, and *I is an injective comodule (by Corollary 2.5), and has no vn- torsion (since it is Ln-local). Since M has no vn-torsion, M embeds in LnM. We expect that injective BP*BP -comodules are not closed under filtered colim- its, though we do not have a counterexample. Those with no vn-torsion, on the other hand, are better behaved. Proposition 2.8. Injective BP*BP -comodules with no vn-torsion are closed under filtered colimits. This proposition depends on the following lemma. Lemma 2.9. Injective E(n)*E(n)-comodules are closed under filtered colimits. Proof.Recall that the category of E(n)*E(n)-comodules is a Grothendieck cate- gory; a set of generators is given by the comodules which are finitely generate* *d and projective over E(n)* [Hov04 , Section 1.4]. There is a version of Baer's crite* *rion for injectivity that works for any Grothendieck category [Ste75]. Let {Gj} be a set* * of generators for the Grothendieck category in question; then an object I is injec* *tive if and only if Hom (Gj, I) -! Hom (Nj, I) is surjective for all j and all subob* *jects Nj of Gj. In particular, if Hom (Gj, -) and Hom (Nj, -) commute with filtered colimits (that is, if Gj and Nj are finitely presented), then injectives are cl* *osed under filtered colimits. In our case, the generators Gj are finitely generated * *and projective over E(n)*. Since E(n)* is Noetherian, the objects Nj are also finit* *ely LOCAL COHOMOLOGY OF BP*BP-COMODULES 9 presented over E(n)*. This means the objects Nj and Gj are also finitely presen* *ted as E(n)*E(n)-comodules by Proposition 1.3.3 of [Hov04 ], completing the proof. Proof of Proposition 2.8.Suppose F :J -! BP*BP -comod is a functor from a filtered category J to injective comodules with no vn-torsion. Then F (j) is L* *n- local for all j 2 J, so we have colimF ~=colim * *F ~= *(colim *F ), by Proposition 1.8. Now *F (j) is an injective E(n)*E(n)-comodule for all j 2 J by Corollary 2.5, so Lemma 2.9 tells us that colim *F is injective. Since * preserves injectives, we conclude that colimF is injective. We can now give a partial structure theorem for injective BP*BP -comodules. Proposition 2.10. Suppose I is an injective BP*BP -comodule, and n 0. Then I ~=I0 I1 . .I.n TnI where (a) Each Ij is an injective BP*BP -comodule with v-1jIj = Ij (and thus Ij is vj-1-torsion). (b) TnI is injective and vn-torsion. In particular, if I is indecomoposable then either I = v-1jI for some j or I is vj-torsion for all j. Proof.Put Ij = Tj-1I=TjI (where T-1I = I). As the comodules TjI are injective (by Theorem 2.3), the filtration TnI Tn-1I . . .T0I I L n must split, giving I = TnI j=0Ij. The comodule Ij is a summand of I and thus is injective. By construction it is vj-1-torsion, and thus vj-divisible by* * Propo- sition 2.6. The definition also implies that there is no vj-torsion, so Ij = v-* *1jIj. It would be nice to have some explicit knowledge of injective BP*BP -comodules I with v-1nI = I. When n = 0, at least, this is easy. Proposition 2.11. Suppose M is a BP*BP -comodule with no p-torsion. Then M is injective if and only if M is a rational vector space. Proof.Proposition 2.6 shows that if M is injective, then it must be a rational * *vector space. Conversely, if M is rational, then M = L0M = * *M. The category of E(0)*E(0)-comodules is the category of rational vector spaces, so *M is inject* *ive. Since * preserves injectives, we conclude that M is injective. The analogue of this proposition is definitely false when n > 0. Still, this * *gives a rationale for why the chromatic resolution is useful. Indeed, suppose we want to find an injective resolution of BP* as a BP*BP -comodule. Proposition 2.11 implies that M0 = p-1BP* is the injective hull of BP* as a BP*BP -comodule. The cokernel N1 is usually written BP*=(p1 ). The injective hull of N1 must be a p-torsion essential extension of N1 on which v1 acts invertibly. The simplest w* *ay to do this is to form M1 = v-11N1, which is the next term in the chromatic resolut* *ion. Sadly, N1 is not actually injective, but it seems to be the closest one can get* * to the injective hull of M1 in a fairly simple way. Iterating this idea leads to the c* *hromatic resolution. 10 MARK HOVEY AND NEIL STRICKLAND 3. The derived functors of Ln Now that we have some knowledge of injective BP*BP -comodules, we can begin to compute derived functors. The goal of this section is to prove Theorem C exc* *ept for part (1), which we deal with in the next section. Recall that Lindenotes the ith right derived functor of Ln. We also let Tnide* *note the ith right derived functor of Tn, where Tn(M) is the subcomodule of vn-torsi* *on elements in M. The first thing to point out is that Linand Tniare closely related. Theorem 3.1. If M is a BP*BP -comodule, we have a natural short exact sequence 0 -!TnM -!M -!LnM -!Tn1M -!0. and natural isomorphisms LinM ~=Tni+1M for i > 0. Proof.Let I* be an injective resolution of M. Then TniM ~=H-i(TnI*), and LinM ~= H-i(LnI*). But LnI* ~= I*=TnI* by Corollary 2.4. Hence we have a natural short exact sequence of complexes 0 -!TnI* -!I* -!LnI* -!0. The long exact sequence in homology gives the desired result. We also point out that computing Linis equivalent to computing the right deri* *ved functors of *. Proposition 3.2. Let M be a BP*BP -comodule, and let Ri * denote the ith right derived functor of *. Then we have a natural isomorphism (Ri *)( *M) ~=LinM. Note that, since * *N ~=N, we can also write this isomorphism as (Ri *)(N) ~=Lin( *N). Proof.Let I* be an injective resolution of M. Then *I* is an injective resolut* *ion of *M, since * is exact and preserves injectives by Corollary 2.5. Hence (Ri *)( *M) ~=H-i( * *I*) ~=H-i(LnI*) ~=LinM. We now begin the computation of Lin. Proposition 3.3. If T is a vn-torsion BP*BP -comodule, then LinT = 0 for all i 0. Furthermore, for an arbitrary comodule M, the map M -!LnM induces an isomorphism LinM -!LinLnM. Proof.Using Proposition 2.2, one can construct an injective resolution I* of T * *that is all vn-torsion. Hence LnI* = 0, so LinT = 0 for all i. For the second statem* *ent, recall that we have short exact sequences 0 -!T -! M -!M=T -! 0 and 0 -!M=T -! LnM -!T 0-!0 where T and T 0are vn-torsion. Applying Ln gives the desired result. LOCAL COHOMOLOGY OF BP*BP-COMODULES 11 The following proposition is part (3) of Theorem C. Proposition 3.4. Suppose M is a BP*BP -comodule. Then LinM is vn-torsion for i > 0. Proof.Let I* be an injective resolution of M. Then Tni(M) = H-iTnI* is obviously vn-torsion. The result follows from Theorem 3.1. We now show that the chromatic resolution is as good as an injective resoluti* *on for computing Lin. The following theorem also proves part (4) of Theorem C. Theorem 3.5. Suppose M is a BP*BP -comodule on which vj acts isomorphically for some j. Then LinM = 0 for all i > 0 and all n. Moreover, we have LnM = 0 if j > n and LnM = M if j n. Proof.We claim that we can choose an injective resolution I* of M for which I* = v-1jI*. To see this, it suffices by induction to show that if N is a BP*BP* * - comodule for which N = v-1jN, there there is a short exact sequence 0 -!N -!I -!N0 -!0 of comodules for which I is injective, v-1jI = I, and v-1jN0 = N0. Since N = v-* *1jN, N is all vj-1-torsion by Proposition 2.9 of [JY80 ], and of course N has no vj-* *torsion. Proposition 2.2 and Theorem 2.7 together imply that the injective hull I of N is vj-1-torsion and has no vj-torsion. Proposition 2.6 then implies that I = v-1j* *I. It follows easily that multiplication by vj is surjective on N0, but we claim i* *t is injective as well. Indeed, suppose x 2 N0 has vjx = 0. Choose a y in I whose image in N0 is x, so that vjy is in N. Since N = v-1jN, there is a z in N such * *that vjz = vjy. It follows that z = y, and so x = 0. We now have our desired injective resolution I* of M for which I* = v-1jI*. T* *he argument now breaks into two cases. If j n, we apply the vn-torsion functor Tn. Since there is no vj-torsion in I*, there is also no vn-torsion by Lemma 2* *.3 of [JY80 ]. Thus TnI* = 0, and so LinM ~=Tni+1M ~=H-i-1TnI* = 0 for all i > 0. For i = 0, use Proposition 1.6. Now suppose j > n. Since v-1jI* = I*, I* is all vj-1-torsion, so also all vn-* *torsion. Hence LnI* = 0, so LinM = H-iLnI* = 0 for i 0. Theorem 3.5 allows us to compute LinM for some important BP*BP -comodules M. The following corollary is Theorem C(5). Corollary 3.6. (a)Suppose k > n. Then Lin(BP*=Ik) = 0 for all i. (b) Lik(BP*=Ik) = 0 for i > 0, whereas L0k(BP*=Ik) = v-1kBP*=Ik. (c) Suppose k < n. Then Lin(BP*=Ik) = 0 unless i = 0 or n - k. We have L0n(BP*=Ik) = BP*=Ik and Ln-kn(BP*=Ik) = BP*=(p, v1, . .,.vk-1, v1k, . .,.v1n). 12 MARK HOVEY AND NEIL STRICKLAND Proof.Let M = BP*=Ik, and consider the chromatic resolution M -! J* of M, where Jt= v-1t+kBP*=(p, v1, . .,.vk-1, v1k, . .,.v1t+k-1). By Theorem 3.5, we have LinJt = 0 for all i > 0. Hence LinM ~=H-iLnJ*. Now, each of the comodules Jt is vk-1-torsion, so LnJ* = 0 if k > n. This completes * *the proof of part (a). If k = n, then LnJt = 0 for t > 0, from which part (b) follows easily. If k <* * n, on the other hand, LnJt = Jt for t < n - k + 1, and is 0 for t n - k + 1, from which part (c) follows. Now suppose M is a BP*BP -comodule that is flat over BP*, and let J* denote the chromatic resolution of BP*. Then M BP* J* is the chromatic resolution of M. Furthermore, vt still acts invertibly on M BP* Jt, so Lin(M BP* Jt) = 0 for all i > 0. Just as in the proof of Corollary 3.6, then, we conclude that Lnn(M) = M=(p1 , v11, . .,.v1n). In particular, Lnn(BP*BP ) is non-zero, showing that relative injectives do not* * suffice to compute Lin. We also discover that Ln has only finitely many right derived functors. Theorem 3.7. Suppose M is a vk-torsion comodule for some -1 k n. Then LinM = 0 for i n - k. In particular, LinN = 0 for i > n for any comodule N. For the purposes of this theorem, we take v-1 = 0, so that every comodule is v-1-torsion. This theorem proves part (2) of Theorem C. Proof.We proceed by downwards induction on k. The base case k = n is Proposi- tion 3.3. So suppose we know the theorem for k, and M is a vk-1-torsion comodul* *e. Let TkM denote the vk-torsion in M. We have a short exact sequence 0 -!TkM -!M -!N -!0 where N has no vk-torsion. By our induction hypothesis, Lin(TkM) = 0 for i n-* *k. It therefore suffices to show that Lin(N) = 0 for i > n - k. Now, since N is vk-1-torsion but has no vk-torsion, we have a short exact se- quence 0 -!N -!v-1kN -!T -! 0, where T is vk-torsion. Our induction hypothesis guarantees that LinT = 0 for i n - k, and Theorem 3.5 guarantees that LinT ~= Li+1nN for i > 0. Hence LinN = 0 for i > n - k, as required. Corollary 3.6 together with the Landweber filtration theorem gives a method f* *or computing LinM for finitely presented BP*BP -comodules M. To compute LinM for more general comodules M, we use the following theorem, which is part (6) of Theorem C. Theorem 3.8. The functors Linpreserve filtered colimits and arbitrary direct su* *ms of BP*BP -comodules. Since Ln itself preserves filtered colimits, this theorem would be easy if fi* *ltered colimits of injective comodules were injective, but we believe that this is fal* *se in gen- eral. However, to compute Linthe only injectives that matter are injectives wit* *h no vn-torsion, and these we know are closed under filtered colimits by Proposition* * 2.8. LOCAL COHOMOLOGY OF BP*BP-COMODULES 13 Proof.It suffices to show that Linpreserves filtered colimits, since arbitrary * *direct sums are filtered colimits of finite direct sums. We use induction on i. When i = 0 we have seen this already in Proposition 1.8. Now suppose Linpreserves filtered colimits for some i 0, and let {Mt} be a filtered diagram of comodul* *es. Then {LnMt} is a filtered diagram of comodules with no vn-torsion, so we can use Theorem 2.7 to find a filtered diagram of injectives {It} with no vn-torsion an* *d a short exact sequence of filtered diagrams {0} -!{LnMt} -!{It} -!{Nt} -!{0}. This gives us a short exact sequence (3.9) 0 -!colimLnMt-! colimIt-! colimNt-! 0, and colimIt is injective by Proposition 2.8. We must now separate the case i = 0 from the case i > 0. If i = 0, by taking the colimit of the exact sequences 0 -!LnMt-! It-! LnNt-! L1nMt-! 0, we get an exact sequence 0 -!colimLnMt-! colimIt-! colimLnNt-! colimL1nMt-! 0. On the other hand, by applying Ln to the short exact sequence 3.9, we get the exact sequence 0 -!colimLnMt-! colimIt-! Ln(colimNt) -!L1n(colimLnMt) -!0. There is a map from the first of these sequences to the second, which is an iso* *mor- phism on every nonzero term except the last one, so we get an isomorphism colimL1nMt~=L1n(colimLnMt). On the other hand, using Proposition 3.3, and the fact that Ln commutes with filtered colimits, we get L1n(colimLnMt) ~=L1nLn(colimMt) ~=L1n(colimMt), as required. If i > 0, the situation is easier. Indeed, using Proposition 3.3 and the fact* * that Ln commutes with filtered colimits, we have colimLi+1nMt~=colimLi+1n(LnMt) ~=colimLinNt ~=Lin(colimNt) ~=Li+1n(colimLnMt) ~=Li+1nLn(colimMt) ~=Li+1n(colimMt), completing the proof. 4.Comparison with ~Cech cohomology The object of this section is to prove part (1) of Theorem B and Theorem C, showing that, for a comodule M, Lin(M) is the same as the ith ~Cech cohomology group ~CHiIn+1M of M with respect to In+1. We also show that ~Cech cohomology ~CH*In+1(-) is the derived functors of localization in the category of BP*-modu* *les with respect to the hereditary torsion theory of In+1-torsion modules. We first remind the reader of the definition of ~Cech cohomology from [GM95 ]. Given an element ff in a commutative ring R, which we will always take to be BP*, we form the cochain complex Ko(ff) which is R in degree 0 and R[1=ff] in 14 MARK HOVEY AND NEIL STRICKLAND degree 1, with the differential being the obvious map R -!R[1=ff]. Given an ide* *al I = (ff0, . .,.ffn), we define Ko(I) to be the cochain complex Ko(I) = Ko(ff0) R Ko(ff1) R . . .RKo(ffn). This stable Koszul complex of course depends on the choice of generators ffi, b* *ut its quasi-isomorphism class does not [GM95 , Corollary 1.2]. There is an obvio* *us surjection Ko(ffi) -! R of complexes, where R is the complex consisting of R concentrated in degree 0. Tensoring these together gives us a map ffl: Ko(I) -!R. We define the flat ~Cech complex ~Co(I) by C~o(I) = (kerffl), where ( B)n = Bn+1 for a cochain complex B. Thus M C~k(I) = R[1=ffS] |S|=k+1 for 0 Q k n, where S runs through the k + 1-element subsets of (0, 1, . .,.n)* * and ffS = i2Sffi. Definition 4.1. The local cohomology H*I(M) of an R-module M with respect to a finitely generated ideal I = (ff0, . .,.ffn) is H*I(M) = H*(Ko(I) R M). The ~Cech cohomology ~CH*I(M) of M with respect to I is ~CH*I(M) = H*(C~o(I) R M). Some of the basic properties of local and Cech cohomology are summarized in the following proposition. Proposition 4.2. Suppose I = (ff0, . .,.ffn) is a finitely generated ideal in a* * com- mutative ring R, and M is an R-module. (a) We have a natural exact sequence 0 -!H0I(M) -!M -!C~H0I(M) -!H1I(M) -!0, and natural isomorphisms ~CHkI(M) ~=Hk+1I(M) for k > 0. (b) ~CHkI(M) = 0 unless 0 k n. (c) HkI(M) is I-torsion for all k, and ~CHkI(M) is I-torsion for all k > 0. * *On the other hand, ~CH0I(M) has no I-torsion. (d) H0I(M) is the submodule of I-torsion elements in M. (e) ~CHkI(M) = 0 for all k if and only if M is I-torsion, and this is true i* *f and only if ~CH0I(M) = 0. (f) A short exact sequence 0 -!M0 -!M -!M00-!0 of R-modules gives rise to natural long exact sequences 0 -!H0I(M0) -!H0I(M) -!H0I(M00) -!H1I(M0) -!. .-.!Hn+1I(M) -!Hn+1I(M00) -!0, LOCAL COHOMOLOGY OF BP*BP-COMODULES 15 and 0 -!C~H0I(M0) -!C~H0I(M) -!C~H0I(M00) -! ~CH1I(M0) -!. .-.!~CHnI(M) -!C~HnI(M00) -!0. (g) Both H0Iand ~CH0Iare left exact idempotent functors. Note that part (f) certainly suggests that HkIis the kth right derived functo* *r of H0I, but it does not prove it, since we also need to know that HkIsends injecti* *ve modules to 0 for all k > 0. We will have to deal with this issue later. Proof.Most of this proposition follows from [GM95 ]. Part (a) appears in Sectio* *n 1 of that paper, and part (b) is obvious from the definition of ~Co(I). The first* * sentence of part (c) is also in Section 1 of [GM95L]. For the second part of part (c), s* *imply note that ~CH0I(M)Lis a submodule of iM[1=ffi]. As M[1=ffi] has no ffi-torsio* *n, it follows that iM[1=ffi] has no I-torsion. Part (d) is clear from the fact t* *hat H0I(M) is the kernel of the map M M -! M[1=ffi]. i For part (e), first suppose that M is I-torsion. Then the complex ~Co(I) R M is the zero complex, and so of course ~CHkI(M) = 0 for all k. Conversely, suppo* *se ~CH0I(M) = 0. Then parts (a) and (d) show that M is I-torsion. For part (f), simply note that the complexes Ko(I) and ~Co(I) are complexes of flat modules. Hence, a short exact sequence of modules gives rise to a short exact sequence of complexes on applying either Ko(I) R (-) or ~Co(I) R (-). The resulting long exact sequence in cohomology gives us part (f). For part (g), note that part (* *f) immediately implies that both H0Iand ~CH0Iare left exact. Part (d) shows that H* *0I is idempotent. To see that ~CH0Iis idempotent, apply part (f) to the short exact sequences 0 -!H0I(M) -!M -!M=H0I(M) -!0 and 0 -!M=H0I(M) -!C~H0I(M) -!H1I(M) -!0. Since both H0I(M) and H1I(M) are I-torsion by part (c), part (e) tells us that * *we get isomorphisms ~CH0I(M) ~=~CH0I(M=H0I(M)) ~=~CH0I(C~H0IM), completing the proof. Corollary 4.3. Suppose I = (ff0, ff1, . .,.ffn) is a finitely generated ideal i* *n a commutative ring R. The functor ~CH0Iis localization in the category of R-modul* *es with respect to the hereditary torsion theory of I-torsion modules. For this corollary to make sense, recall that a class of objects in an abelia* *n cate- gory A is a hereditary torsion theory if it is closed under subobjects, extensi* *ons, quotient objects, and arbitrary coproducts. If T is a hereditary torsion theory* *, we define a map f to be a T -equivalence if its kernel and cokernel are in T . An object X is called T -local if A(f, X) is an isomorphism for all T -equivalence* *s f. A T -localization of an object M is a T -local object X together with a T -equiva* *lence M -! X. When T -localizations exist, they are unique up to unique isomorphism and are functorial. 16 MARK HOVEY AND NEIL STRICKLAND Proof.One can easily check that the class T of I-torsion modules is a hereditary torsion theory. It is clear from parts (a) and (c) of Proposition 4.2 that the * *map M -!C~H0I(M) has I-torsion kernel and cokernel, so is a T -equivalence. It rema* *ins to show that ~CH0I(M) is T -local. By factoring a T -equivalence into an inject* *ion followed by a surjection, we see that this boils down to showing that ~CH0I(M) * *has no I-torsion and that Ext1R(T, ~CH0I(M)) = 0 for all I-torsion modules T . The first part is part (c) of Proposition 4.2. For the second part, suppose we have* * an extension 0 -!C~H0I(M) -!X -!T -! 0 where T is I-torsion. Applying the left exact idempotent functor ~CH0I, we get * *an isomorphism ~CH0I(M) -!C~H0I(X). Thus the composite X -!C~H0I(X) ~=~CH0I(M) defines a splitting of our extension. Thus ~CH0I(M) is T -local as required. We also need to know that ~CH*Iare the right derived functors of ~CH0I. This seems to require some hypotheses on I. Theorem 4.4. Suppose I is an ideal in a commutative ring R, generated by a regular sequence (ff0, . .,.ffn), in which each element ffi is not a zero-divis* *or. Then HkI(M) = colimJExtkR(R=J, M), p __ p _ where J runs over ideals J I with J = I. Moreover, HkIis the k'th right derived functor of H0Iand ~CHkIis the k'th right derived functor of ~CH0I. Proof.Put Ir = (ffr0, . .,.ffrn); these ideals are evidently cofinal among the * *J's. Let Korbe the usual (unstable) Koszul complex for Ir, which is the tensor product o* *ver ffrj j of the complexes (R --! R). As our sequence of generators is regular, this is* * a finite resolution of R=Ir by finitely generated free modules. Now let DKorbe the dual of Kor, which is naturally thought of as the tensor product of the complex* *es R -! R . ff-rj. (In fact DKoris isomorphic to Kor, up to a degree shift in the graded case.) It is clear that the stable Koszul complex Ko(I) is the colimit o* *f the complexes DKor, so H*I(M)= colimrH*(DKor R M) = colimrH* Hom R(Kor, M) = colimrExt*R(R=Ir, M) = colimJExt*R(R=J, M). It is immediate from this that HiI(M) = 0 if i > 0 and M is injective. Using pa* *rt (a) of Proposition 4.2, we see that ~CHiI(M) = 0 as well. It now follows formally f* *rom the long exact sequences in Proposition 4.2 that HkIis the k'th right derived f* *unctor of H0I, and ~CHkIis the k'th right derived functor of ~CH0I. We can now investigate the functors H*Iand ~CH*Irestricted to the category of comodules, proving part (1) of Theorem B amd Theorem C. Theorem 4.5. Suppose M is a BP*BP -comodule. Then there are natural isomor- phisms Tnk(M) ~=HkIn+1(M) and LknM ~=~CHkIn+1(M). LOCAL COHOMOLOGY OF BP*BP-COMODULES 17 Proof.We first show that HkIn(M) = 0 for all injective BP*BP -comodules M and k > 0. This does not follow from Theorem 4.4 because injective comodules need not be injective as BP*-modules. We proceed by induction on n, using the spectr* *al sequence HsvnHtIn(M) ) Hs+tIn+1(M). discussed in [GM95 , Section 2]. By induction, the E2-term of this spectral seq* *uence is HsvnH0In(M). In degree s = 1, this is v-1nH0In(M)=H0In(M). But Theorem 2.3 shows that H0InM = Tn-1M is an injective BP*BP -comodule, which of course is vn-1-torsion. Proposition 2.6 then shows that H0InM is vn- divisible. Hence the E2-term of our spectral sequence is H0vnH0In(M) = H0In+1(M) concentrated in bidegree (0, 0), completing the proof. Now suppose M is an arbitrary BP*BP -comodule. Take an resolution I* of M by injective BP*BP -comodules. By definition, Tnk(M) ~=H-k(TnI*). On the other hand, applying H*In+1, which we have just seen vanishes on injective comodules, shows that HkIn+1(M) ~=H-k(H0In+1I*) ~=H-k(TnI*) as well. Similarly, Lkn(M) ~=H-k(LnI*), which is isomorphic to H-k(I*=TnI*) by Corol- lary 2.4. Now suppose N is an injective BP*BP -comodule. The exact sequence 0 -!H0In+1(N) -!N -!C~H0In+1(N) -!H1In+1(N) -!0 of Proposition 4.2 together with the fact that HkIn+1(N) = 0 for k > 0 implies * *that ~CH0In+1(N) ~=N=TnN. Also, C~HkIn+1(N) ~=Hk+1In+1(N) = 0 for k > 0. Hence, applying ~CH*In+1to I*, we find that ~CHkIn+1(M) ~=H-k(I*=TnI* **), completing the proof. We can now give the promised converse to Lemma 1.3. Corollary 4.6. A BP*BP -comodule M is Ln-local if and only if Hom *BP*(BP*=In+1, M) = Ext1,*BP*(BP*=In+1, M) = 0. Proof.The if direction is Lemma 1.3. For the only if direction, suppose M is Ln-local. Then M is also local with respect to the hereditary torsion theory of In+1-torsion BP*-modules, in view of Theorem 4.5 and Corollary 4.3. This means that, for any In+1-torsion module T , we have Hom BP*(T, M) = Ext1BP*(T, M) = 0. Applying this to BP*=In+1 and all its suspensions gives the desired result. 18 MARK HOVEY AND NEIL STRICKLAND 5.The spectral sequence The object of this section is to prove Theorem A. That is, we construct a spectral sequence converging to BP*LnX whose E2-term consists of the derived functors Lsn(BP*X). Analogously, let CnX denote the fiber of X -! LnX. We construct a spectral sequence converging to BP*CnX whose E2-term consists of the derived functors Tns(BP*X). Our method is based on the construction of the modified Adams spectral sequence due to Devinatz and Hopkins [Dev97 , Section 1* *]. Definition 5.1. Define a functor D from injective BP*BP -comodules to (the ho- motopy category of) spectra as follows. Given an injective BP*BP -comodule I, consider the functor DI from spectra to abelian groups defined by DI(X) = Hom BP*BP(BP*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 reason for the letter D is that DI is a sort of duality functor, built al* *ong the lines of Brown-Comenetz duality [BC76 ]. Also note that we are considering cohomology functors as exact functors to ungraded abelian groups; we recover the usual graded cohomology functor by DtI(X) = DI( tX). The following theorem is a special case of Theorem 1.5 of [Dev97 ]. Theorem 5.2. Suppose I is an injective BP*BP -comodule. Then there is a natural isomorphism BP*D(I) ~=I. This isomorphism of course corresponds to the identity map of D(I) under the isomorphism [D(I), D(I)] ~=Hom BP*BP(BP*D(I), I). We need to know how the D(I) behave under localization. Proposition 5.3. Suppose I is an injective BP*BP -comodule. Then the natural map I -!LnI induces an isomorphism LnD(I) -!D(LnI). Proof.Recall that LnI is again injective, by Corollary 2.4. We first note that D(LnI) is E(n)-local. Indeed, if E(n)*(X) = 0, then BP*(X) is all vn-torsion. Since LnI has no vn-torsion, we have [X, D(LnI)] ~=Hom BP*BP(BP*X, LnI) = 0. Thus D(LnI) is indeed Ln-local. On the other hand, the map D(I) -! D(LnI) induces the map I -! LnI on BP*-homology, by Theorem 5.2. Since LnI ~=I=TnI by Corollary 2.4, this map becomes an isomorphism after applying *, and so D(I) -! D(LnI) is an E(n)- equivalence. Corollary 5.4. Suppose I is an injective BP*BP -comodule. Then the natural map TnI -!I induces an isomorphism D(TnI) -!CnD(I). LOCAL COHOMOLOGY OF BP*BP-COMODULES 19 Proof.Note that D(TnI) makes sense since TnI is an injective comodule by The- orem 2.3. Since BP*(D(TnI)) ~=TnI, one easily sees that D(TnI) is E(n)-acyclic. Therefore, the map D(TnI) -!D(I) induced by the inclusion TnI -!I induces the desired map D(TnI) -!CnD(I). This map is an isomorphism on BP*(-) by Proposition 5.3, and one can check that both sides are BP -local, so it is an isomorphism. We can now build our spectral sequences, following the standard approach used by Ravenel in [Rav86 , Section 2.1]. Suppose X is a spectrum, and let C = BP*X. Choose an injective resolution 0 -!C j-!I0 o0-!I1 o1-!. . . of C in the category of BP*BP -comodules. Let js: Cs -!Is denote the kernel of os, so that j0 = j. The following lemma is easily proved by induction on s, and is implicit in [D* *ev97 , Section 1]. Lemma 5.5. Let X be a spectrum and choose an injective resolution of BP*X as above. Then there is a tower X = X0 --g0--X1 --g1-- X2 --g2-- . . . ?? ? yf0 ?yf1 K0 K1 over X satisfying the following properties. (a) Ks = -sD(Is). (b) Xs+1 is the fiber of fs. (c) BP*Xs ~= -sCs. (d) The map fs is induced by the inclusion Cs -!Is. (e) BP*gs = 0, and the boundary map Ks -! Xs+1 induces the surjection -sIs -! -sCs+1 on BP*-homology. We can now construct our spectral sequences. The following theorem is Theo- rem A except for the statements about convergence. Theorem 5.6. Let X be a spectrum. There is a natural spectral sequence E***(X) with dr: Es,tr-!Es+r,t+r-1rand E2-term Es,t2(X) ~=(LsnBP*X)t. This is a spectral sequence of BP*BP -comodules, in the sense that Es,*ris a graded BP*BP -comodule for all r 2 and dr: Es,*r-!Es+r,*ris a BP*BP -comodule map of degree r - 1. Furthermore, every element in E0,*2that comes from BP*X is a permanent cycle. Proof.Begin with the tower of Lemma 5.5 and apply Ln. We get the tower below. LnX = LnX0 -Lng0---LnX1-Lng1---LnX2 -Lng2---. . . ? ? (5.7) ?yLnf0 ?yLnf1 LnK0 LnK1 By applying BP*-homology, we get an associated exact couple and spectral se- quence. That is, we let Ds,t1= BPt-sLnXs and Es,t1= BPt-sLnKs. We take i1 = BPt-sLngs: Ds+1,t+11-!Ds,t1andj1 = BPt-sLnfs: Ds,t1-!Es,t1 20 MARK HOVEY AND NEIL STRICKLAND and we take k1: Es,t1-!Ds+1,t1 to be BPt-s of the boundary map LnKs -! LnXs+1. Note that this is an exact couple in the category of BP*BP -comodules, in that each Ds,*1and Es,*1is a graded BP*BP -comodule and the maps i1, j1, k1 are maps* * of comodules. It follows that the spectral sequence is a spectral sequence of BP*B* *P - comodules. Now, by combining Theorem 5.2 and Proposition 5.3, we find Es,t1~=BPt-s(Ln -sD(Is)) ~=BPtD(LnIs) ~=(LnIs)t. To compute the first differential d1, note that we have the commutative diagram below. Ks ----! Xs+1 ----! Ks+1 ?? ? ? y ?y ?y LnKs ----! Ln( Xs+1) ----! Ln( Ks+1) The map on BPt-s-homology induced by the bottom composite is d1. The map on BPt-s-homology induced by the top composite is os, by Lemma 5.5. The outside vertical maps are surjective in BP*-homology, by Proposition 5.3 and Corollary * *2.4. It follows that d1 = Lnos. Therefore, the E2-term of our spectral sequence is Es,t2~=Hs(LnI*)t~=(LsnBP*X)t, as required. The naturality of the spectral sequence follows in the usual way. That is, a * *map of spectra X -!Y induces a map BP*X -!BP*Y . This can be lifted, nonuniquely, to a map of injective resolutions and so to a map of the towers of Lemma 5.5. T* *his map induces a map of spectral sequences which is the evident map Lsn(BP*X) -!Lsn(BP*Y ) on the E2-terms. This map is independent of the choice of map of injective reso* *lu- tions, and so is functorial. This also shows that our spectral sequence is inde* *pendent of the choice of injective resolution (from E2 on). To complete the proof, we must show that every element in E0,*2that comes from BP*X is a permanent cycle. To see this, note that there is a map from the tower* * of Lemma 5.5 to the tower 5.7 induced by Ln. Applying BP*-homology to the tower of Lemma 5.5 gives us a spectral sequence with Es,t2= 0 if s > 0 and E0,t2= BPt* *X. The map from this spectral sequence to our spectral sequence immediately gives the desired result. We have an analogous theorem for Cn. Theorem 5.8. Let X be a spectrum. There is a natural spectral sequence E***(X) with dr: Es,tr-!Es+r,t+r-1rand E2-term Es,t2(X) ~=(TnsBP*X)t. This is a spectral sequence of BP*BP -comodules, in the sense that Es,*ris a graded BP*BP -comodule for all r 2 and dr: Es,*r-!Es+r,*ris a BP*BP -comodule map of degree r - 1. LOCAL COHOMOLOGY OF BP*BP-COMODULES 21 Proof.Begin with the tower of Lemma 5.5 and apply Cn to get the tower below. CnX = CnX0 -Cng0---CnX1-Cng1---CnX2 -Cng2---. . . ? ? (5.9) ?yCnf0 ?yCnf1 CnK0 CnK1 Apply BP*-homology to get an associated exact couple and spectral sequence, as in the proof of Theorem 5.6. This time the E1 term will be Es,t1~=BPt-sCnKs ~=(TnIs)t, using Corollary 5.4. The identification of the E2-term uses the commutative dia- gram below. CnKs ----! Cn( Xs+1) ----! Cn( Ks+1) ?? ? ? y ?y ?y Ks ----! Xs+1 ----! Ks+1 The vertical maps are injective on BP*-homology by Corollary 5.4 and Theorem 2.* *3. Thus d1, which is the effect on BPt-s-homology of the top horizontal composite,* * is Tnos. Hence we get the desired E2-term and naturality, as in Theorem 5.6. We must now prove that our spectral sequences converge strongly. This essen- tially boils down to showing that the homotopy inverse limits of the towers 5.7 and 5.9 are trivial. The plan of the proof is very simple; in the original towe* *r of Lemma 5.5, we have BP*gs = 0. Hence E(n)*(Lngs) = E(n)*gs = 0 as well by Landweber exactness. Now we just apply the following theorem. Theorem 5.10. Given n 0, there exists an N such that every composite g = fN O fN-1 O . .O.f1 of maps of spectra such that E(n)*fi= 0 for all i has Lng = 0. This theorem was certainly known to Hopkins and probably others. Proof.Use the modified Adams spectral sequence Es,t2= Exts,tE(n)*E(n)(E(n)*X, E(n)*Y ) ) [X, LnY ]t-s of Devinatz [Dev97 ]. It was proved in [HS99 , Proposition 6.5] that there are * *integers r0 and s0, independent of X and Y , such that Es,tr= 0 whenever r r0 and s * *s0. Take N = s0. Then the composite g is represented by an element in Es,t2for some s N. Therefore g must be represented by some element in Es,t1for some s N, so g = 0. The following corollary completes the proof of Theorem A. Corollary 5.11. The spectral sequence of Theorem 5.6 converges strongly to BP*LnX. Proof.In view of Theorem 5.10, the composites LnXk+s -!LnXk in the tower 5.7 are trivial for large s. Hence limsBP*LnXs = lim1sBP*LnXs = 0, and so the spectral sequence converges conditionally [Boa99, Definition 5.10]. On the oth* *er hand, it is clear that lim1rEs,tr= 0, since we have a horizontal vanishing line* *. Thus, the spectral sequence converges strongly to BP*LnX [Boa99, Theorem 7.3]. 22 MARK HOVEY AND NEIL STRICKLAND We also want to know that the other spectral sequence we have constructed converges. Corollary 5.12. The spectral sequence of Theorem 5.8 converges strongly to BP*CnX. Proof.We have a cofiber sequence CnXs -! Xs -! LnXs of towers, where Xs denotes the tower of Lemma 5.5. By applying BP*, we get an exact sequence of towers BP*+1LnXs -!BP*CnXs -!BP*Xs -!BP*LnXs. We have just seen, in Corollary 5.11, that the towers BP*+1LnXs and BP*LnXs are pro-trivial. It follows that the tower BP*CnXs is pro-isomorphic to the tow* *er BP*Xs. But the tower BP*Xs is obviously pro-trivial by Lemma 5.5, so the tower BP*CnXs is also pro-trivial. Hence limsBP*CnXs ~=lim1sBP*CnXs = 0, and so the spectral sequence of Theorem 5.8 converges conditionally. Since it has a horizontal vanishing line, it converges strongly to BP*CnX [Boa99, Theorem 7.3]. We close the paper by considering the spectral sequence of Theorem A in case X = S0 and n > 0. In that case, we have E0,*2~=BP* and En,*2= BP*=I1n+1, by Corollary 3.6. The only possible differential is dn, but this must be trivial * *since E0,*2must consist of permanent cycles by Theorem A. Thus our spectral sequence degenerates to the short exact sequence of comodules 0 -! -n BP*=I1n+1-!BP*LnS0 -!BP* -!0. A splitting of this sequence is given by the map BP* -!BP*LnS0 induced by S0 -! LnS0. Hence we recover Ravenel's computation of BP*LnS0 [Rav84 , Theorem 6.2]. References [AM69]M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Ad* *dison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 39 #4129 [BC76]Edgar H. Brown, Jr. and Michael Comenetz, Pontrjagin duality for generali* *zed homology and cohomology theories, Amer. J. Math. 98 (1976), no. 1, 1-27. MR 53 #91* *96 [Boa99]J. Michael Boardman, Conditionally convergent spectral sequences, Homoto* *py invariant algebraic structures (Baltimore, MD, 1998), Contemp. Math., vol. 239, Ame* *r. Math. Soc., Providence, RI, 1999, pp. 49-84. MR 2000m:55024 [Dev97]Ethan S. Devinatz, Morava modules and Brown-Comenetz duality, Amer. J. M* *ath. 119 (1997), no. 4, 741-770. MR 98i:55008 [GM95]J. P. C. Greenlees and J. P. May, Completions in algebra and topology, Ha* *ndbook of algebraic topology, North-Holland, Amsterdam, 1995, pp. 255-276. MR 96j:5* *5011 [Gre92]J. P. C. Greenlees, A remark on local cohomology for non-Noetherian ring* *s, preprint, 1992. [Gre93]J. P. C. Greenlees, K-homology of universal spaces and local cohomology * *of the repre- sentation ring, Topology 32 (1993), no. 2, 295-308. MR 94c:19007 [Hov04]Mark Hovey, Homotopy theory of comodules over a Hopf algebroid, Homotopy* * theory: relations with algebraic geometry, group cohomology, and algebraic K-theo* *ry (Evanston, IL, 2002), Contemp. Math., vol. 346, Amer. Math. Soc., Providence, RI, 20* *04, pp. 261- 304. [HS99]Mark Hovey and Neil P. Strickland, Morava K-theories and localisation, Me* *m. Amer. Math. Soc. 139 (1999), no. 666, viii+100. MR 99b:55017 [HS03]_____ , Comodules and Landweber exact homology theories, preprint, 2003. [JY80]David Copeland Johnson and Zen-ichi Yosimura, Torsion in Brown-Peterson h* *omol- ogy and Hurewicz homomorphisms, Osaka J. Math. 17 (1980), no. 1, 117-136.* * MR 81b:55010 LOCAL COHOMOLOGY OF BP*BP-COMODULES 23 [Lam99]T. Y. Lam, Lectures on modules and rings, Springer-Verlag, New York, 199* *9. MR 99i:16001 [Lan79]Peter S. Landweber, New applications of commutative algebra to Brown-Pet* *erson ho- mology, Algebraic topology, Waterloo, 1978 (Proc. Conf., Univ. Waterloo, * *Waterloo, Ont., 1978), Lecture Notes in Math., vol. 741, Springer, Berlin, 1979, pp* *. 449-460. MR 81b:55011 [Rav84]Douglas C. Ravenel, Localization with respect to certain periodic homolo* *gy theories, Amer. J. Math. 106 (1984), no. 2, 351-414. MR 85k:55009 [Rav86]____ , Complex cobordism and stable homotopy groups of spheres, Pure and* * Applied Mathematics, vol. 121, Academic Press Inc., Orlando, FL, 1986. MR 87j:550* *03 [Rav92]____ , Nilpotence and periodicity in stable homotopy theory, Annals of M* *athematics Studies, vol. 128, Princeton University Press, Princeton, NJ, 1992, Appen* *dix C by Jeff Smith. MR 94b:55015 [Ste75]Bo Stenstr"om, Rings of quotients, Springer-Verlag, New York, 1975, Die * *Grundlehren der Mathematischen Wissenschaften, Band 217, An introduction to methods of ri* *ng theory. MR 52 #10782 Department of Mathematics, Wesleyan University, Middletown, CT 06459 E-mail address: hovey@member.ams.org Department of Pure Mathematics, University of Sheffield, Sheffield S3 7RH, En* *g- land E-mail address: N.P.Strickland@sheffield.ac.uk