COMODULES AND LANDWEBER EXACT HOMOLOGY THEORIES MARK HOVEY AND NEIL STRICKLAND Abstract.We show that, if E is a commutative MU-algebra spectrum such that E* is Landweber exact over MU*, then the category of E*E-comodules is equivalent to a localization of the category of MU*MU-comodules. This localization depends only on the heights of E at the integer primes p. * *It follows, for example, that the category of E(n)*E(n)-comodules is equiva* *lent to the category of (v-1nBP)*(v-1nBP)-comodules. These equivalences give simple proofs and generalizations of the Miller-Ravenel and Morava change of rings theorems. We also deduce structural results about the category * *of E*E-comodules. We prove that every E*E-comodule has a primitive, we give a classification of invariant prime ideals in E*, and we give a version * *of the Landweber filtration theorem. Introduction Suppose E*(-) and R*(-) are reduced homology theories with commutative products defined on finite CW complexes. Then E*(-) is said to be Landweber exact over R*(-) if there is a natural isomorphism E*(X) ~=E* R* R*(X) for all finite CW complexes X. It then follows that this natural isomorphism ex* *tends to all CW complexes X, and indeed to all spectra X. Because of this, we usually just say that the spectrum E is Landweber exact over the spectrum R. Examples of this phenomenon abound in stable homotopy theory, and were first studied in some generality by Landweber [Lan76]. Example 0.1. (a) Conner and Floyd [CF66 ] showed that complex K-theory K is Landweber exact over complex cobordism MU, and also that real K- theory KO is Landweber exact over symplectic cobordism MSp. Hopkins and the first author [HH92 ] showed that KO is also Landweber exact over Spin cobordism MSpin. (b) The various elliptic cohomology theories [LRS95 ] are all Landweber exact over MU. (c) The Brown-Peterson spectrum BP at a prime p is Landweber exact over MU. Furthermore, the p-localization MU(p)of MU is also Landweber exact over BP [Lan76]. (d) The Johnson-Wilson spectrum E(n) [JW73 ] as well as the Morava E-theory spectrum En used in the work of Hopkins and Miller [HM ] are Landweber exact over BP . (e) The Morava K-theory spectrum K(n) is Landweber exact over the spec- trum P (n) = BP In [Yos76]. ____________ Date: May 1, 2004. 1 2 MARK HOVEY AND NEIL STRICKLAND In all of the examples of spectra E above, the module E*E is flat over E*, so (E*, E*E) is a Hopf algebroid, or equivalently, a groupoid object in the op- posite of the category of graded-commutative rings. For compatibility with the usual conventions in topology, we set up this correspondence so that the maps jL, jR :E* -! E*E represent the maps sending a morphism to its target and source respectively. We refer to [Rav86 , Appendix 1] for basic facts about Ho* *pf algebroids. The reduced homology E*X is a comodule over the Hopf algebroid (E*, E*E) [Rav86 , Proposition 2.2.8]. One of the main reasons this is importan* *t is because the E2-term of the Adams spectral sequence of X based on E is Ext**E*E(E*, E*X), and this Extis taken in the category of E*E-comodules. To help compute these E2- terms, various authors have constructed isomorphisms of the form Ext**(M, N) ' Ext**(M0, N0) under various hypotheses on the algebroids and , and the co- modules M, N, M0 and N0. This includes the change of rings theorems of Miller- Ravenel [MR77 ], Morava [Mor85 ], and the first author and Sadofsky [HS99 ]. The main result of this paper is that these isomorphisms come from equivalences of comodule categories, and that such equivalences are much more common and sys- tematic than was previously suspected. The definition of Landweber exactness given above for homology theories has an analogue for Hopf algebroids. Given a Hopf algebroid (A, ) and an A-algebra B, we define B to be Landweber exact over (A, ), or, by abuse of notation, over A, if the functor B A (-) is exact on the category of -comodules. If E*(* *-) and R*(-) are homology theories as above, E is an R-module spectrum, and E* is Landweber exact over R*, then a well-known argument shows that E is Landweber exact over R. In the examples listed above, K*, Ell*, and BP* are all Landweber exact over MU*, and (MU*)(p), E(n)*, and En* are Landweber exact over BP*, and K(n)* is Landweber exact over P (n)*. On the other hand, it is not known whether KO* is Landweber exact over MSp* or MSpin*. In the above situation, it is important to understand the relationship between E*E-comodules and R*R-comodules. Given a Hopf algebroid (A, ) and a ring map f :A -! B, we put B = B A A B. The pair (B, B ) has a natural structure as a Hopf algebroid; we recall some details in Section 2. A central o* *bject of this paper is to make a detailed study of the relationship between the categ* *ory of B -comodules and the category of -comodules when B is Landweber exact over A. We prove the following theorem as Theorem 2.5. Theorem A. Suppose (A, ) is a flat Hopf algebroid and B is a Landweber exact A-algebra. Then the category of B -comodules is equivalent to the localization* * of the category of -comodules with respect to the hereditary torsion theory T = {M | B A M = 0}. To apply this to cases of interest in algebraic topology, we give a partial c* *lassifi- cation of graded hereditary torsion theories of BP*BP -comodules in Theorem 3.1. Theorem B. Let T be a graded hereditary torsion theory of BP*BP -comodules, and suppose that T contains a nonzero finitely presented comodule. Then either T is the whole category of comodules, or there is an n such that T is the collect* *ion of vn-torsion comodules. COMODULES AND LANDWEBER EXACT HOMOLOGY THEORIES 3 This theorem is analogous to the classification of thick subcategories of fin* *ite p-local spectra [HS98 ]. This then leads to our main result, which is Theorem 4.2. Theorem C. Define the height of a Landweber exact BP*-algebra E* to be the largest n such that E*=In is nonzero. If E* and E0*are Landweber exact BP*- algebras of the same height, then the category of graded E*E-comodules is equiv- alent to the category of graded E0*E0-comodules. In particular, the categories* * of E(n)*E(n)-comodules, En*En-comodules, and (v-1nBP )*(v-1nBP )-comodules are all equivalent. As mentioned previously, this gives a simple explanation for the change of ri* *ngs theorems of Miller-Ravenel, Morava, and Hovey-Sadofsky, all of which say that two Ext groups computed over different Hopf algebroids are isomorphic. Namely, the Ext groups are isomorphic because the categories they are computed in are equivalent. When E* is Landweber exact over BP*, the category of E*E-comodules is a localization of the category of BP*BP -comodules, by Theorem A. This allows us to extend the standard structure theorems for BP*BP -comodules of Landwe- ber [Lan76] to E*E-comodules. The following theorem is a summary of the results of Section 5. Theorem D. Suppose E* is a Landweber exact BP*-algebra of height n 1. (a) Every nonzero E*E-comodule has a nonzero primitive. (b) If I is an invariant radical ideal in E*, then I = Ik for some k n. (c) Every E*E-comodule M that is finitely presented over E* admits a finite filtration by subcomodules 0 = M0 M1 . . .Ms = M for some s, with Mr=Mr-1 ~=stE*=Ij for some j n and some t, both depending on r. Remark. Baker [Bak95 ] has constructed a counterexample to a statement closely related to (a), in the case where E is the Morava E-theory spectrum En. This is* * not in fact a contradiction, because of the difference between E*E = ss*(E ^ E) (wh* *ich is used in our work) and ss*LK(n)(E ^ E) (which is more closely related to the Morava stabiliser group, and is used in Baker's work). The topological comodule categories considered by Baker are well-known to be important, but they do not * *fit into our present framework; we hope to return to this in future. The theorems we have just discussed all have global versions, where we replace BP* by MU*, and more local versions, where we replace BP* by BP*=J for a nice regular sequence J. We discuss these versions briefly at the end of the paper. As mentioned above, the category of E(n)*E(n)-comodules is a localization of the category of BP*BP -comodules. The resulting localization functor on BP*BP - comodules is denoted Ln, and is analogous to the chromatic localization functor* * Ln much used in stable homotopy theory [Rav92 ]. The algebraic Ln is very interest* *ing in its own right; it is left exact, and has interesting right derived functors * *Lin, which are closely related to local cohomology. The functor Ln and its derived functo* *rs are studied in [HS03 ]. We also point out that, to give a good algebraic model for stable homotopy theory, one wants a triangulated category rather than an abelian category. So t* *here 4 MARK HOVEY AND NEIL STRICKLAND should be analogues of the theorems in this paper for some kind of derived cate* *gories of BP*BP -comodules and E*E-comodules. There are problems with the ordinary derived category; the first author has constructed a well-behaved replacement f* *or it in [Hov04 ]. The authors have proved analogues of some of the theorems of th* *is paper for these derived categories in [Hov02a ]. The authors would like to thank the Universitat de Barcelona, the Universi- tat Aut`onoma de Barcelona, the Centre de Recerca Matematica, and the Isaac Newton Insitute for Mathematical Sciences for their support during this project. They would also like to thank John Greenlees and Haynes Miller for several help* *ful discussions about this paper. 1.Abelian localization In this section, we summarize Gabriel's theory of localization of abelian cat* *e- gories from an algebraic topologist's point of view for the convenience of the * *reader. The original source for this material is [Gab62 ]; a standard source for locali* *zation in module categories is [Ste75]. The book [VOV79 ] gives a quick summary of the theory in an arbitrary Grothendieck category. The following definition is standard in homotopy theory. Definition 1.1. Suppose E is a class of maps in a category C. An object X of C * *is said to be E-local if C(f, X) is an isomorphism of sets for all f 2 E. We denot* *e the full subcategory of E-local objects by LEC. An E-localization of an object M 2 C is a map M -! LM in E where LM 2 LEC. If every M 2 C has an E-localization, we say that E-localizations exist. It is also possible to define localizations without reference to the class E. Definition 1.2. A localization functor on a category C is a functor L: C -! C and a natural transformation 'M : M -! LM such that L'M = 'LM and this map is an isomorphism. The following proposition is reasonably well-known; a version of it can be fo* *und in [HPS97 , Section 3.1] and in other places. Proposition 1.3. Suppose L is a localization functor on a category C. Let E denote the class of all maps f such that Lf is an isomorphism. Then 'M is an E-localization of M for all M 2 C. Conversely, if E is a class of maps on C such that an E-localization 'M : M -! LM exists for all M 2 C, then L is a localizat* *ion functor. Furthermore, in either case L, thought of as a functor L: C -! LEC, is left adjoint to the inclusion functor. We refer to the localization functor of Proposition 1.3 as localization with * *re- spect to E. A common way for localizations to arise is displayed in the following proposi* *tion. Proposition 1.4. Suppose F :C -! D is a functor with right adjoint G, and the counit of the adjunction fflM : F GN -! N is an isomorphism for all M 2 D. Then GF is the localization functor on C with respect to E = {f|F f is an isomorphis* *m}. Furthermore, G defines an equivalence of categories G: D -!LEC. Proof.Let L = GF . The natural transformation 'M : M -! LM is the unit jM of the adjunction. The two triangular relations of the adjunction say, respecti* *vely, COMODULES AND LANDWEBER EXACT HOMOLOGY THEORIES 5 that fflFM O (F jM ) = 1FM and (GfflM ) O jGM = 1GM . In particular, GF jM = (GfflFM )-1 = jGFM . This means that L'M = 'LM and that this map is an isomorphism, as required. By Proposition 1.3, L is localization * *with respect to E = {f|Lf is an isomorphism}. Since F G is naturally isomorphic to t* *he identity, one can easily check that GF f is an isomorphism if and only if F f i* *s an isomorphism. Since F G is naturally isomorphic to the identity, G defines an equivalence of categories from D to its image. Adjointness shows that GN is E-local for all N * *2 D. Conversely, the image of G contains LM for all M 2 C, so is a skeleton of LEC. The result follows. In point of fact, every localization functor arises from an adjunction as in * *Propo- sition 1.4; if L is a localization functor on C, we can think of it as a functor L: C -! LEC, where it is left adjoint to the inclusion and satisfies the hypoth* *e- ses of Proposition 1.4. Now suppose that our category C is abelian. It is natural, then, to consider localization functors arising from adjunctions F :C AE D :G as in Proposition 1* *.4 where D is also abelian and F is exact. Definition 1.5. Suppose T is a full subcategory of an abelian category C. Then T is called a hereditary torsion theory if T is closed under subobjects, quotie* *nt objects, extensions, and arbitrary coproducts. When T is a hereditary torsion theory, we define the class ET of T -equivalences to consist of those maps whose cokernel and kernel are in T . We define an object to be T -local if and only i* *f it is ET -local. We let LT C denote the full subcategory of T -local objects. Note that a hereditary torsion theory is just a Serre class that is closed un* *der coproducts. Also, one can form the smallest hereditary torsion theory containin* *g a specified class of objects by taking the intersection of all hereditary torsion* * theories containing that class. Proposition 1.6. Suppose C and D are abelian categories, F :C -!D is an exact functor with right adjoint G, and the counit of the adjunction fflM : F GN -! N* * is an isomorphism for all M 2 D. Then GF is the localization functor on C with respect to the hereditary torsion theory T = kerF = {M|F M = 0}. Furthermore, G defines an equivalence of categories G: D -!LT C. Proof.Proposition 1.4 implies that GF is localization with respect to E = {f|F f is an isomorphism}. But, since F is exact, F f is an isomorphism if and only if F (kerf) = F (coker* *f) = 0, which is true if and only if f is a T -equivalence. The main result of Gabriel on abelian localizations is the following theorem.* * Re- call that a Grothendieck category is a cocomplete abelian category with a gener* *ator in which filtered colimits are exact. Theorem 1.7. Suppose T is a hereditary torsion theory in a Grothendieck abelian category C. Then T -localizations exist. We outline the proof of Gabriel's theorem 1.7, as we will need some of the id* *eas from this proof later. We first recall the characterization of T -local objects. 6 MARK HOVEY AND NEIL STRICKLAND Lemma 1.8. Suppose T is a hereditary torsion theory in an abelian category C. An object X of C is T -local if and only if C(T, X) = Ext1C(T, X) = 0 for all T* * 2 T . Recall that one can define Extin an arbitrary abelian category without recour* *se to either projectives or injectives [ML95 ]. In particular, Ext1C(M, N) is jus* *t the collection of all equivalence classes of short exact sequences 0 -!N -!E -!M -!0. The usual exact sequences for Ext work in this generality. Proof.Suppose first that X is T -local, and T 2 T . Since 0 -!T is a T -equival* *ence, we conclude that C(T, X) = 0. Given a short exact sequence 0 -!X f-!Y -! T -! 0, the map f is a T -equivalence, so f* :C(Y, X) -!C(X, X) is an isomorphism. Thus the identity map of X comes from a map g :Y -! X, and g defines a splitting of the given sequence. Hence Ext1C(T, X) = 0. Conversely, suppose C(T, X) = Ext1C(T, X) = 0 for all T 2 T , and f :A -!B is a T -equivalence. Consider the two short exact sequences 0 -!kerf -!A -!im f -!0 and 0 -!im f -!B -!cokerf -!0. By applying the functor C(-, X) to these short exact sequences, using the fact * *that C(kerf, X) = C(cokerf, X) = Ext1C(cokerf, X) = 0, we see that C(f, X) is an isomorphism. Now, in order to construct the localization LT (X) of X with respect to a her* *ed- itary torsion theory T in a Grothendieck category C, we first form the union T X of all the subobjects of X that are in T (these form a set because we are in a Grothendieck category). This gives us a T -equivalence X -! X=T X. Then we taken an injective envelope I of X=T X (injective envelopes exist in a Grothend* *ieck category), producing an exact sequence 0 -!X=T X -!I -!Q -!0. Finally, we let LT (X) be the pullback I xQ T Q. The induced embedding X=T X -! LT (X) is a T -equivalence, and one can check that LT (X) is T -local. Remark. In our case we will be working with graded abelian categories C. This means that we have a given self-equivalence s: C -!C, which in fact is an isomo* *r- phism of categories in our examples. In this case, we define a full subcategory* * D to be graded when X 2 D if and only if sX 2 D. Similarly, a class of maps E in C is said to be graded when f 2 E if and only if sf 2 E. The results of this sect* *ion all have corresponding graded versions. COMODULES AND LANDWEBER EXACT HOMOLOGY THEORIES 7 2.Landweber exact algebras In this section, we apply localization techniques to comodules over Hopf alge- broids. Recall that a Hopf algebroid is a pair of (possibly graded) commutative rings (A, ) so that Rings(A, R) and Rings( , R) are the objects and morphisms of a groupoid that is natural in the (graded) commutative ring R. The precise structure maps and relations they satisfy can be found in [Rav86 , Appendix 1]. The reason we are interested in them is that (E*, E*E) is a Hopf algebroid for many homology theories E, as explained in [Rav86 , Proposition 2.2.8]. We will always assume our Hopf algebroids are flat; this means that the left * *unit jL :A -! corepresenting the target of a morphism is a flat ring extension. This is the same as assuming that the right unit jR :A -! corepresenting the source of a morphism is flat. We note that in working with Hopf algebroids it is important to remember that M A N always denotes the tensor product of A-bimodules, using the right A-module structure on M and the left A-module structure on N. This mostly matters for , which is a right A-module via the right unit jR and a left A-mod* *ule via the left unit jL. Recall that a comodule over a Hopf algebroid (A, ) is a left A-module M equipped with a coassociative and counital coaction map _ :M -! A M. The category of -comodules is abelian when (A, ) is flat [Rav86 , Theorem A1.1.3]. We now recall the definition of Landweber exactness, mentioned in the introdu* *c- tion. Definition 2.1. Suppose (A, ) is a flat Hopf algebroid, and f :A -!B is a ring homomorphism. We say that B is Landweber exact over (A, ), or just over A, if the functor M 7! B A M from -comodules to B-modules is exact. We next recall the construction of the Hopf algebroid B , and use it to refo* *r- mulate the notion of Landweber exactness. The definition is motivated by the following construction on groupoids. Consider a groupoid with object set X and morphism set G. Given a set Y and a map f :Y -! X, we define a new groupoid (Y, Gf) as follows: the object set is Y , and the morphisms in Gf from y1 to y0* * are the morphisms in G from f(y1) to f(y0), so as a set we have Gf = Y xX,fG xX,fY. The map f induces a full and faithful functor F :(Y, Gf) -!(X, G). To understand when this is an equivalence, consider the set U = {(y, g) | y 2 Y , g 2 G , f(y) = target(g)} = Y xX G. There is a map ss :U -!X given by (y, g) 7! source(g). Our functor F is essenti* *ally surjective, and thus an equivalence, iff ss is surjective. Now suppose we have a Hopf algebroid (A, ) and an A-algebra B. For any ring T , we have a groupoid (Rings (A, T ), Rings( , T )) and a map Rings(B, T ) -!Rings (A, T ). We can apply the construction above to obtain a new groupoid (Rings (B, T ), Rings( B , T )), where B = B A A B as before. The groupoid structure is natural in T , so Yoneda's lemma gives (B, B* * ) the structure of a Hopf algebroid. For further details, see [Hov02b , p. 1315]. The* *re is a morphism = (f, "f): (A, ) -!(B, B ) of Hopf algebroids, where "f(u) = 1 u 1; 8 MARK HOVEY AND NEIL STRICKLAND this corresponds to the functor F . The morphism induces a functor * from - comodules to B -comodules, given by M 7! B A M; by definition, this is exact * *iff B is Landweber exact over A. The map ss :U -! X corresponds to the ring map f jR :A -!B A given by a 7! 1 jR (a). Lemma 2.2. Suppose (A, ) is a flat Hopf algebroid, and f :A -! B is a ring homomorphism. Then B is Landweber exact if and only if the map f jR makes B A into a flat A-algebra. Proof.Suppose that B is Landweber exact, and M -! N is a monomorphism of A-modules. Then A f is a monomorphism as well, since is flat as a right A- module. But A M and A N are both -comodules, with the coaction coming from the diagonal on . (This is called the extended comodule structure on A M). This makes A f a comodule map. Since B is Landweber exact, we conclude that B A A f is a monomorphism. Conversely, suppose that B A is flat over A, and u: M -! N is a monomor- phism of comodules. The coaction map _M is a split monomorphism of A-modules; the splitting is given by ffl 1, where ffl is the counit of (A, ). Hence u i* *s a retract of A u as a map of A-modules. It follows that B A u is a retract of B A A* * u as a map of B-modules. Since B A is flat over A, we conclude that B A u is a monomorphism, as required. Corollary 2.3. If B is Landweber exact over A, then B is flat over B (so the category of B -comodules is abelian). Proof.We have seen that B A is flat as a right A-module; now take tensor products with B on the right. For any morphism of flat Hopf algebroids, the functor * obviously preserves colimits, so it should have a right adjoint *; we next check that this works. Lemma 2.4. Suppose : (A, ) -!(B, ) is a map of flat Hopf algebroids. Then the functor *: -comod -! -comod defined by *M = B A M has a right adjoint *. This lemma is proved in [Hov04 , Section 1], but it is central to our work, s* *o we recall the proof here. Proof.First note that any -comodule N is the kernel of a map of extended co- modules. Indeed, the structure map _N :N -! B N is a comodule map if we give B N the extended comodule structure, and an embedding because it is split over B by the counit of . If we let q : B N -!Q denote the quotient, then we get a diagram of comodules N _N--! B N _Qq---! B Q expressing N as the kernel of a map of extended comodules. Adjointness forces us to define *( B P ) = A P for any B-module P . Once we define * on maps between extended comodules such as _Q q, we can then define *N as the kernel of *(_Q q). So suppose we have a map f : B P -! B P 0. We need to define *f : A P -! A P 0. COMODULES AND LANDWEBER EXACT HOMOLOGY THEORIES 9 By adjointness, it suffices to define a map of A-modules A P -! P 0. We defi* *ne this map as the composite A P -! B P -f! B P 0-!P 0. Here the first map is induced by the map -! and the last map is induced by the counit of . Remark. It can be shown that when N is a -comodule, the group N A has compatible structures as a -comodule and a -comodule. This makes the - primitives Prim (N A ) into a -comodule, which turns out to be isomorphic to *N. One can give a proof of the existence of * based on this formula, but we * *do not need it so we omit the details. Mark Behrens pointed out to the authors that *N ~=( A B) N, where the symbol denotes the cotensor product; see [Rav86 , Lemma A1.1.8]. We can now prove the main result of this section, which is also Theorem A of the introduction. Theorem 2.5. Suppose (A, ) is a flat Hopf algebroid, and B is a Landweber exact A-algebra. Let : (A, ) -! (B, B ) denote the corresponding map of Hopf algebroids, inducing *: -comod -! B -comodwith right adjoint *. Then the counit of the adjunction ffl: * *M -! M is a natural isomorphism. Hence * * is localization with respect to the hereditary torsion theory T = ker *, and * defines an equivalence of categories from B -comodto LT ( -comod). Proof.Since B is Landweber exact, * is exact, so ffl is a natural transformati* *on of left exact functors. Since every comodule is a kernel of a map of extended comodules, it suffices to check that fflN is an isomorphism for extended comodu* *les N = B B V . But then we have * *N ~=B A ( A V ) ~=B A A B B V ~= B B V ~=M, as required. Proposition 1.6 completes the proof. 3. Torsion theories of BP*BP -comodules Suppose (A, ) is a flat Hopf algebroid, and B is a Landweber exact A-algebra. Theorem 2.5 shows that the category of B -comodules is equivalent to the local* *iza- tion of the category of -comodules with respect to some hereditary torsion the* *ory T . Thus we would like to classify all hereditary torsion theories of -comodul* *es. This is of course impossible in general, but it turns out to be tractable in th* *e cases of most interest in algebraic topology. In this section, we concentrate on the * *case (A, ) = (BP*, BP*BP ), where BP is the Brown-Peterson spectrum. Recall that BP* ~=Z(p)[v1, v2, . .]., where |vi| = 2(pi-1). We choose the vit* *o be the Araki generators [Rav86 , Section A2.2] for definiteness, but all that matt* *ers is that vn is primitive modulo In = (p, v1, . .,.vn-1). The ideals In are independ* *ent of the choice of generators. For notational purposes, we take v0 = p and v-1 = 0. * *We also write s for the shift functor on BP*BP -comodules, so that (sM)n = Mn-1. Let Tn denote the class of all graded BP*BP -comodules that are vn-torsion, in the sense that each element is killed by some power of vn, depending on the ele* *ment. 10 MARK HOVEY AND NEIL STRICKLAND By Lemma 2.3 of [JY80 ], M is vn-torsion if and only if M is In+1-torsion, so T* *n is independent of the choice of generators. The following theorem is Theorem B of the introduction. Theorem 3.1. Let T be a graded hereditary torsion theory of graded BP*BP - comodules, and suppose that T contains some nonzero comodule that is finitely presented over BP*. Then T = Tn for some n -1. The reader should compare Theorem 3.1 to the classification of Serre classes of finitely presented BP*BP -comodules in [JLR96 ] (which they call thick sub- categories). Given a hereditary torsion theory T , the collection F of all fin* *itely presented comodules in it is a Serre class (of all the finitely presented comod* *ules); combining Theorem 3.1 with the result of [JLR96 ] says that as long as F is non* *zero, then T is uniquely determined by F. We do not know what happens when there are no nonzero finitely presented comodules in T . In this case, Proposition 3.3 below implies that every comodule in T is vn-torsion for all n. Ravenel [Rav84 , Section 2] conjectures that th* *ere are uncountably many different Bousfield classes of spectra BP I where I is an infinite regular sequence in BP*. One might similarly conjecture that there are uncountably many different hereditary torsion theories T containing no nonzero finitely presented comodules. Theorem 3.1 will follow from the two propositions below. Proposition 3.2. Tn is the graded hereditary torsion theory generated by the BP*BP -comodule BP*=In+1. Proposition 3.3. Suppose that T is a graded hereditary torsion theory of graded BP*BP -comodules such that BP*=In 62 T . Then T Tn. Given these two propositions, Theorem 3.1 follows easily. Proof of Theorem 3.1.Suppose T is a graded hereditary torsion theory containing the nonzero finitely presented comodule M. The Landweber filtration theorem for BP*BP -comodules [Lan76, Theorem 2.3] guarantees that M has a subcomodule of the form stBP*=Ij for some j and some t. Thus BP*=Ij 2 T . Let n + 1 = min{j | BP*=Ij 2 T }. Then T Tn by Proposition 3.2. On the other hand, BP*=In 62 T , so T Tn by Proposition 3.3. Hence T = Tn, as required. We owe the reader proofs of Proposition 3.2 and Proposition 3.3. We need the following lemma. Lemma 3.4. Suppose M is a nonzero vn-torsion graded BP*BP -comodule. Then M has a nonzero primitive x such that In+1 Ann(x). p_____ Proof.Let y be a nonzero element of M, and let I = Ann y. Since y is vn-torsio* *n, it is also In+1-torsion, and so In+1 I. Theorem 2 of [Lan79] guarantees that there is a primitive x with Ann(x) = I. Proof of Proposition 3.2.Let T denote the graded hereditary torsion theory gen- erated by BP*=In+1. Since one can easily check that Tn is a graded hereditary torsion theory, and BP*=In+1 2 Tn, we see that T Tn. Conversely, suppose M is vn-torsion. We construct a transfinite increasing sequence Mffof subcomodules COMODULES AND LANDWEBER EXACT HOMOLOGY THEORIES 11 of M such that each Mffis in T . This sequence will be strictly increasing unle* *ss Mfi= M for some fi, so we conclude that M = Mfi2 T . To construct M0, we use Lemma 3.4 to find a nonzero primitive x 2 M such that In+1x = 0. This gives a subcomodule M0 ~=stBP*=I of M such that I In+1. Hence M0 is isomorphic to a quotient of stBP*=In+1, so M0 2 T . This begins the transfinite induction. The limit ordinal step of the induction is simple.SIf we* * have defined Mfffor all ffL< fi for some limit ordinal fi, we define Mfi= ff n. The Landweber filtration theorem [Lan76, Theorem 2.3] tells us that every finitely presented -comodule T is an iterated extension of suspensions of comodules of the form A=Ir; if T is vn-torsion, one can easily check that r > n. Thus Ext1(T, M) = 0 for all finite* *ly presented vn-torsion comodules. Now every comodule T is a filtered colimit of finitely presented comodules, by Lemma 1.15 of [JY80 ]. If T is vn-torsion, the* *n the finitely presented vn-torsion comodules mapping to T are cofinal, so T is a fil* *tered colimit colimTffof finitely presentedLvn-torsion comodules. This gives a short * *exact sequence T 00-!T 0-!T , where T 0= ffTff(so Ext1,*(T 0, M) = 0) and T 00 T 0 (so T 00is vn-torsion and thus Hom *(T 00, M) = 0). It follows that Ext1,*(T, M* *) = 0 as required. In [HS03 ], we strengthen Corollary 4.3 by showing that 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, COMODULES AND LANDWEBER EXACT HOMOLOGY THEORIES 13 with the Ext group computed over the ring BP* rather than the Hopf algebroid BP*BP . This Ext group can easily be computed from a Koszul resolution and so is much more accessible than the previous one. Corollary 4.4. Let (A, ) = (BP*, BP*BP ), and suppose B -! B0 is a map of Landweber exact A-algebras. Let T denote the graded hereditary torsion theory of B -comodules generated by B=IhtB0+1 if htB0 < 1 and (0) if htB0 = 1. Then the functor M 7! B0 B M defines an equivalence of categories between the local- ization of the category of graded B -comodules with respect to T and the categ* *ory of graded B0-comodules. In particular, if htB = htB0, then this functor is its* *elf an equivalence of categories. This corollary is a special case of the general fact that maps between locali* *zations are themselves localizations; see [HPS97 , Lemma 3.1.5]. Example 4.5. There are well-known maps v-1nBP* -!E(n)* -!En* of Landweber exact BP*-algebras of height n. These maps induce equivalences of the associated categories of comodules. Note that they certainly do not induce equivalences of the associated categories of modules; in particular, E(n)* is N* *oe- therian and v-1nBP* is not. We can now give a straightforward and systematic account of some well-known change of rings theorems, as mentioned in the introduction. The following is our general result; it follows immediately from Corollary 4.4. Theorem 4.6. Let (A, ) = (BP*, BP*BP ), and suppose B -! B0 is a map of Landweber exact A-algebras such that htB = htB0. Then Ext**(BM, N) ~=Ext** B0(B0 B M, B0 B N). The Morava change of rings theorem [Mor85 ] is often stated in precisely this form. We give a graded version of it, as opposed to the ungraded version given in [Rav86 , Theorem 6.1.3]. Corollary 4.7. Suppose (A, ) = (BP*, BP*BP ), and let I denote the ideal in A generated by p and all the vi except vn. Let B denote the completion of v-1nA a* *t I, and let B0 denote the completion of E(n)* at In. Then Ext**(BM, N) ~=Ext** B0(B0 B M, B0 B N) for all B -comodules M and N. Note that the Morava change of rings theorem was only known before in case M = B. Here is our version of the Miller-Ravenel change of rings theorem [MR77 , The* *o- rem 3.10]. Corollary 4.8. Let (A, ) = (BP*, BP*BP ), B = v-1nBP*, and B0 = E(n)*. Then Ext**(BM, N) ~=Ext** B0(B0 B M, B0 B N) for all B -comodules M and N. 14 MARK HOVEY AND NEIL STRICKLAND The Miller-Ravenel change of rings theorem is usually stated as (4.9) Ext**BP*BP(BP*, N) ~=Ext**E(n)*E(n)(E(n)*, E(n)* BP* N) for all BP*-comodules N on which vn acts invertibly. This is a consequence of Corollary 4.8, arguing as in Lemmas 3.11 and 3.12 of [MR77 ]. The point is esse* *n- tially as follows: if vn acts invertibly on N, then nothing changes if we inver* *t vn in BP* and BP*BP . Moreover, N is necessarily In-torsion, so vn is asymptotically primitive on anything involving N, so we need not distinguish between inverting jL(vn) or jR (vn). We have also generalized the statement 4.9 of the Miller-Ravenel change of ri* *ngs theorem in [Hov02a ], expressing it as an isomorphism between morphism sets in appropriate derived categories. Similarly, if m n, we can apply Theorem 4.6 to the map v-1nBP* -!v-1nE(m)* to get a version of the change of rings theorem of [HS99 , Theorem 3.1]. Corollary 4.10. Suppose (A, ) = (BP*, BP*BP ), and let B = v-1nBP* and B0= v-1nE(m)* for m n. Then Ext**(BM, N) ~=Ext** B0(B0 B M, B0 B N) for all B -comodules M and N. Again, the methods of Miller and Ravenel allow one to derive the original cha* *nge of rings theorem of the first author and Sadofsky from Corollary 4.10. 5.The structure of E*E-comodules This section is devoted to proving analogues for E*E-comodules of the Landwe- ber structure theorems for BP*BP -comodules, when E* is Landweber exact over BP*. Theorem 5.1. Let (A, ) = (BP*, BP*BP ), and suppose B is a Landweber exact A-algebra. Then every nonzero B -comodule has a nonzero primitive. Proof.Let : (A, ) -! (B, B ) denote the evident map of Hopf algebroids. Let *: -comod -! B -comoddenote the induced functor, with right adjoint *. Sup- pose M is a nonzero B -comodule. We must show that Hom * B(B, M) is nonzero. But adjointness implies that Hom * B(B, M) = Hom * B( *A, M) ~=Hom *(A, *M). Since * *M ~= M by Theorem 2.5, *M is a nonzero -comodule. It is well- known that every nonzero -comodule has a primitive; it follows for example from Lemma 3.4. Thus Hom *(A, *M) is nonzero, as required. We now compute the primitives in B=In for all n. The case B = BP* is well- known [Rav86 , Theorem 4.3.2]. Theorem 5.2. Let (A, ) = (BP*, BP*BP ), and suppose B is a nonzero Landwe- ber exact A-algebra. (a) If htB > 0, then Hom * B(B, B) ~=Z(p). (b) If htB = 0, then Hom * B(B, B) ~=Q. (c) If htB > n > 0, then Hom * B(B, B=In) ~=Fp[vn]. (d) If 1 > htB = n > 0, then Hom * B(B, B=In) ~=Fp[vn, v-1n]. (e) If htB = 1 then Hom * B(B, B=I1 ) ~=Fp. COMODULES AND LANDWEBER EXACT HOMOLOGY THEORIES 15 (f) If n > htB then B=In = 0 and so Hom * B(B, B=In) = 0. The simplest way to prove this theorem is to use the following computation. R* *e- call that Ln denotes the localization functor on the category of BP*BP -comodul* *es with respect to the hereditary torsion theory of vn-torsion comodules. Lemma 5.3. For n < 1 we have Ln(BP*=In) ~=v-1nBP*=In and Ln(BP*=Im ) = BP*=Im for m < n. As usual, we let v0 = p and I0 = (0) in interpreting the statement of this le* *mma. Recall also that L1 is the identity functor, so the n = 1 case is trivial. Proof.Let M denote either BP*=Im (for m < n) or v-1nBP*=In (for m = n). It suffices to show that M is Ln-local, since the map BP*=In -!v-1nBP*=In has vn- torsion cokernel, so is an Ln-equivalence. For this, we use Corollary 4.3. Sinc* *e M has no vn-torsion, it suffices to show that Ext1,*BP*BP(BP*=In+1, M) = 0. We first show that Ext1,*BP*(BP*=In+1, M) = 0. So suppose we have a short exa* *ct sequence of BP*-modules (5.4) 0 -!M -i!X g-!stBP*=In+1 -!0. Let x 2 X be such that g(x) is the generator of stBP*=In+1. The argument now depends on whether M = BP*=Im or M = v-1nBP*=In. In case M = v-1nBP*=In, note that vnx 2 M. Since vn acts invertibly on M, there is a y 2 M such that vny = vnx. Then vn(x - y) = 0, and also vi(x - y) 2 M for i < n. Hence vnvi(x - y) = 0, and so vi(x - y) = 0 since M has no vn-torsion. Thus x - y def* *ines a splitting of our given exact sequence 5.4. Now suppose M = BP*=Im for some m < n. Then vm x and vm+1 x are both in M. Since vm+1 (vm x) = vm (vm+1 x) and M is a unique factorization domain, we conclude that vm x = vm y for some y 2 M, and that vm+1 x = vm+1 y. Now a similar argument as we used in case M = v-1nBP*=In shows that vix = viy for all i n. Hence x - y defines a splitting of 5.4. Now suppose the short exact sequence 5.4 is a sequence of BP*BP -comodules. Write X ~=M stBP*=In+1 as BP*-modules. We claim that this splitting must be a splitting of BP*BP -comodules as well. Indeed, let Y be the vn-torsion in X, which is a subcomodule and is obviously just 0 stBP*=In+1. Hence the map Y -! X -! BP*=In+1 is an isomorphism, and its inverse gives a splitting of the sequence. Proof of Theorem 5.2.As usual, let * denote the functor from -comodules to B -comodules that takes M to B A M. Then we have Hom * B(B, B=Im ) = Hom * B( *A, *A=Im ) ~=Hom * (A, * *(A=Im )) ~=Hom *(A, Ln(A=Im )). Parts (a) to (d) of the theorem now follow from Lemma 5.3 and [Rav86 , Propo- sition 5.1.12]. Part (e) follows from the observation that Hom *(A, BP*=I1 ) = BP*=I1 = Fp. Part (f) is just the definition of htB, recorded for ease of compa* *ri- son. 16 MARK HOVEY AND NEIL STRICKLAND We now consider the analogue of Landweber's classification of invariant radi- cal ideals in BP* [Lan76, Theorem 2.2]. For this to work smoothly, we need to modify the problem slightly. Consider an abelian category A with a symmetric monoidal tensor product, and let k be the unit for the tensor product. We define a categorical ideal in A to be a subobject of k; given ideals I and J, we let IJ denote the image of the evident map I J -! k. We say that I is categorically radical if J2 I implies J I. This notion is evidently invariant under monoi* *dal equivalences of abelian categories, such as those in Theorem 4.2. We now specialize to the case A = (B, )-comod. The categorical ideals are then the invariant ideals in B. An invariant radical ideal is categorically rad* *ical, but the converse need not be true. For example, take (A, ) = (BP*, BP*BP ) as before, and B = BP*[x]=x2, and = B . Then InB is categorically radical, but not radical. Indeed, the only invariant ideals are those of the form IB for so* *me invariant ideal I BP*, and IB is never radical. One can easily check that the proof of Landweber's classification of invariant radical ideals in BP* in [Rav86 , Theorem 4.3.2] also classifies categorically * *radical ideals. That is, we have the following theorem. Theorem 5.5. The ideal I BP* is a categorically radical invariant ideal if and only if I = In for some 0 n 1. Almost the same theorem holds for categorically radical ideals in Landweber exact BP*-algebras. Theorem 5.6. Suppose (A, ) = (BP*, BP*BP ) and B is a Landweber exact A- algebra. Then the categorically radical invariant ideals in B are {IkB | 0 k htB}. In particular, this set contains all the invariant radical ideals. Proof.Put n = htB. We assume that n > 0, leaving the rational case to the reade* *r. As n > 0, we have *B = LnBP* = BP* = A. Note also that * is left exact, so it preserves monomorphisms, so it sends invarant ideals in B to invariant ideal* *s in A. Consider a categorically radical invariant ideal J B, and put K = *J A, so J = *K = KB. If K = A this means that J = B, which we have implicitly excluded from consideration; so K is a proper ideal in A. We claim that K is al* *so categorically radical. Indeed, suppose we have an invariant ideal K0 with K20 * *K. Put J0 = BK0 and apply * to the maps K0 K0 -!K -!A to see that J20 J. As J is categorically radical, we have J0 J, so LnK0 = * *K0 = *J0 *J = K. Moreover, as A is an integral domain, K0 has no In+1-torsion, so K0 LnK0, so K0 K as required. Since K is categorically radical, we must have K = Ik for some 0 k 1. If k > htB, then J = *K = B. Hence 0 k htB. This classification leads to the analogue of the Landweber filtration theorem, proved by Landweber [Lan76] for BP*BP -comodules. Theorem 5.7. Suppose (A, ) = (BP*, BP*BP ), and let B be a Landweber exact A-algebra. Then every B -comodule M that is finitely presented over B admits a finite filtration by subcomodules 0 = M0 M1 . . .Ms = M for some s, with Mr=Mr-1 ~=strB=Ijrwith jr htB for all r. COMODULES AND LANDWEBER EXACT HOMOLOGY THEORIES 17 Proof.First note that M is finitely presented over B if and only if M is a fini* *tely presented object of B -comod; that is, if and only if Hom * B(M, -) commutes w* *ith all filtered colimits. This is proved in [Hov04 , Proposition 1.3.3]. It foll* *ows that the statement of Theorem 5.7 is invariant under the equivalences of categories * *in Theorem 4.2. Hence we can assume that either B = BP* or B = E(n)*. The case B = BP* is the classical Landweber filtration theorem. So now suppose B = E(n)* and M is an arbitrary graded B -comodule. We construct a subcomodule of M isomorphic to stB=Im for some p and some m n. Indeed, choose a nonzero primitive y0 in M. If Ann(y0) = (0), we are done. If n* *ot, piy0 = 0 for some i by Theorem 5.2. Choose a minimal such i and let y1 = pi-1y0. Then Ann (y1) is a proper invariant ideal containing (p). If it is (p), we are * *done. If not, then Theorem 5.2 implies that vj1y1 = 0 for some minimal j > 0. Let y2 = vj-11y1. Then y2 is primitive (since v1 is primitive mod p), and Ann(y2) i* *s an invariant ideal containing I2. We continue in this fashion until we reach an m * *such that Ann(ym ) = Im . This must happen before we reach n + 1. Now we construct Mi by induction, by applying this construction to M=Mi-1. Since M is finitely generated and B is Noetherian, M is a Noetherian module, so Ms = M for some s. 6.Weak equivalences of Hopf algebroids In this section, we show that our equivalences of comodule categories are ind* *uced by weak equivalences of Hopf algebroids. Definition 6.1. Suppose : (A, ) -!(B, ) is a map of Hopf algebroids. We say that is a weak equivalence if the induced functor *: -comod -! -comod, where *M = B A M, is an equivalence of categories. We have the following characterization of weak equivalences of Hopf algebroid* *s. Theorem 6.2. A map = ( 0, 1): (A, ) -!(B, ) of flat Hopf algebroids is a weak equivalence if and only if the composite A jR--! -0-1-!B A is a faithfully flat ring extension and the map B A A B -! that takes b x b0to jL(b) 1(x)jR (b0) is a ring isomorphism. One can rephrase this theorem using sheaves of groupoids. A Hopf algebroid (A, ) has an associated sheaf of groupoids Spec(A, ) with respect to the flat topology on Aff, the opposite category of commutative rings (see [Hov02b ]). Ho* *l- lander [Hol01] has constructed a Quillen model structure on (pre)sheaves of gro* *up- oids in a Grothendieck topology, and Theorem 6.2 says that is a weak equivale* *nce if and only if Spec is a weak equivalence of sheaves of groupoids. Proof.The "if" half of this theorem is the main result of [Hov02b ]. Conversel* *y, suppose is a weak equivalence. Then * is in particular exact, so that B is Landweber exact over A. Lemma 2.2 then guarantees that B A is flat over A. On the other hand, if B A A M = 0, then *( A M) = 0, so, since * is an equivalence of categories, A M = 0. Since A is an A-module retract of , we * *see that M = 0. Hence B A is faithfully flat over A. 18 MARK HOVEY AND NEIL STRICKLAND Now, if * is an equivalence of categories, then the counit * *N -! N must be an isomorphism for all -comodules N. In particular, * * -! must be an isomorphism. But * * ~=B A *( B B) ~=B A A B, completing the proof. For rings R and S, we can have equivalences of categories between R-modules and S-modules that are not induced by maps R -!S; this is, of course, the conte* *nt of Morita theory. However, two commutative rings are Morita equivalent if and o* *nly if they are isomorphic. We view our Hopf algebroids as fundamentally commutative objects, so we do not expect any non-trivial Morita theory. Conjecture 6.3. Suppose (A, ) and (B, ) are flat Hopf algebroids such that the category of -comodules is equivalent to the category of -comodules. Then (A, * * ) and (B, ) are connected by a chain of weak equivalences. If this conjecture is going to be true, then in particular the equivalences o* *f co- module categories in Theorem 4.2 must be induced by chains of weak equivalences. We will now prove this. Let (A, ) = (BP*, BP*BP ) as usual. If we have two Landweber exact A- algebras B and B0 of the same heights, and a map (B, B ) -! (B0, B0) under (A, ), then it is a weak equivalence by Corollary 4.4. In general there may be* * no such map, though. We therefore record another obvious source of equivalences. Suppose we have a groupoid with object set X and morphism set G. Given another set Y and a map f :Y -! X, we previously constructed a groupoid (Y, Gf), where the morphisms in Gf from y1 to y0 are the morphisms in G from f(y1) to f(y0). Now suppose we have another map g :Y -! X, and thus another groupoid (Y, Gg); we want to know when this is equivalent to (Y, Gf). Suppose we have a map h: Y -!G such that targetO h = f and sourceO h = g, so that h(y) is a morphism from g(y) to f(y) in G. We can then define a functor H :Gg -!Gf by H(y) = y on objects, and -1 H(y0 u-y1) = (f(y0) -h(y0)--g(y0)- ug(y1) -h(y1)----f(y1)) on morphisms (for u 2 Gg(y1, y0) = G(g(y1), g(y0))). Equivalently, let H0 be the map Y xX,gG xX,gY -hx1x(invertOh)---------!G xX G xX G compose-----!G. Then H(y0, u, y1) = (y0, H0(y0, u, y1), y1). It is easy to see that the functor* * H is an isomorphism of groupoids. The analogue for Hopf algebroids is as follows. Lemma 6.4. Let (A, ) be a Hopf algebroid, and suppose h: -! B is a ring homomorphism. Let f = hjL and g = hjR . Then there is an isomorphism of Hopf algebroids from (B, g) to (B, f). Proof.The pair (A, ) represents a functor from graded rings to groupoids, and * *the conclusion follows from the above discussion by Yoneda's lemma. Alternatively, * *we can give a formula for the map f -! g as follows. A map f -! g of B-bimodules is equivalent to a map -!B g g B of A-bimodules, where the target has the A-bimodule structure coming from f. This map is the composite -! A --1-! A A h-1-(hOO)-----!B g g B COMODULES AND LANDWEBER EXACT HOMOLOGY THEORIES 19 (corresponding to H0 in the previous discussion). Theorem 6.5. Let (A, ) = (BP*, BP*BP ), and suppose B and B0are Landweber exact A-algebras such that htB = htB0. Then the Hopf algebroids (B, B ) and (B0, B0) are connected by a chain of weak equivalences. Proof.Let C = B A A B0. Let us denote C together with the ring homo- morphism f :A -! B -! C by Cf, and C together with the ring homomorphism g :A -!B0-! C by Cg. Our desired chain of weak equivalences is (B, B ) -!(Cf, f) ~=(Cg, g)- (B0, B0). The middle isomorphism comes from the evident map h: -!C such that hjL = f and hjR = g and Lemma 6.4. We now claim that Cf, and therefore also Cg, is Landweber exact. Indeed, Lemma 2.2 implies that B A and B0 A are flat over A. But then C A = (B A ) A (B0 A ) is also flat over A, and so Cf is Landweber exact. Thus, it suffices to show that htCf = htCg = htB. Because In is invariant, we have C=In ~=(B=In) A A (B0=In), and therefore B0=In = 0 implies C=In = 0. Conversely, suppose C=In = 0, but B0=In 6= 0. This means that htB = htB0 n. Since B A ( A B0=In) = 0, we conclude that A B0=In is vhtB-torsion, and therefore vn-torsion. But B0=In is a retract of A B0=In as an A-module, so B0=In is vn-torsion. Since B0 is Landweber exact, this means B0=In = 0, which is a contradiction. 7. The global case The object of this section is to show that our results about Landweber exact algebras over BP* extend to Landweber exact algebras over the complex cobordism ring MU*. Recall that MU* ~=Z[x1, x2, . .].for some generators xiof degree 2i. * *All we require of these generators is that the Chern numbers of xpn-1 are all divis* *ible by p, as in [Lan76]. In this case, the ideals Ip,n= (p, xp-1, . .,.xpn-1-1)Sare in* *variant and independent of the choice of generators. These ideals and Ip,1 = n Ip,nare the only invariant prime ideals in MU* [Lan76]. Our first goal is to understand the relation between graded hereditary torsion theories of MU*MU-comodules and graded hereditary torsion theories of BP*BP - comodules. We use the notation A(p)for A ZZ(p), and we recall the well-known fa* *ct that (MU*)(p)is a Landweber exact BP*-algebra of infinite height. Theorem 4.2 then gives us an equivalence of categories between graded (MU*MU)(p)-comodules and graded BP*BP -comodules. Lemma 7.1. Let T be a graded proper hereditary torsion theory of graded MU*MU- comodules, and,Lfor a prime p, let T (p)denote the class ofLp-torsion comodules* * in T . Then T = pT (p); that is, M 2 T if and only if M = M(p)for M(p)2 T (p). Furthermore, there is a one-to-one correspondence between graded hereditary tor- sion theories of graded p-torsion MU*MU-comodules and graded proper hereditary torsion theories of graded BP*BP -comodules. 20 MARK HOVEY AND NEIL STRICKLAND Here we refer to a hereditary torsion theory as proper if it is not the entire category. Proof.First of all, if T is proper, then T must consist entirely of comodules t* *hat are torsion as abelian groups. Indeed, suppose M 2 T is non-torsion. Let T (M) deno* *te the torsion in M, which is easily seen to be a comodule. Let x be a nonzero pri* *mitive in M=T (M) 2 T . Then x is non-torsion. The annihilator ideal I of x is invaria* *nt, and we claim it is 0. Indeed, if I is nonzero, it must contain a nonzero invari* *ant element of MU*, which must be an integer m. But then mx = 0, contradicting the fact that x is non-torsion. The subcomodule of M=T (M) generated by x is thus isomorphic to stMU* for some t, so MU* 2 T . This implies that T is the entire category of MU*-comodules. Indeed, we then get MU*=I 2 T for all invariant ideals I. The Landweber filtration theorem implies that every finitely present* *ed MU*MU-comodule is in T , and every comodule is a filtered colimit of finitely presented comodules. L Now it is easy to check that every torsion comodule M can be written as (p)M(p), where M(p)is the p-localization of M and therefore is just the p-tor* *sion in M. The correspondence between graded hereditary torsion theories of p-torsion MU*MU-comodules and proper graded hereditary torsion theories of BP*BP - comodules follows from the equivalence of categories between BP*BP -comodules and (MU*MU)(p)-comodules. We then let Tn(p)denote the hereditary torsion theory of p-torsion MU*MU- comodules corresponding to Tn. Thus Tn(p)is generated by MU*=Ip,n+1. For nota- tional reasons, we let T1(p)= (0). Definition 7.2. Given an MU*-module B and a prime p, define the height of B at p, written htpB, to be the largest n such that B=Ip,nis nonzero, or 1 if B=I* *p,n is nonzero for all n. We then have the integral analogue of Theorem 4.2. Theorem 7.3. Let (A, ) = (MU*, MU*MU), and suppose B and B0 are two graded Landweber exact A-algebras with htpB = htpB0 for all primes p. Then the category of graded B -comodules is equivalent to the category of graded B* *0- comodules, and both categories are equivalent to the localizationLof the catego* *ry of graded -comodules with respect to the torsion theory pTh(p)tpB. This localiz* *ation is the full subcategory of graded -comodules consisting of all those M such th* *at Hom *A(A=Ip,htpB+1, M) = Ext1,*(A=Ip,htpB+1, M) = 0 for all p such that htpB < 1. Note that this theorem implies Theorem 4.2, since if B is a Landweber exact BP*-algebra, it is also a Landweber exact MU*-algebra. Proof of Theorem 7.3.Theorem 2.5 implies that graded B -comodules are equiva- lent to the localization of the category of graded -comodules with respect to * *the kernel T of the functor M 7! B A M. Given a prime p, let T (p)denote the col- lection of p-torsion comodules in T . If B is zero, there is nothing to prove, * *so we can assume B is nonzero and therefore T is proper. Lemma 7.1 then implies that we need only check that T (p)= Th(p)tpB. COMODULES AND LANDWEBER EXACT HOMOLOGY THEORIES 21 Suppose first that htpB = 1. Then we claim that T (p)= (0). Indeed, suppose M is a nonzero comodule in T (p). By choosing a primitive in M, we find that A=I 2 T (p)for some proper invariant ideal I in A such that pr 2 I for some r. But then I is an invariant ideal in A(p). The equivalence of categories between graded (p)-comodules and graded BP*BP -comodules preserves invariant ideals. Since every proper invariant ideal in BP* is contained in I1 , we see that I * *Ip,1. Thus A=Ip,1 2 T . Hence B=Ip,1 = 0. This means that 1 2 Ip,1B, so 1 2 Ip,nB for some n. But then B=Ip,n= 0, violating our assumption that htpB = 1. Now suppose that htpB = n < 1. Then A=Ip,nis not in T (p)but A=Ip,n+12 T (p). Propositions 3.2 and 3.3 imply that T (p)corresponds to the hereditary t* *orsion theory Tn of BP*BP -comodules. L The characterization of local objects follows from the fact that T = pT (p* *), Lemma 1.8, and Corollary 4.3. We then get analogues of the results of Sections 4-6 for MU*MU-comodules. We will state only the structure theorem for comodules. We have the same difficulty with the classification of invariant prime ideals that we have with invariant r* *adical ideals in the BP*-case. We fix it analogously. That is, if A is a symmetric mon* *oidal abelian category with unit k for the tensor product, we define a categorical id* *eal I in A to be categorically prime if JK I for categorical ideals J and K implies that J I or K I. One checks that Landweber's classification of invariant pr* *ime ideals in MU* [Lan73] actually classifies categorically prime invariant ideals. Theorem 7.4. Let (A, ) = (MU*, MU*MU), and suppose B is a Landweber exact A-algebra. (a) Every nonzero graded B -comodule has a nonzero primitive. (b) The categorically prime invariant ideals in B are {Ip,nB | 0 n htpB}. In particular, this set contains all the invariant prime ideals. (c) If B is Noetherian, then every graded B -comodule that is finitely gene* *rated over B admits a finite filtration by subcomodules 0 = M0 M1 . . .Ms = M for some s, with Mr=Mr-1 ~=stB=Ip,jfor some s,p, and j depending on r with j htpB. In particular, this theorem applies to K*K-comodules or Ell*Ell-comodules, where K is complex K-theory, and Ell is one of the many versions of (periodic, complex oriented) elliptic cohomology. We leave the proof of this theorem to the interested reader, except for a few comments that illustrate the differences between this theorem and Theorem D. First of all, in part (b) we need to assume that I is categorically prime, rath* *er than just categorically radical. This is already true in MU*, since the ideal (* *6), for example, is an invariant radical ideal in MU* not of the desired form. Secondly, in the proof of part (c), we need a Noetherian hypothesis that is n* *ot present in the corresponding fact for (A, ) = (BP*, BP*BP ). The reason for th* *is is that, in the BP*BP case, the category of B -comodules is either equivalent * *to the category of BP*BP -comodules or to the category of E(n)*E(n)-comodules. In the first case, we already know the Landweber filtration theorem, and in the second case E(n)* is Noetherian. We believe that Theorem 7.4(c) is true without the Noetherian hypothesis, however. 22 MARK HOVEY AND NEIL STRICKLAND 8. BP J*BP J-comodules Throughout this section, we let J be a fixed invariant sequence pi0, vi11, . * *.,.vik-1k-1 in BP* of length k. The spectrum BP J is constructed from BP by killing this re* *g- ular sequence, as in [JY80 ], or, in a more modern fashion, in [EKMM97 , Chap* *ter V] or [Str99]. Then BP J is an associative ring spectrum, with BP J* = BP*=J. We will assume that the product on BP J has been chosen to be commutative. This is always possible if p > 2, and we believe that it is possible for a cofinal set * *of ideals J when p = 2 although we have not checked the details. However, it is not possi* *ble when p = 2 and J = Ik. The co-operation ring BP J*BP J is not evenly graded if k > 0, but is still f* *ree over BP J*, so that (BP J*, BP J*BP J) is a Hopf algebroid. (When BP J is not commutative, the structure is more complicated.) The object of this section is * *to extend our results to Landweber exact BP J*-algebras B. The most important case is when J = In; the spectrum BP In is often called P (n). The Morava K-theory coefficient ring K(n)* is Landweber exact over P (n)* [Yos76]. Our first job is to classify the herditary torsion theories of BP J*BP J-como* *dules. As before, we let Tn denote the class of all graded BP J*BP J-comodules that are vn-torsion. By Lemma 2.3 of [JY80 ], M is vn-torsion if and only if M is In+1- torsion. Of course, any BP J*-module is automatically Ik-torsion, so this is o* *nly interesting for n k - 1. The following theorem is our generalization of Theorem 3.1. Theorem 8.1. Let T be a graded hereditary torsion theory of graded BP J*BP J- comodules, and suppose that T contains some nonzero comodule that is finitely presented over BP J*. Then T = Tn for some n k - 1. This theorem is proved just as Theorem 3.1, except that the results of [Lan79* *], which we used in the proof of Lemma 3.4, are not written so as to apply to BP J*BP J-comodules. So one can either reprove the results of [Lan79] in this case, or construct a direct proof of Lemma 3.4 for BP J*BP J-comodules using the results of [JY80 ]. We can then define the height for BP J*-algebras. Definition 8.2. Suppose B is a nonzero graded BP J*-module. We define the height of B, written htB, to be the largest n such that B=In is nonzero, or 1 if B=In is nonzero for all n. Note that every nonzero BP J*-module B has htB k. Here is the analogue of Theorem 4.2. Theorem 8.3. Let (A, ) = (BP J*, BP J*BP J), and suppose B and B0 are two graded Landweber exact BP J*-algebras with k htB = htB0 = n 1. Then the category of graded B -comodules is equivalent to the category of graded B* *0- comodules. If n = 1, these categories are equivalent to the category of graded -comodules. If n < 1, these categories are equivalent to the localization of * *the category of graded -comodules with respect to the torsion theory Tn. We then get analogues of the results of Sections 4-6 for BP J*BP J-comodules. These depend on the results of [JY80 ] on the structure of BP J*BP J-comodules * *to replace the results of Landweber on the structure of BP*BP -comodules. In particular, we get versions of the Miller-Ravenel, Morava, and Hovey-Sadof* *sky change of rings theorems. These use the spectra v-1nBP J and E(n, J) for n k * *in COMODULES AND LANDWEBER EXACT HOMOLOGY THEORIES 23 place of v-1nBP and E(n). Here E(n, J)* is Landweber exact over BP J* with E(n, J)* ~=v-1n(BP J*=(vn+1, vn+2, . .).). Here is the structure theorem for comodules. Theorem 8.4. Let (A, ) = (BP J*, BP J*BP J), and suppose B is a Landweber exact A-algebra. (a) Every nonzero graded B -comodule has a nonzero primitive. (b) The categorically radical invariant ideals in B are {InB | k n htB}. (c) Every graded B -comodule that is finitely presented over B admits a fin* *ite filtration by subcomodules 0 = M0 M1 . . .Ms = M for some s, with Mr=Mr-1 ~=stB=Ij for some s,p, and j depending on r with k j htB. References [Bak95] Andrew Baker, A version of Landweber's filtration theorem for vn-perio* *dic Hopf al- gebroids, Osaka J. Math. 32 (1995), no. 3, 689-699. MR 97h:55007 [CF66] P. E. Conner and E. E. Floyd, The relation of cobordism to K-theories,* * Lecture Notes in Mathematics, No. 28, Springer-Verlag, Berlin, 1966. MR 35 #7344 [EKMM97] A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. 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Math. 13 (1976), no. 2, 289-* *309. MR 54 #3686 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