GALOIS THEORY OF THICK SUBCATEGORIES IN MODULAR REPRESENTATION THEORY

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1 GALOIS THEORY OF THICK SUBCATEGORIES IN MODULAR REPRESENTATION THEORY MARK HOVEY AND JOHN H. PALMIERI 1. Introduction Let B be a finite-dimensional algebra over a field K. The basic question of representation theory is to classify B-modules up to isomorphism. This is too hard in general; one way to weaken the question is to ask for a classification of all subcategories of the category of B-modules satisfying certain conditions. We will focus on so-called thick subcategories. Definition 1.1. A full subcategory C of the category of finitely generated B- modules is thick if the following conditions hold: (a) If M is a direct summand of N and N C,thenM C. (b) If 0 M 1 M 2 M 3 0 is a short exact sequence and two out of the three modules M 1, M 2,andM 3 are in C, then so is the third. A thick subcategory is nontrivial if it contains a nonzero module. The thick subcategory generated by some set of finitely generated modules is the smallest thick subcategory containing those modules. This definition is a translation of the notion of a thick, or épaisse, subcategory of a triangulated category, which was introduced by Verdier [Ver77], and has been studied in stable homotopy theory (see [HPS97], for example) and more recently in the modular representation theory of finite groups [BCR97]. Note that, in the literature, a thick subcategory of an abelian category sometimes means an abelian subcategory that is closed under extensions, but our thick subcategories need not be abelian. Convention. We fix a field K. All unadorned tensor products in this paper are taken over K. All subcategories in this paper are full, so we will describe them by specifying their objects. We will concentrate on the case when B is a finite-dimensional cocommutative Hopf algebra over K. In this case, there is a B-module structure on the tensor product M N of B-modules. Definition 1.2. Suppose B is a cocommutative Hopf algebra over K. We define a thick subcategory C of B-modules to be tensor-closed if, for all M Cand all finite-dimensional B-modules N, the tensor product M N is in C. Example 1.3. Suppose that B is afinite-dimensionalcocommutativehopfalgebra over a field K, in which case projective and injective modules coincide. Here are some examples of thick subcategories of finitely generated B-modules. Date: August 11, Mathematics Subject Classification. 20C05, 20C20, 20J05, 18E30, 55P42. 1

2 2 MARK HOVEY AND JOHN H. PALMIERI (a) The subcategory of all finitely generated modules: this is clearly the largest thick subcategory and is tensor-closed. (b) The subcategory of finitely generated projective modules: this is thick because projective and injective modules coincide. It is also tensor-closed; a standard Hopf algebra argument shows that for any B-module M, B M with the usual diagonal action is isomorphic to B M with the left action, which is a free module. Note that this thick subcategory is not abelian. (c) Given a sub-hopf algebra A of B, the subcategory of finitely generated modules which are projective when restricted to A: this is thick and tensor-closed for the same reason that Example (b) is. (d) Given a module X, the subcategory of finitely generated modules M so that M X is projective: this is thick because the functor M M X is exact. It is tensor-closed since projective modules are so. Example (c) is a special case of this, with X = B A K. Indeed, using the theory of finite localization from stable homotopy theory see [HPS97], for instance one can see that every tensor-closed thick subcategory is of this form, for some (possibly infinitely generated) module X. (e) The subcategory of finitely generated modules satisfying specific homological criteria may be thick. For example, when B is graded, then Ext B (K, ) is bigraded; for any fixed number m, this collection of finitely generated modules forms a thick subcategory: {M b : Ext s,t B (K, M) =0whens mt + b}. This is thick by the long exact Ext sequence and the 5-lemma, but may not be tensor-closed in general. (f) Every nontrivial tensor-closed thick subcategory C contains all finitely generated projective modules. Indeed, it suffices to show that C contains a free module. But we have already seen that B M is free for any module M. (g) If the trivial module K istheonlysimpleb-module, then every thick subcategory C is tensor-closed. This follows by induction on the composition series of N, in which all of the composition factors must be direct sums of copies of K, by assumption. (h) One can easily see that (tensor-closed) thick subcategories of the abelian category of B-modules correspond to (tensor-closed) thick subcategories of the triangulated category of stable B-modules [HPS97, Section 9.6]. In this paper, we focus on the case of the example: we assume that B is a finitedimensional cocommutative Hopf algebra over K. The main examples to keep in mind are the mod p group algebras of finite groups, and the finite-dimensional sub-hopf algebras of the mod 2 Steenrod algebra. Together with Neil Strickland, the authors have given a conjectured classification of the thick subcategories of finitely generated B-modules, when B is a finitedimensional cocommutative Hopf algebra over a field K with K being the only simple module. This is stated in [HPS97, Conjecture and Theorem 6.3.7] in the language of axiomatic stable homotopy theory; here is a paraphrase that also removes the condition that K be the only simple B-module. By results of Wilkerson [Wil81] when B is graded connected and Friedlander- Suslin [FS97] when B is ungraded, Ext B (K, K) is a Noetherian graded commutative ring. Write R for Ext B(K, K). The K-algebra R has a unique maximal homogeneous ideal: the ideal consisting of all elements in positive gradings. When B is

3 THICK SUBCATEGORIES 3 ungraded, we write Proj R for the set of non-maximal homogeneous prime ideals of R; whenb is graded, so that R is bigraded, then Proj R is the set of all nonmaximal bihomogeneous primes. A subset T of Proj R is closed under specialization if it is a union of Zariski-closed sets; that is, if p is in T and q p, thenq is in T. For each homogeneous prime ideal p of R, we construct a finitely generated B-module S(p) sothatext B (K, S(p)) approximates Ext B (K, K)/p see Section 3. Conjecture 1.4 (Hovey-Palmieri-Strickland). There is a bijection between nontrivial tensor-closed thick subcategories C of finitely generated B-modules and nonempty subsets T of Proj Ext B(K, K) closed under specialization: given a tensorclosed thick subcategory C, define T to be the set of primes {p S(p) C}. Given a subset T of prime ideals, let C be the tensor-closed thick subcategory generated by {S(p) p T }. In [HPS97], we show that the assignment sending a set T of prime ideals, closed under specialization, to the thick subcategory generated by {S(p) p T } is oneto-one (in the case that K istheonlysimpleb-module). The difficulty is showing that it is onto. Definition 1.5. Suppose that B is a finite-dimensional cocommutative Hopf algebra over a field K. Wesaythatprime ideals determine thick subcategories over B if Conjecture 1.4 holds for B. The conjecture has been verified in two particular cases: for group algebras of finite groups over an algebraically closed field, by Benson, Carlson, and Rickard; and for finite sub-hopf algebras of the mod 2 Steenrod algebra, extended to an algebraically closed field, by the authors. Recall that the mod p Steenrod algebra A is the Hopf algebra of stable additive operations on the mod p cohomology of any topological space X. Given a field K containing F p, we refer to A Fp K as the mod p Steenrod algebra defined over K. Theorem 1.6. (a) [BCR97] If G is a finite group and K is an algebraically closed field, then prime ideals determine thick subcategories over KG. (b) [HP99] If B is a finite sub-hopf algebra of the mod 2 Steenrod algebra defined over an algebraically closed field of characteristic 2, then prime ideals determine thick subcategories over B. The goal of this paper is to remove the algebraically closed condition from the previous result. Here is our main result. Theorem 1.7. Suppose that B is a finite-dimensional cocommutative Hopf algebra over a field K. Let L be a normal field extension of K. If prime ideals (in Proj Ext B L(L, L)) determine thick subcategories over B L, then prime ideals (in Proj Ext B(K, K)) do so for B. In this theorem, if B is graded, then we are to understand the phrase K is the only simple B-module to mean that every simple B-module is isomorphic to a regrading of K. Recall that a normal field extension L is an algebraic field extension such that any irreducible polynomial in K[x] that has one root in L splits in L[x] into a product of linear factors. Note, by the way, that if K is the only simple B-module, then L istheonlysimpleb L-module; this is easy to see, and is proved in Lemma 3.6. In this case, we are then classifying all thick subcategories. Combining this theorem with Theorem 1.6, we obtain the following corollary.

4 4 MARK HOVEY AND JOHN H. PALMIERI Corollary 1.8. Prime ideals determine thick subcategories for group algebras of finite groups over any field, and for finite sub-hopf algebras of the mod 2 Steenrod algebra. Any normal field extension is the composite of a Galois extension followed by a purely inseparable field extension. We say as much as possible about general algebraic extensions in Section 2, we discuss Galois extensions in Section 3, and finally we discuss purely inseparable extensions in Section 5. More precisely, Theorem 3.5 proves Theorem 1.7 in the case when L is Galois over K, and Theorem 5.1 proves the purely inseparable case. Theorem 1.7 follows immediately. We also prove in Section 4 that if L is Galois over K, then there is a bijection, quite generally, between (tensor-closed) thick subcategories over B and Galois invariant (tensor-closed) thick subcategories over B L. This result is independent of the rest of the paper; in particular, it does not assume any given classification of thick subcategories of B L-modules. A notion related to thick subcategory is a localizing subcategory: this is a subcategory of the category of all B-modules which is thick and closed under arbitrary direct sums. One can also consider tensor-closed localizing subcategories C, where now we must assume that if M Cand N is any module, then M N C. One type of tensor-closed localizing subcategory is the Bousfield class X of a module X, defined by X = {M M X is projective}. The results in [HP99] give a classification of the Bousfield classes, for certain Hopf algebras B defined over algebraically closed fields: they are in bijection with arbitrary subsets of Proj Ext B (K, K). We do not know how to prove an analogue of Theorem 1.7 for Bousfield classes, though, or for localizing subcategories. We also point out that, together with Strickland, we have conjectured that every tensorclosed localizing subcategory is a Bousfield class. We do not know of any non-trivial Hopf algebra B for which that conjecture has been settled; nonetheless, it would be nice to understand how localizing subcategories behave when one works over different fields. This paper is written using as little of the terminology of axiomatic stable homotopy theory [HPS97] as possible, so as to improve accessibility. But the authors would never have been able to prove the results without the conceptual clarity provided by the stable homotopy theoretic approach, and strongly recommend it to the reader. The authors would like to thank Dave Benson, who pointed out the likely necessity of separating Galois and purely inseparable extensions. The authors also thank Bill Graham, Tom Hagedorn, and Jim Reid for assistance with Galois theory. 2. Algebraic extensions Suppose that B is an algebra over K, andl is an extension field of K. Then B L is an algebra over L of the same dimension as B. IfB is a (cocommutative) Hopf algebra, so is B L. There is an obvious restriction functor from B L- modules to B-modules, and this restriction functor Res L K =Reshasbothaleftand a right adjoint. The left adjoint is induction Ind L K = Ind, and takes M to M L. The right adjoint, which we do not use, is coinduction, and takes M to Hom(L, M).

5 THICK SUBCATEGORIES 5 If L is finite over K, induction and coinduction coincide. Note that Res Ind M is isomorphic to a direct sum of copies of M, one for each basis element of L over K. We will usually assume that B is a cocommutative Hopf algebra, in which case the category of B-modules is symmetric monoidal under the tensor product (over K). The B-action on M N is defined using the coproduct of B. The induction functor is symmetric monoidal, but the restriction functor is not. Definition 2.1. We use the functors Ind and Res to define functions I and R between the set of tensor-closed thick subcategories of B-modules and the set of tensor-closedthick subcategoriesof B L-modules, as follows. If C is a tensor-closed thick subcategory of finitely generated B-modules, denote by I(C) the tensor-closed thick subcategory of B L-modules generated by the objects {Ind M M C}. We cannot make exactly the same definition with restriction, since Res does not preserve finite-dimensionality in general. Nevertheless, given a tensor-closed thick subcategory D of finitely generated B L-modules, we define R(D) to be the tensor-closed thick subcategory of B-modules generated by the finite-dimensional summands of Res N for N D. The following lemma shows that the definition of R is reasonable. Lemma 2.2. Suppose B is a finite-dimensional algebra over a field K, andl is an algebraic field extension of K. Then for every finite-dimensional B L-module N, there is a subfield L of L, finite-dimensional over K, andab L -module N, such that N = Ind L L N. In particular, Res N is isomorphic to a direct sum of copies of the finite-dimensional B-module Res L K N. Proof. Let {b i } be a basis of B over K, andlet{n j } be a basis of N over L. Then the B L-module structure is determined by elements α ijk of L such that b i n j = ijk α ijkn k. There are only finitely many of these α ijk, and since L is algebraic over K, the subfield L of L generated by the α ijk is finite-dimensional over K. LetN denote the L -vector space spanned by the n i. Then the α ijk define a B L -module structure on N, and it is clear that Ind L L N = N. Hence Res N = Res L K Res L L IndL L N, which is isomorphic to a direct sum of copies of Res L K N. To understand the functions I and R better, we introduce the action of the Galois group. Let G =Gal(L/K) denote the Galois group of L over K. Forσ G and N a B L-module, we define a new B L-module N σ as follows. Note that a B L- module N is just an L-vector space together with a B-module structure on Res N, so we define N σ by modifying the L-vector space structure on N: we define the L- vector space structure on N σ by α x = α σ x.thenresn σ =ResN, so this defines a B L-module. This construction defines a functor σ : (B L)-Mod (B L)-Mod, with Res I σ =ResI. The functor σ is an exact symmetric monoidal isomorphism of categories, with inverse given by σ 1. Note that for any B-module M, there is a natural isomorphism Ind M ρ (Ind M) σ, defined by ρ(m α) = m α σ. Since ( ) σ is an exact symmetric monoidal equivalence of categories, if D is a tensor-closed thick subcategory of B L-modules, then so is D σ = {N σ N D}. Definition 2.3. A (tensor-closed) thick subcategory D of B L-modules is called Galois invariant if D σ = D for all σ Gal(L/K).

6 6 MARK HOVEY AND JOHN H. PALMIERI Proposition 2.4. Suppose B is an algebra over a field K, L is an algebraic field extension of K, C is a tensor-closed thick subcategory of B-modules, and D is a tensor-closed thick subcategory of B L-modules. Then: (a) I(C) is Galois invariant. (b) R(D) =R(D σ ). (c) RI(C) =C. Proof. Part (a): I(C) σ is a tensor-closed thick subcategory containing (Ind C) σ = Ind C. Hence I(C) σ I(C). Using σ 1 to reverse the argument, we get the desired equality. Part (b): The tensor-closed thick subcategory R(D) is generated by the finitedimensional summands of Res N for N D. Since Res N σ =ResN, theresult follows. Part (c): RI(C) is the tensor-closed thick subcategory generated by the finitedimensional summands of Res Ind M, wherem C. But Res Ind M is a direct sum of copies of M, one for each basis element of L over K. ThusRI(C) containsc; but also any finite-dimensional summand of Res Ind M is a summand of a finite direct sum of copies of M, soc contains RI(C). Proposition 2.4 establishes a one-to-one correspondence between tensor-closed thick subcategories of B-modules and certain Galois invariant tensor-closed thick subcategories of B L-modules, namely, the image of I. To characterize the image of I, we study the purely inseparable case and the Galois case separately. 3. Galois extensions In this section we prove that the thick subcategory theorem descends through Galois extensions. We start by defining modules S(p), one for each prime ideal p of Ext B (K, K), and we study their behavior under induction; this requires some axiomatic stable homotopy theory. We then combine this with some basic algebraic geometry to show that the thick subcategory theorem descends. First, we examine the modules S(p). We assume that B is a finite-dimensional cocommutative Hopf algebra over K, and we write R for the graded commutative K-algebra Ext B (K, K), and Ri for the ith homogeneous piece Ext i B (K, K). Note that R is a Noetherian ring, by [Wil81] when B is graded connected and [FS97] when B is ungraded. Given a homogeneous element x R i, we can form a B- module S(x) as follows. Choose an injective resolution P for K such that each P j is finite-dimensional. Let M i denote the kernel of the map P i P i+1. Then x is realized by a map K M i, which is necessarily injective if x is nontrivial. We then let S(x) denote the cokernel of this map. One can show that S(x) is well-defined up to injective summands; in particular, any choice for S(x) generates the same tensor-closed thick subcategory. (If the reader is willing to think in the triangulated category of stable modules the quotient category obtained by identifying two maps when their difference factors through an injective then x is a self-map of degree i of K. The module S(x) is the cofiber of that self-map.) Now, given a homogeneous (necessarily finitely generated) ideal a in R, wechoose asetx 1,...,x k of homogeneous generators for a, and define S(a) =S(x 1 ) S(x k ). Here B acts on the tensor product using the diagonal on B; thisiswhywe need to assume B is a Hopf algebra.

7 THICK SUBCATEGORIES 7 Of course S(a) will depend on the choice of generators, but the following proposition shows that the tensor-closed thick subcategory generated by S(a) is independent of that choice. Proposition 3.1. Suppose B is a finite-dimensional cocommutative Hopf algebra over a field K. Given a homogeneous ideal a of Ext B (K, K), let C(a) denote the tensor-closed thick subcategory generated by S(a). LetT denote the set of minimal homogeneous primes containing a, and let C(T ) denote the tensor-closed thick subcategory generated by the S(p) for p T.Then: (a) C(a) is well-defined. (b) C(a) =C(T ). This proposition depends on several results from axiomatic stable homotopy theory. Lemma 3.2. Fix notation as in the proposition. (a) [HPS97, Lemma 6.0.9] C(a) is independent of the choice of generators of a. (b) [HPS97, Lemma 6.0.9] If a b, thens(b) is in C(a). (c) [HPS97, Theorem 3.3.3] Given a nontrivial tensor-closed thick subcategory C, there is a module L f CK so that for any finitely generated module M, M C if and only if L f CK M is projective. (d) For every homogeneous prime p, there is a module K p defined as in [HPS97, Proposition 6.0.7] satisfying the following conditions: (i) (ii) [HPS97, Proposition 6.1.7(d)] If p a, thens(a) K p is projective. [HPS97, Theorem 6.1.9] AmoduleM is projective if and only if M S(p) K p is projective for all p. Proof. There are several issues we must deal with here. First, the cited results are about a Noetherian stable homotopy category, which is triangulated, rather than the category of B-modules, which is abelian. A Noetherian stable homotopy category is a closed symmetric monoidal triangulated category such that the unit K is a small weak generator, and such that the graded self-maps of K form a Noetherian ring. The category of B-modules is abelian, not triangulated. However, if we form the stable category of B-modules by identifying maps f,g: M N if f g factors through a projective module, we do get a closed symmetric monoidal triangulated category. (A good reference for the stable category is [Ben98]). Furthermore, nontrivial (tensor-closed) thick subcategories of B-modules correspond precisely to nonempty (tensor-closed) thick subcategories of the stable category; that is, full triangulated subcategories of finite objects closed under summands and tensoring with an arbitrary finite object. The simple B-modules form a set of small weak generators; for the moment, let us assume that K is the only simple B-module, so that there is only one such. Unfortunately, graded self-maps of K in the stable module category do not form a Noetherian ring; there are negative-dimensional elements that behave badly. But the stable module category is a well-behaved localization of a Noetherian stable homotopy category, as explained in [HPS97, Section 9.6], so the results of [HPS97, Section 6] do apply to it. We also note that the statement of [HPS97, Lemma 6.0.9] assumes that the ideals in question are prime, but the proof does not. Now, in general, we will have more than one simple B-module. However, all of the results of [HPS97, Section 6] go through with a slightly relaxed definition of a Noetherian stable homotopy category. So let us redefine a Noetherian stable

8 8 MARK HOVEY AND JOHN H. PALMIERI homotopy category to be closed symmetric monoidal triangulated category with a set of small weak generators, including the unit K, such that the graded selfmaps [K, K] of K form a Noetherian ring and such that [K, M] is a finitely generated module over [K, K] for all small objects M. Thishypothesisdoeshold in the situation at hand, since Ext B(K, M) is a finitely generated module over Ext B (K, K) whenm is finite-dimensional, by [FS97] (see also [BS94] for the graded connected case). Then, in [HPS97, Section 6], one can replace every occurrence of thick subcategory by tensor-closed thick subcategory and every occurrence of π (X) by π (X DM) for all generators M, to get correct statements with virtually identical proofs. Proof of Proposition 3.1. Part (a) follows immediately from part (a) of the lemma. For part (b), Lemma 3.2(b) implies that if T is the set of minimal homogeneous primes containing a, andift is the set of all homogeneous primes containing a, then C(T )=C(T ). For the same reason, if a p, thens(p) isinc(a). Therefore, C(a) C(T ). To show the other inclusion, it suffices, by Lemma 3.2(c), to show that L f C(T ) K S(a) is projective, or equivalently, L f C(T ) K S(a) S(p) K p is projective for every prime p. Ifp a, then Lemma 3.2(d) implies that S(a) K p is projective. If p a, thens(p) C(T ), so L f C(T ) K S(p) is projective. Since tensoring with a projective yields a projective, we are done. As above, let R denote Ext B(K, K). Observe that Ext B L(L, L) = R L. OnewaytoseethisistotakeaninjectiveresolutionP of the B-module K, so that each P n is finitely-generated; then P L is an injective resolution of L as a B L-module, and Hom L (L, P L) = Hom K (K, P L). Since each P n is finite-dimensional over K, Hom K (K, P L) = Hom K (K, P ) L. SinceL is flat over K, Ext B L (L, L) = H (Hom K (K, P ) L) = H (Hom K (K, P )) L = R L. This description makes it clear that Ind S(x) isachoicefortheb L-module S(x), where x is the image of x under the identification of R with the obvious subalgebra of R L. Since induction preserves tensor product, we find that Ind S(a) isachoice for S(a e ), where a e denotes the ideal of R L generated by a. The Galois group G of L over K acts on R L by graded ring automorphisms, and we can describe this action as follows. Suppose x Ext i B L (L, L). Let P be an injective resolution of K by finite-dimensional B-modules, so that P L is an injective resolution of L. The element x corresponds to a map L M i L, where M i is the kernel of P i P i+1.thenforσ G, the element x σ corresponds to the map L σ M i L σ, using the isomorphism L σ = L. It follows that S(x) σ is a choice for S(x σ ). The action of G on R L induces an action of G on the set of ideals of R L: given an ideal b of R L, thenforσ G, b σ is the ideal b σ = {x σ x b}. By the preceding computations, for any ideal b in R L, S(b) σ is a choice for S(b σ ). We have then proved the following lemma.

9 THICK SUBCATEGORIES 9 Lemma 3.3. Suppose B is a finite-dimensional cocommutative Hopf algebra over a field K, and suppose L is an extension field of K with Galois group G. (a) Given a homogeneous ideal a of Ext B(K, K), let C(a) denote the tensor-closed thick subcategory generated by S(a). ThenI(C(a)) = D(a e ),wherei denotes the map given in Definition 2.1. (b) Given a homogeneous ideal b of Ext B L(L, L), let D(b) denote the tensorclosed thick subcategory generated by S(b). Then for σ G, D(b) σ = D(b σ ). We need one more lemma before stating and proving Theorem 3.5. Note that if p is a prime ideal of R L, thensoisp σ for any σ G =Gal(L/K). Hence G acts on Proj(R L). Lemma 3.4. Suppose K is a field, L is a Galois extension field of K with Galois group G, andr is a graded connected Noetherian graded-commutative K-algebra. Then Proj R = Proj(R L)/G as topological spaces. Proof. The result for Proj follows immediately from the corresponding result for Spec. Since R L is faithfully flat over R, themapspec(r L) Spec R dual to the inclusion R R L is surjective. This map takes p to p R, so is clearly constant on orbits of the Galois group. Now suppose p 1 and p 2 map to the same prime p of Spec R. SinceR, and hence R L, is Noetherian, we can find a subfield L of L which is a finite Galois extension of K such that both p 1 and p 2 are generated by elements of R L. Then [AM69, Ex.13, p.68] implies that there is some element σ of the Galois group of L over K that sends p 1 (R L )top 2 (R L ). This element σ extends to an element σ of G, andthenp σ 1 = p 2. This proves that the map Spec(R L)/G Spec R is bijective and continuous; to prove that it is closed, use the fact that R L is integral over R so satisfies the going-up theorem [AM69, Ex. 11, p.79]. We are now ready for the main theorem of this section. Theorem 3.5. Suppose B is a finite-dimensional cocommutative Hopf algebra over a field K, andl is a Galois extension field of K. Suppose that prime ideals determine the thick subcategories over B L. Then prime ideals determine the thick subcategories over B. Proof. Let R denote the ring Ext B (K, K), so that R L is Ext B L (L, L). Suppose C is a tensor-closed thick subcategory of B-modules. Then I(C) is a Galois invariant tensor-closed thick subcategory of B L-modules. Since the thick subcategory theorem holds for B L, I(C) =D(T ) for some set T of homogeneous primes of R L. Recall that D(T ) is the tensor-closed thick subcategory generated by the S(p) forp T. Since I(C) is Galois invariant, T can be taken to be a union of orbits T i = {q σ i σ G} of the Galois group G, by Lemma 3.3. Let p i = q i R, a homogeneous prime of R, andlett be the set of the p i. We claim that C = C(T ). To see this, note first that Lemma 3.4 implies that T i is the set of minimal primes containing p e i. Hence Proposition 3.1 and Lemma 3.3 imply that I(C(p i )) = D(p e i )=D(T i). It follows that I(C(T )) = D(T )=I(C). Applying R, we find that C(T )=C. Recall that when K is the only simple B-module, every thick subcategory is tensor-closed. To apply Theorem 3.5 to this case, we would like to know that the same condition holds for B L-modules.

10 10 MARK HOVEY AND JOHN H. PALMIERI Lemma 3.6. Suppose B is a finite-dimensional cocommutative Hopf algebra over a field K such that K is the only simple B-module, and suppose L is an extension field of K with Galois group G. ThenL is the only simple B L-module. Proof. Since K is the only simple B-module, then for every B-module M, there is an element x of M such that bm = ε(b)m, whereε is the counit of the Hopf algebra B. Indeed, since B is finite-dimensional, any B-module has a finite-dimensional submodule; we then proceed by induction on the dimension. If N is a B L-module, we can find such an x by considering Res N. It then follows that (b α)(x) = (ε(b)α)x = ε L (b α)x. ThusL is the only simple B L-module. 4. More on Galois extensions In this section, we prove that the injection I from thick subcategories of B- modules to Galois invariant thick subcategories of B L-modules is in fact a bijection, regardless of whether prime ideals determine thick subcategories of B L- modules. Our proof holds for either thick subcategories or tensor-closed thick subcategories. This result is independent of our main results. We begin with two lemmas. Lemma 4.1. Suppose B is an algebra defined over a field K, and suppose L is a finite Galois field extension of K with Galois group G. If N is a B L-module, there is a natural isomorphism Ind Res N = σ G N σ. Proof. For each σ in G, we have an isomorphism Res N = Res N σ. The adjoint of this isomorphism is a map of B L-modules Ind Res N N σ. Putting these together gives us a natural map Ind Res N σ G N σ.thismapisshowntobe an isomorphism of vector spaces in [Bou90, Proposition V.10.8]. Lemma 4.2. Suppose B is an algebra defined over a field K, L is a Galois extension of K, andl L is a subextension of L which is Galois over K. Let G =Gal(L/K) and H =Gal(L/L ), so that H is a normal subgroup of G and G/H = Gal(L /K). Then for any B L -module M and any σ G mapping to σ G/H, there is a natural isomorphism Ind L L (M σ ) = (Ind L L M)σ. Proof. The easiest way to see this is to calculate the adjoints. A map of B L- modules (Ind L L M)σ N is the same thing as a map of B L -modules M Res L L (N σ 1 ). A map of B L-modules Ind L L M σ N is the same thing as a map of B L -modules M (Res L L N)σ 1. But there is an obvious isomorphism Res L L (N σ 1 ) = (Res L L N)σ 1. We can now prove the desired correspondence between thick subcategories of B-modules and Galois invariant thick subcategories of B L-modules. Theorem 4.3. Suppose B is a finite-dimensional algebra over a field K, andl is a Galois extension field of K. Then the maps I and R of Proposition 2.4 define a one-to-one correspondence between tensor-closed thick subcategories of finitely generated B-modules and Galois invariant tensor-closed thick subcategories of finitely generated B L-modules.

11 THICK SUBCATEGORIES 11 Proof. It suffices to show that IR(D) is the smallest Galois invariant tensor-closed thick subcategory containing D for all tensor-closed thick subcategories D of B Lmodules. For each N D, choose a finite extension L of L and a finite B L - module N such that Ind L L N = N. Since L is Galois, we can assume that L is Galois. Then R(D) is generated by the modules Res L K N. Hence IR(D) is generated by the modules Ind L L IndL K ResL K N. Using Lemmas 4.1 and 4.2, we find that Ind L L IndL K ResL K N is a finite direct sum of Galois twists of N, including N itself. Hence IR(D) containsd and is contained in the smallest Galois invariant tensor-closed thick subcategory containing D. SinceIR(D) is Galois invariant, the result follows. Note that we can drop the tensor-closed hypothesis from the definitions of I and R and from the statement of Theorem 4.3 and the theorem will still be true, with the same proof. 5. Purely inseparable extensions In this brief section, we show that the thick subcategory theorem descends through purely inseparable extensions L. Recall that L is a purely inseparable field extension of a field K of characteristic p if, for every element α of L, thereis some n such that α pn K. We have the following theorem. Theorem 5.1. Suppose B is a finite-dimensional cocommutative Hopf algebra over a field K. Suppose L is a purely inseparable field extension of K. If the thick subcategories over B L are determined by prime ideals, then so are the thick subcategories over B. Proof. As usual, let R denote the ring Ext B(K, K), so that R L = Ext B L(L, L). Suppose C is a tensor-closed thick subcategory of B-modules. Then there is some set T of homogeneous prime ideals of R L such that I(C) =D(T ), since the thick subcategory theorem holds for B L. Forq T,letp = q R, andlett denote the set of such ideals p. Then q = p e, the radical of p e. Indeed, every element of q is in p e,sincelispurely inseparable. On the other hand, every element of p e must be in q, sinceq is prime. Hence, by Lemma 3.3 and Proposition 3.1, I(C(p)) = D(q). Hence I(C(T )) = D(T )=I(C), and so C = C(T ). We would also like a theorem analogous to Theorem 4.3, asserting in this case that tensor-closed thick subcategories of B L-modules are in one-to-one correspondence with tensor-closed thick subcategories of B-modules, without having a classification of thick subcategories of B L-modules. We are not able to do this, though, because we do not know a replacement for Lemma 4.1. References [AM69] M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison- Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., [BS94] A. Bajer and H. Sadofsky, Cohomology of finite-dimensional connected cocommutative Hopf algebras, J. Pure Appl. Algebra 94 (1994), [BCR97] D. J. Benson, Jon F. Carlson, and Jeremy Rickard, Thick subcategories of the stable module category, Fund. Math. 153 (1997), [Ben98] D. J. Benson, Representations and cohomology. I, second ed., Cambridge University Press, Cambridge, [Bou90] N. Bourbaki, Algebra. II. Chapters 4 7, Springer-Verlag, Berlin, 1990, Translated from the French by P. M. Cohn and J. Howie.

12 12 MARK HOVEY AND JOHN H. PALMIERI [FS97] E. M. Friedlander and A. Suslin, Cohomology of finite group schemes over a field, Invent. Math. 127 (1997), [HP99] M. Hovey and J. H. Palmieri, Stably thick subcategories of modules over Hopf algebras, preprint, [HPS97] M. Hovey, J. H. Palmieri, and N. P. Strickland, Axiomatic stable homotopy theory, Mem. Amer. Math. Soc. 128 (1997), no. 610, x+114. [Ste75] B. Stenström, Rings of quotients, Die Grundlehren der mathematischen Wissenschaften, vol. 217, Springer-Verlag, Berlin, [Ver77] J.-L. Verdier, Catégories dérivées, Cohomologie Etale (SGA 4 1 ) (P. Deligne, ed.), 1977, 2 Springer Lecture Notes in Mathematics 569, pp [Wil81] C. Wilkerson, The cohomology algebras of finite dimensional Hopf algebras, Trans. Amer. Math. Soc. 264 (1981), Department of Mathematics, Wesleyan University, Middletown, CT address: hovey@member.ams.org Department of Mathematics, University of Washington, Seattle, WA address: palmieri@member.ams.org