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 wi* *ll 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 2 C, then M 2 C. (b) If 0 -! M1 -! M2 -! M3 -! 0 is a short exact sequence and two out of the three modules M1, M2, and M3 are in C, then so is the third. A thick subcategory is nontrivial if it contains a nonzero module. The thick su* *b- category generated by some set of finitely generated modules is the smallest th* *ick subcategory containing those modules. This definition is a translation of the notion of a thick, orepaisse, subcate* *gory of a triangulated category, which was introduced by Verdier [Ver77], and has be* *en studied in stable homotopy theory (see [HPS97 ], for example) and more recently in the modular representation theory of finite groups [BCR97 ]. Note that, in t* *he literature, a thick subcategory of an abelian category sometimes means an abeli* *an subcategory that is closed under extensions, but our thick subcategories need n* *ot 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 the* *m 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 2 C and all finite-dimensional B-modules N, the tensor product M N is in C. Example 1.3. Suppose that B is a finite-dimensional cocommutative Hopf algebra over a field K, in which case projective and injective modules coincide. Here a* *re some examples of thick subcategories of finitely generated B-modules. ____________ Date: August 11, 1999. 1991 Mathematics Subject Classification. 20C05, 20C20, 20J05, 18E30, 55P42. 1 2 MARK HOVEY AND JOHN H. PALMIERI (a) The subcategory of all finitely generated modules: this is clearly the lar* *gest thick subcategory and is tensor-closed. (b) The subcategory of finitely generated projective modules: this is thick be* *cause projective and injective modules coincide. It is also tensor-closed; a sta* *ndard 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 mod- ules which are projective when restricted to A: this is thick and tensor-c* *losed 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 7! 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 so* *me (possibly infinitely generated) module X. (e) The subcategory of finitely generated modules satisfying specific homologi* *cal 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 mo* *dules forms a thick subcategory: {M | 9 b : Ext s;tB(K; M) = 0 when s 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 g* *en- erated projective modules. Indeed, it suffices to show that C contains a f* *ree module. But we have already seen that B M is free for any module M. (g) If the trivial module K is the only simple B-module, then every thick subc* *at- egory C is tensor-closed. This follows by induction on the composition ser* *ies of N, in which all of the composition factors must be direct sums of copie* *s 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 fin* *ite- dimensional 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 classific* *ation of the thick subcategories of finitely generated B-modules, when B is a finite- dimensional cocommutative Hopf algebra over a field K with K being the only simple module. This is stated in [HPS97 , Conjecture 6.1.3 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 commutati* *ve ring. Write R for Ext*B(K; K). The K-algebra R has a unique maximal homoge- neous ideal: the ideal consisting of all elements in positive gradings. When B* * is THICK SUBCATEGORIES 3 ungraded, we write ProjR for the set of non-maximal homogeneous prime ideals of R; when B is graded, so that R is bigraded, then ProjR is the set of all non- maximal bihomogeneous primes. A subset T of ProjR is closed under specialization if it is a union of Zariski-closed sets; that is, if p is in T and q p, then q* * is in T . For each homogeneous prime ideal p of R, we construct a finitely generated B-module S(p) so that Ext*B(K; S(p)) approximates Ext*B(K; K)=p_see Section 3. Conjecture 1.4 (Hovey-Palmieri-Strickland).There is a bijection between non- trivial tensor-closed thick subcategories C of finitely generated B-modules and* * non- empty subsets T of ProjExt*B(K; K) closed under specialization: given a tensor- closed thick subcategory C, define T to be the set of primes {p | S(p) 2 C}. Gi* *ven a subset T of prime ideals, let C be the tensor-closed thick subcategory generate* *d by {S(p) | p 2 T }. In [HPS97 ], we show that the assignment sending a set T of prime ideals, clo* *sed under specialization, to the thick subcategory generated by {S(p) | p 2 T } is * *one- to-one (in the case that K is the only simple B-module). The difficulty is show* *ing that it is onto. Definition 1.5.Suppose that B is a finite-dimensional cocommutative Hopf alge- bra over a field K. We say that prime 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 Ricka* *rd; 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 alge* *bra A is the Hopf algebra of stable additive operations on the mod p cohomology of any topological space X. Given a field K containing Fp, 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 de- fined over an algebraically closed field of characteristic 2, then prime i* *deals 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 alge- bra over a field K. Let L be a normal field extension of K. If prime ideals (* *in ProjExt*BL (L; L)) determine thick subcategories over B L, then prime ideals (in ProjExt*B(K; K)) do so for B. In this theorem, if B is graded, then we are to understand the phrase "K is t* *he 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 t* *hat any irreducible polynomial in K[x] that has one root in L splits in L[x] into a pro* *duct of linear factors. Note, by the way, that if K is the only simple B-module, the* *n L is the only simple B 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 theo* *rem with Theorem 1.6, we obtain the following corollary. 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 Ste* *enrod 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, a* *nd finally we discuss purely inseparable extensions in Section 5. More precisely, * *Theo- rem 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* * gen- erally, between (tensor-closed) thick subcategories over B and "Galois invarian* *t" (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 o* *f thick subcategories of B L-modules. A notion related to thick subcategory is a localizing subcategory: this is a * *sub- category of the category of all B-modules which is thick and closed under arbit* *rary direct sums. One can also consider tensor-closed localizing subcategories C, wh* *ere now we must assume that if M 2 C and N is any module, then M N 2 C. One type of tensor-closed localizing subcategory is the Bousfield class of a mo* *dule X, defined by = {M | M X is projective}: The results in [HP99 ] give a classification of the Bousfield classes, for cert* *ain Hopf algebras B defined over algebraically closed fields: they are in bijection with* * arbi- trary subsets of ProjExt*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 tensor- closed localizing subcategory is a Bousfield class. We do not know of any non-t* *rivial 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 * *ho- motopy theory [HPS97 ] as possible, so as to improve accessibility. But the aut* *hors would never have been able to prove the results without the conceptual clarity * *pro- vided 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 neces- sity of separating Galois and purely inseparable extensions. The authors also t* *hank Bill Graham, Tom Hagedorn, and Jim Reid for assistance with Galois theory. 2.Algebraic extensions Suppose that B is an algebra over K, and L is an extension field of K. Then B L is an algebra over L of the same dimension as B. If B 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 ResLK= Reshas both a left and a right adjoint. The left adjoint is induction IndLK= Ind, and takes M to M L. The right adjoint, which we do not use, is coinduction, and takes M to Hom (L; * *M). THICK SUBCATEGORIES 5 If L is finite over K, induction and coinduction coincide. Note that ResInd 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-closed thick subcategories of BL-modules, as follows. If C is a tensor-c* *losed thick subcategory of finitely generated B-modules, denote by I(C) the tensor-cl* *osed thick subcategory of B L-modules generated by the objects {IndM | M 2 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 ResN for N 2 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, and L is an algebraic field extension of K. Then for every finite-dimensional B L-module N, there is a subfield L0 of L, finite-dimensional over K, and a B L0-module N0, such that N ~=IndLL0N0. In particular, ResN0is isomorphic to a direct sum of copies of the finite-dimensional B-module ResLKN0. Proof.Let {bi} be a basis of B over K, and let {nj} be a basis of N over L. Then thePB L-module structure is determined by elements ffijkof L such that binj = ijkffijknk. There are only finitely many of these ffijk, and since L * *is algebraic over K, the subfield L0 of L generated by the ffijkis finite-dimensio* *nal over K. Let N0 denote the L0-vector space spanned by the ni. Then the ffijkdefi* *ne a B L0-module structure0on N0, and it is clear that IndLL0N0 ~=N. Hence ResN ~=0ResLKResLL0IndLL0N0, which is isomorphic to a direct sum of copies of ResLKN0. |___| To understand the functions I and R better, we introduce the action of the Ga* *lois group. Let G = Gal(L=K) denote the Galois group of L over K. For oe 2 G and N a B L-module, we define a new B L-module Noeas follows. Note that a B L- module N is just an L-vector space together with a B-module structure on ResN, so we define Noeby modifying the L-vector space structure on N: we define the L- vector space structure on Noeby ff.x = ffoex. Then ResNoe= ResN, so this define* *s a BL-module. This construction defines a functor oe :(BL)-Mod -! (BL)-Mod , with ResIoe= ResI. The functor oe is an exact symmetric monoidal isomorphism of categories, with inverse given by oe-1. Note that for any B-module M, there is a natural isomorphism IndM ae-! (IndM)oe, defined by ae(m ff) = m ffoe. Since (-)oeis an exact symmetric monoidal equivalence of categories, if D is a tensor-closed thick subcategory of B L-modules, then so is Doe= {Noe| N 2 D}. Definition 2.3.A (tensor-closed) thick subcategory D of B L-modules is called Galois invariant if Doe= D for all oe 2 Gal(L=K). 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(Doe). (c) RI(C) = C. Proof.Part (a): I(C)oeis a tensor-closed thick subcategory containing (IndC)oe= IndC. Hence I(C)oe I(C). Using oe-1 to reverse the argument, we get the desired equality. Part (b): The tensor-closed thick subcategory R(D) is generated by the finite- dimensional summands of ResN for N 2 D. Since ResNoe= ResN, the result follows. Part (c): RI(C) is the tensor-closed thick subcategory generated by the finit* *e- dimensional summands of ResIndM, where M 2 C. But ResIndM is a direct sum of copies of M, one for each basis element of L over K. Thus RI(C) contains C; * *but also any finite-dimensional summand of ResIndM is a summand of a finite direct_ sum of copies of M, so C 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 thi* *ck 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 ExtiB(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 2 Ri, we can form a B- module S(x) as follows. Choose an injective resolution P* for K such that each * *Pj is finite-dimensional. Let Mi denote the kernel of the map Pi -!Pi+1. Then x is realized by a map K -! Mi, which is necessarily injective if x is nontrivial. W* *e 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 tria* *ngulated category of stable modules_the quotient category obtained by identifying two ma* *ps when their difference factors through an injective_then x is a self-map of degr* *ee -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, we ch* *oose a set x1; : :;:xk of homogeneous generators for a, and define S(a) = S(x1) . .* * . S(xk). Here B acts on the tensor product using the diagonal on B; this is why we need to assume B is a Hopf algebra. THICK SUBCATEGORIES 7 Of course S(a) will depend on the choice of generators, but the following pro* *po- sition shows that the tensor-closed thick subcategory generated by S(a) is inde* *pen- dent 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). Let T denote the set of min* *i- mal homogeneous primes containing a, and let C(T ) denote the tensor-closed thi* *ck subcategory generated by the S(p) for p 2 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, then S(b) is in C(a). (c) [HPS97 , Theorem 3.3.3] Given a nontrivial tensor-closed thick subcategory* * C, there is a module LfCK so that for any finitely generated module M, M 2 C if and only if LfCK M is projective. (d) For every homogeneous prime p, there is a module Kp defined as in [HPS97 , Proposition 6.0.7] satisfying the following conditions: (i) [HPS97 , Proposition 6.1.7(d)] If p 6 a, then S(a) Kp is projective. (ii)[HPS97 , Theorem 6.1.9] A module M is projective if and only if M S(p) Kp 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 cate- gory 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 Noetheri* *an ring. The category of B-modules is abelian, not triangulated. However, if we fo* *rm the stable category of B-modules by identifying maps f; g :M -! N if f -g facto* *rs through a projective module, we do get a closed symmetric monoidal triangulat- ed category. (A good reference for the stable category is [Ben98]). Furthermore, nontrivial (tensor-closed) thick subcategories of B-modules correspond precisel* *y to nonempty (tensor-closed) thick subcategories of the stable category; that is, f* *ull 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 * *gen- erators; for the moment, let us assume that K is the only simple B-module, so t* *hat there is only one such. Unfortunately, graded self-maps of K in the stable modu* *le 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 ide* *als 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 defini* *tion of a Noetherian stable homotopy category. So let us redefine a Noetherian stable 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 self- maps [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. This hypothesis does hold in the situation at hand, since Ext*B(K; M) is a finitely generated module over Ext*B(K; K) when M 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 "ss*(X)" by "ss*(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 lemm* *a. For part (b), Lemma 3.2(b) implies that if T is the set of minimal homogeneous primes containing a, and if T 0is the set of all homogeneous primes containing * *a, then C(T ) = C(T 0). For the same reason, if a p, then S(p) is in C(a). Theref* *ore, C(a) C(T ). To show the other inclusion, it suffices, by Lemma 3.2(c), to show that LfC(T* *)K S(a) is projective, or equivalently, LfC(T)K S(a) S(p) Kp is projective for every prime p. If p 6 a, then Lemma 3.2(d) implies that S(a) * * Kp is projective. If p a, then S(p) 2 C(T ), so LfC(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*BL (L; L) ~= R L. One way to see this is to take an injective resolution P* of the B-module K, so that each Pn 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 Pn is finite-dimensional over K, Hom K(K; P* L) ~=Hom K(K; P*) L. Since L is flat over K, Ext*BL(L; L) ~=H*(Hom K (K; P*) L) ~=H*(Hom K (K; P*)) L ~=R L: This description makes it clear that IndS(x) is a choice for the B L-module S(_* *_x), where __xis the image of x under the identification of R with the obvious subal* *gebra of RL. Since induction preserves tensor product, we find that IndS(a) is a choi* *ce for S(ae), where ae 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 2 ExtiBL (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 -!Mi L, where Mi is the kernel of Pi-! Pi+1. Then for oe 2 G, the element xoecorresponds to t* *he map Loe-! Mi Loe, using the isomorphism Loe~=L. It follows that S(x)oeis a choice for S(xoe). 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, then for oe 2 G, boeis the ideal boe= {xoe| x 2 b}: By the preceding computations, for any ideal b in RL, S(b)oeis a choice for S(b* *oe). We have then proved the following lemma. 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-clo* *sed thick subcategory generated by S(a). Then I(C(a)) = D(ae), where I denotes the map given in Definition 2.1. (b) Given a homogeneous ideal b of Ext*BL (L; L), let D(b) denote the tensor- closed thick subcategory generated by S(b). Then for oe 2 G, D(b)oe= D(boe* *). We need one more lemma before stating and proving Theorem 3.5. Note that if p is a prime ideal of R L, then so is poefor any oe 2 G = Gal(L=K). Hence G ac* *ts on Proj(R L). Lemma 3.4. Suppose K is a field, L is a Galois extension field of K with Galois group G, and R is a graded connected Noetherian graded-commutative K-algebra. Then ProjR ~=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, the map Spec(R L) -! SpecR dual to the inclusion R ,! R L is surjective. This map takes p to p \ R, so is clea* *rly constant on orbits of the Galois group. Now suppose p1 and p2 map to the same prime p of SpecR. Since R, and hence RL, is Noetherian, we can find a subfield * *L0 of L which is a finite Galois extension of K such that both p1 and p2 are gener* *ated by elements of RL0. Then [AM69 , Ex.13, p.68] implies that there is some element oe of the Galois group of L0over K that sends p1 \ (R L0) to p2 \ (R L0). This element oe extends to an element eoeof G, and then peoe1= p2. This proves that * *the map Spec(RL)=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_[AM* *69 , 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, and L is a Galois extension field of K. Suppose that prime ideals de- termine 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 RL is Ext*BL(L; L). Suppose C is a tensor-closed thick subcategory of B-modules. Then I(C) is a Galois invari* *ant 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) for p 2 T . Since I(C) is Galois invariant, T can be taken to be a union * *of orbits Ti= {qoei| oe 2 G} of the Galois group G, by Lemma 3.3. Let pi= qi\ R, a homogeneous prime of R, and let T 0be the set of the pi. We claim that C = C(T * *0). To see this, note first that Lemma 3.4 implies that Tiis the set of minimal p* *rimes containing pei. Hence Proposition 3.1 and Lemma 3.3 imply that I(C(pi)) = D(pei) = D(Ti): It follows that I(C(T 0)) = D(T ) = I(C). Applying R, we find that C(T 0) = 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 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. Then L 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 = "(b)m, where " is the counit of the Hopf algeb* *ra 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 BL-module, we can find such an x by considering ResN. It then follows that (b ff)(x) =_ ("(b)ff)x = "L(b ff)x. Thus L 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 bi- jection, regardless of whether prime ideals determine thick subcategories of B * * L- modules. Our proof holds for either thick subcategories or tensor-closed thick * *sub- categories. 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 GaloisLgroup G. If N is a B L-module, there is a natural isomorphism IndResN ~= oe2GNoe. Proof.For each oe in G, we have an isomorphism ResN ~= ResNoe. The adjoint of this isomorphism is a map of B L-modulesLIndRes N -! Noe. Putting these together gives us a natural map IndResN -! oe2GNoe. This map is shown to be * *__ 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 ex- tension of K, and L0 L is a subextension of L which is Galois over K. Let G = Gal(L=K) and H = Gal(L=L0), so that H is a normal subgroup of G and G=H_~=Gal(L0=K). Then for any B L0-module M and any oe 2 G mapping to oe2 G=H, there is a natural isomorphism __oe L oe IndLL0(M ) ~=(IndL0M) : Proof.The easiest way to see this is to calculate the adjoints. A map of B L- modules (IndLL0M)oe-! N is the same thing as a_map of B L0-modules M -! ResLL0(Noe-1). A map of B L-modules_IndLL0Moe-! N is the same thing as a map of B L0-modules M -!_(ResLL0N)oe-1._But there is an obvious isomorphism ResLL0(Noe-1) ~=(ResLL0N)oe-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, and L 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* * gen- erated B-modules and Galois invariant tensor-closed thick subcategories of fini* *tely generated B L-modules. THICK SUBCATEGORIES 11 Proof.It suffices to show that IR(D) is the smallest Galois invariant tensor-cl* *osed thick subcategory containing D for all tensor-closed thick subcategories D of B* * L- modules. For each N 2 D, choose a finite extension L0 of L and a finite B L0- module N0 such that IndLL0N0 ~=N. Since L is Galois, we0can assume that L0 is Galois. Then R(D) is generated by0the0modules ResLKN0. Hence IR(D) is generated by the0modules0IndLL0IndLKResLKN0. Using Lemmas 4.1 and 4.2, we find that IndLL0IndLKResLKN0 is a finite direct sum of Galois twists of N, including* * N itself. Hence IR(D) contains D and is contained in the smallest Galois invariant tensor-closed thick subcategory containing D. Since IR(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 anfield K of characteristic p if, for every element ff of L,* * there is some n such that ffp 2 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 thi* *ck 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*BL(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 thi* *ck subcategory theorem holds for B L.pFor_q 2 T , let p = q \ R, and let T 0denote the set of suchpideals_p. Then q = pe, the radical of pe. Indeed, every eleme* *nt of qpis_in pe, since L is purely inseparable. On the other hand, every eleme* *nt of pemust be in q, since q is prime. Hence, by Lemma 3.3 and Proposition 3.1,* *__ I(C(p)) = D(q). Hence I(C(T 0)) = D(T ) = I(C), and so C = C(T 0). |* *__| 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 corre- spondence with tensor-closed thick subcategories of B-modules, without having a classification of thick subcategories of B L-modules. 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Palmieri, Stably thick subcategories of modules over * *Hopf algebras, preprint, 1999. [HPS97]M. Hovey, J. H. Palmieri, and N. P. Strickland, Axiomatic stable homotop* *y theory, Mem. Amer. Math. Soc. 128 (1997), no. 610, x+114. [Ste75]B. Stenstr"om, Rings of quotients, Die Grundlehren der mathematischen Wi* *ssenschaften, vol. 217, Springer-Verlag, Berlin, 1975. [Ver77]J.-L. Verdier, Categories derivees, Cohomologie Etale (SGA 41_2) (P. Del* *igne, ed.), 1977, Springer Lecture Notes in Mathematics 569, pp. 262-311. [Wil81]C. Wilkerson, The cohomology algebras of finite dimensional Hopf algebra* *s, Trans. Amer. Math. Soc. 264 (1981), 137-150. Department of Mathematics, Wesleyan University, Middletown, CT 06459 E-mail address: hovey@member.ams.org Department of Mathematics, University of Washington, Seattle, WA 98155 E-mail address: palmieri@member.ams.org