GALOIS THEORY OF THICK SUBCATEGORIES IN MODULAR REPRESENTATION THEORY

MARK HOVEY AND JOHN H. PALMIERI

Abstract. Benson, Carlson, and Rickard [BCR97] classified the tensor-closed thick subcategories of finite-dimensional representations of finite groups over algebraically closed fields. In this paper, we remove the algebraically closed hypothesis, by applying some Galois theory. Our methods apply more generally to finite-dimensional cocommutative Hopf algebras over a field. Thus they allow us to drop the algebraically closed hypothesis in the classification [HP99] of thick subcategories of modules over finite-dimensional sub-Hopf algebras of the Steenrod algebra as well.

1. Introduction

Let B be a finite-dimensional algebra over a field K. The basic problem 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 Bmodules is thick if the following conditions hold:

(a) If M is a direct summand of N and N ∈ C, then M ∈ 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}, and M_{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.

Date: January 24, 2000.

1991 Mathematics Subject Classification. 20C05, 20C20, 20J05, 18E30, 55P42.

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 ∈ C and all finite-dimensional B-modules N, the tensor product M ⊗ N is in C.

The collection of all tensor-closed thick subcategories of finitely generated Bmodules forms a lattice which we denote L_{thick}^{⊗}(B). Our goal is to determine L_{thick}^{⊗}(B) in certain cases.

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 are some examples of thick subcategories of finitely generated B-modules.

(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] or Lemma 3.2(c), 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_{B}^{s,t}(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 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 is the only simple B-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 a field 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 description of L_{thick}^{⊗}(B) when B is a finite-dimensional cocommutative Hopf algebra over a field K with K being the only simple B-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, R = Ext_{B}^{∗}(K, K) is a Noetherian graded commutative ring. The K-algebra R has a unique maximal homogeneous ideal: the ideal consisting of all elements in positive gradings. When B is ungraded, we write Proj R for the set of non-maximal homogeneous prime ideals of R; when B is graded, so that R is bigraded, then Proj R is the set of all non-maximal bihomogeneous primes. A subset T of Proj R is closed under specialization if it is a union of Zariskiclosed sets; that is, if 𝔭 is in T and 𝔮 ⊇ 𝔭, then 𝔮 is in T. Let J(R) denote the lattice of subsets of Proj R that are closed under specialization. In Section 3, we define a map f = f_{B}: J(R) ⟶ L_{thick}^{⊗}(B).

Conjecture 1.4 (Hovey-Palmieri-Strickland). If B is a finite dimensional cocommutative Hopf algebra over a field K, the map f: J(Ext_{B}^{∗}(K, K)) ⟶ L_{thick}^{⊗}(B) is an isomorphism.

In [HPS97, Theorem 6.2.3], we show that f is injective, at least when K is the only simple B-module. In fact, the proof given in [HPS97] will also work for arbitrary B, as we explain in Section 3. Thus, the difficulty is to show that f is surjective.

Definition 1.5. Suppose that B is a finite-dimensional cocommutative Hopf algebra 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 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 ⊗_{F_{p}} 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 Ext_{B⊗L}^{∗}(L, L)) determine thick subcategories over B ⊗ L, then prime ideals (in Ext_{B}^{∗}(K, K)) determine thick subcategories over B.

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 is the only simple B ⊗ L-module; this is easy to see, and is proved in Lemma 3.7. In this case, every thick subcategory is tensor-closed. Combining Theorem 1.7 with Theorem 1.6, we obtain the following corollary.

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 3.6 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 ∈ C and 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 Wai Kiu Chan, 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. In this section, we construct maps

L_{thick}^{⊗}(B) ⟶^{I} L_{thick}^{⊗}(B ⊗ L) ⟶^{R} L_{thick}^{⊗}(B)

such that RI is the identity.

These maps are based on the induction and restriction functors, which we now recall. The restriction functor Res_{K}^{L} = Res from B ⊗ L-modules to B-modules has both a left and a right adjoint. The left adjoint is induction Ind_{K}^{L} = Ind, and takes M to M ⊗ L. The right adjoint, which we do not use, is coinduction, and takes M to Hom(L, M). If L is finite-dimensional 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. Define I: L_{thick}^{⊗}(B) ⟶ L_{thick}^{⊗}(B ⊗ L) as follows. Given a tensorclosed thick subcategory C ∈ L_{thick}^{⊗}(B), define I(C) to be 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 finitedimensionality in general. Nevertheless, given D ∈ L_{thick}^{⊗}(B ⊗ L), we define R(D) to be the tensor-closed thick subcategory of B-modules generated by the finitedimensional summands of Res N for N ∈ D.

Note that I and R are both order-preserving maps, and that I preserves arbitrary joins. The definition of R may seem strange, but the following lemma shows that it 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 L′ of L, finite-dimensional over K, and a B ⊗ 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_{K}^{L′} N′.

Proof. Let {b_{i}} be a basis of B over K, and let {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} = ∑_{k} α_{ijk}n_{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. Let N′ 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_{K}^{L′} Res_{L′}^{L} Ind_{L′}^{L} N′, which is isomorphic to a direct sum of copies of Res_{K}^{L′} 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. For σ ∈ G, we define an (L, L)-bimodule L^{σ}: as a left L-module, L^{σ} = L, but the right action of L is defined by x · α = α^{σ}x. Then we have isomorphisms of bimodules L^{1} ≅ L and L^{σ} ⊗_{L} L^{σ′} ≅ L^{σσ′}. Note that L ≅ L^{σ} as a right L-module as well, since they are both one-dimensional vector spaces over L; the isomorphism takes α to α^{σ}. If σ ≠ 1, though, L is not isomorphic to L^{σ} as an (L, L)-bimodule.

Now, a B ⊗ L-module N is the same thing as a (B, L)-bimodule, and so we can define N^{σ} = N ⊗_{L} L^{σ}, where the tensor product is of course a tensor product of bimodules. In concrete terms, suppose that {b_{i}} is a basis for B over K and {n_{j}} is a basis for N over L, and let the B ⊗ L-module structure of N be defined by b_{i}n_{j} = ∑_{k} α_{ijk}n_{k}, where α_{ijk} ∈ L. Then the B ⊗ L-module structure of N^{σ} is defined by b_{i}n_{j} = ∑_{k} α_{ijk}^{σ}n_{k}. This construction defines an exact symmetric monoidal isomorphism of categories σ: (B ⊗ L)-Mod ⟶ (B ⊗ L)-Mod, with inverse given by σ^{−1}. Furthermore, Res ∘σ = Res.

Note that for any B-module M, there is a natural isomorphism

(Ind M)^{σ} = (M ⊗ L) ⊗_{L} L^{σ} ≅ M ⊗ L^{σ} ≅ M ⊗ L = Ind M

since L^{σ} ≅ L as a (K, L)-bimodule. Also note, 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).

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} = Res N, the result follows.

Part (c): RI(C) is the tensor-closed thick subcategory generated by the finitedimensional summands of Res Ind M, where M ∈ C. But Res Ind M 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 Res Ind M 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 thick subcategories of B⊗L-modules, namely, those in the image of I. When L is a Galois extension of K, we will see in Section 4 that the image of I consists of all Galois invariant tensor-closed thick subcategories. We expect this holds for all normal extensions L, but we have been unable to prove this. We will see in Section 5 that the image of I is all of L_{thick}^{⊗}(B ⊗ L) if L is purely inseparable over K and prime ideals determine thick subcategories over B ⊗ L.

Remark 2.5. All of the constructions, results, and proofs in this section work with thick subcategories which are not assumed to be tensor-closed. For example, one can imitate Definition 2.1 to get functions I and R between the lattices of thick subcategories of B-modules and B ⊗ L-modules; then for any thick subcategory C of B-modules, RI(C) = C. The tensor-closed hypothesis becomes necessary in the next section.

3. Galois extensions

In this section we prove that detection of tensor-closed thick subcategories by prime ideals descends through Galois extensions. Recall that for any Noetherian graded commutative ring R, J(R) denotes the lattice of subsets of Proj R that are closed under specialization. We start this section by defining a map f = f_{B}: J(Ext_{B}^{∗}(K, K)) ⟶ L_{thick}^{⊗}(B). We show that f is always injective, and we study the compatibility of this map with induction. This requires some axiomatic stable homotopy theory. We then combine this with some basic algebraic geometry to show that detection of thick subcategories descends.

Let B be a finite-dimensional cocommutative Hopf algebra over K, and let R denote the graded commutative K-algebra Ext_{B}^{∗}(K, K). We write R^{i} for the ith homogeneous piece Ext_{B}^{i}(K, K). Note that R is a Noetherian ring, by [Wil81] when B is graded connected and [FS97] when B is ungraded. The map f is the restriction of a map f: 2^{Ideals(R)} ⟶ L_{thick}^{⊗}(B), where Ideals(R) is the collection of homogeneous ideals of R.

To define f: 2^{Ideals(R)} ⟶ L_{thick}^{⊗}(B), we work from the bottom up: given a homogeneous element x ∈ R, we will define a module S(x); then given a homogeneous ideal 𝔞 of R, we choose homogeneous generators x_{1}, …, x_{n} for 𝔞 and let

S(𝔞) = S(x_{1}) ⊗ S(x_{2}) ⊗ ⋯ ⊗ S(x_{n}),

using the cocommutative Hopf algebra structure on B to make this tensor product into a B-module. Then f(𝔞) is defined to be the tensor-closed thick subcategory generated by S(𝔞), and for any set T of homogeneous ideals of R, f(T) is the tensor-closed thick subcategory generated by {f(𝔞) | 𝔞 ∈ T }.

We point out that the module S(𝔞) will depend on the set of homogeneous generators x_{1}, …, x_{n} of 𝔞, but we will see that f(𝔞) is independent of this generating set.

It remains to define the module S(x), when x is a homogeneous element x ∈ R^{i}. 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.)

Proposition 3.1. Suppose B is a finite-dimensional cocommutative Hopf algebra over a field K. Suppose 𝔞 is a homogeneous ideal of R = Ext_{B}^{∗}(K, K), and let T denote the set of minimal homogeneous primes containing 𝔞. Then:

(a) The map f: 2^{Ideals(R)} ⟶ L_{thick}^{⊗}(B) is well-defined.

(b) f(𝔞) = f(T).

(c) The restriction of f to a map f: J(R) ⟶ L_{thick}^{⊗}(B) is injective.

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] f(𝔞) is independent of the choice of generators of 𝔞.

(b) [HPS97, Lemma 6.0.9] If 𝔞 ≤ 𝔟, then S(𝔟) is in f(𝔞).

(c) [HPS97, Theorem 3.3.3] Given a nontrivial tensor-closed thick subcategory C, there is a module L_{C}K so that for any finitely generated module M, M ∈ C if and only if L_{C}K ⊗ 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) [HPS97, Proposition 6.1.7(b)] S(𝔭) ⊗ K_{𝔭} is not projective.

(ii) [HPS97, Proposition 6.1.7(d)] If 𝔭 ≱ 𝔞, then S(𝔞) ⊗ K_{𝔭} is projective. (iii) [HPS97, Theorem 6.1.9] A module M 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 abelian category of B-modules. 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. We can construct a triangulated category, called the stable category of Bmodules, by identifying maps f, g: M ⟶ N if f − g factors through a projective module. This is 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, the stable category is not a Noetherian stable homotopy category because 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 homotopy category to be a 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. 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 “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, and if T′ is the set of all homogeneous primes containing 𝔞, then f(T) = f(T′). For the same reason, if 𝔞 ≤ 𝔭, then S(𝔭) is in f(𝔞). Therefore, f(𝔞) ⊇ f(T).

To show the other inclusion, it suffices, by Lemma 3.2(c), to show that L_{f(T)}K ⊗ S(𝔞) is projective, or, equivalently, that

L_{f(T)}K ⊗ S(𝔞) ⊗ S(𝔭) ⊗ K_{𝔭}

is projective for every prime 𝔭. If 𝔭 ≱ 𝔞, then Lemma 3.2(d) implies that S(𝔞) ⊗ K_{𝔭} is projective. If 𝔭 ≥ 𝔞, then S(𝔭) ∈ f(T), so L_{f(T)}K ⊗ S(𝔭) is projective. Since tensoring with a projective yields a projective, we are done.

For part (c), we define a right inverse g: L_{thick}^{⊗}(B) ⟶ J(R) as follows. For a tensor-closed thick subcategory C in L_{thick}^{⊗}(B), let

g(C) = {𝔭 ∈ Proj R | S(𝔭) ∈ C}.

By Lemma 3.2(b), g(C) is in J(R). We claim that gf(T) = T, for T ∈ J(R). It is clear from the definitions that gf(T) ⊇ T. Conversely, suppose 𝔭 ∉ T. Then for 𝔮 ∈ T, 𝔭 ≱ 𝔮. Thus Lemma 3.2(d) implies that S(𝔮) ⊗ K_{𝔭} is projective. Since projective modules form a tensor-closed thick subcategory, this means that M ⊗K_{𝔭} is projective for all M ∈ f(T). Since S(𝔭) ⊗ K_{𝔭} is not projective, S(𝔭) ∉ f(T). Thus 𝔭 ∉ gf(T).

We now wish to understand how f behaves under induction. Let L be a field extension of K. Observe that Ext_{B⊗L}^{∗}(L, L) ≅ R ⊗ L. One way to see this is to take an injective resolution P_{∗} of the B-module K, so that each P_{n} is finitelygenerated; 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. Since L 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) is a choice for the B ⊗L-module S(\overline{x}), where \overline{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) is a choice for S(𝔞^{e}), where 𝔞^{e} denotes the ideal of R ⊗ L generated by 𝔞.

Note that the inclusion i: R ⟶ R ⊗ L induces a map i_{∗}: J(R) ⟶ J(R ⊗ L): given a homogeneous prime 𝔭, we define i_{∗}(𝔭) to be the set of all primes 𝔮 of R ⊗ L containing 𝔭^{e}. Then, given T ∈ J(R), we define i_{∗}T to be the union of the i_{∗}𝔭 for 𝔭 ∈ T. We have the following lemma.

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. Let R = Ext_{B}^{∗}(K, K). Then the diagram below is commutative.

J(R) ⟶^{i_{∗}} J(R ⊗ L)
f_{B} ↓  ↓^{f_{B⊗L}}
L_{thick}^{⊗}(B) ⟶_{I} L_{thick}^{⊗}(B ⊗ L)

Proof. Suppose T ∈ J(R). Then If_{B}(T) is the tensor-closed thick subcategory generated by Ind S(𝔭) for 𝔭 ∈ T. But the comments above imply that this is the tensor-closed thick subcategory generated by S(𝔭^{e}), for 𝔭 ∈ T. Proposition 3.1(b) then implies that this is f_{B⊗L}i_{∗}T, as required.

We now investigate the compatibility of f_{B⊗L} with the action of the Galois group G.

Lemma 3.4. 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. Let R = Ext_{B}^{∗}(K, K). Then there is an action of G on J(R ⊗ L), and f: J(R ⊗ L) ⟶ L_{thick}^{⊗}(B ⊗ L) is G-equivariant.

Proof. The Galois group G of L over K obviously acts on R ⊗ L by graded ring automorphisms, but we need to describe this action using the isomorphism

Ext_{B⊗L}^{∗}(L, L) ≅ R ⊗ L.

Suppose x ∈ Ext_{B⊗L}^{i}(L, L). Let P_{∗} be an injective resolution of K by finitedimensional 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}. Then for σ ∈ G, the element x^{σ} corresponds to the map L^{σ} ⟶ M_{i} ⊗ L^{σ}, using the isomorphism of (K, L)-bimodules 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 Ideals(R ⊗ L): given an ideal 𝔟 of R ⊗ L, then for σ ∈ 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}). It is clear that this action restricts to an action on Proj R, and therefore defines an action on J(R ⊗ L). The preceding comments imply that f is equivariant.

Note that, if T ∈ J(R), i_{∗}T is Galois invariant in J(R ⊗ L). We now use some basic algebraic geometry to show that every Galois invariant element of J(R ⊗ L) is in the image of i_{∗}.

Lemma 3.5. 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 Proj R ≅ Proj(R ⊗ L)/G as topological spaces. In particular, any Galois invariant element of J(R ⊗ L) is in the image of i_{∗}: J(R) ⟶ J(R ⊗ L).

Proof. The second statement follows immmediately from the first, since a Galois invariant element of J(R ⊗ L) will be a union of G-orbits of Proj R. 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) ⟶ 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. Since R, 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′) to 𝔭_{2} ∩ (R ⊗ L′). This element σ extends to an element σ̃ of G, and then 𝔭σ̃ ; 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 3.6. 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 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). We need only show that f: J(R) ⟶ L_{thick}^{⊗}(B) is surjective. Suppose C is a tensorclosed thick subcategory in L_{thick}^{⊗}(B). Then I(C) is a Galois invariant element of L_{thick}^{⊗}(B ⊗ L). Since f_{B⊗L} is an isomorphism by hypothesis, I(C) = f_{B⊗L}(T) for a unique T ∈ J(R ⊗ L). Since f_{B⊗L} is Galois equivariant by Lemma 3.4, and I(C) is Galois invariant by Proposition 2.4, T must be a Galois invariant element of J(R ⊗ L). Lemma 3.5 then guarantees that T = i_{∗}(U) for some U ∈ J(R). We then have

f_{B}(U) = RIf_{B}(U) = Rf_{B⊗L}i_{∗}(U) = Rf_{B⊗L}(T) = RI(C) = C,

and so f_{B} is surjective.

Note that this proof implies that the image of I is precisely the Galois invariant elements of L_{thick}^{⊗}(B ⊗ L) when prime ideals determine thick subcategories over B ⊗ L. We will see that this is true without restriction in the next section.

Recall that when K is the only simple B-module, every thick subcategory is tensor-closed. To apply Theorem 3.6 to this case, we would like to know that the same condition holds for B ⊗ L-modules.

Lemma 3.7. 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. 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 bx = ε(b)x, where ε 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. 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 Bmodules to Galois invariant thick subcategories of B ⊗ L-modules is in fact a bijection, regardless of whether prime ideals determine thick subcategories of B ⊗ Lmodules. 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. Note that Ind Res N ≅ N ⊗ L ≅ N ⊗_{L} (L ⊗ L). So we must understand L ⊗ L as an (L, L)-bimodule. Write L = K[x]/(f(x)) for some monic irreducible polynomial f(x). Then L ⊗ L = L[y]/(f(y)), where the right action of x ∈ L coincides with multiplication by y. Since L is Galois over K, f(y) = ∏_{σ∈G}(y −x^{σ}). Hence the Chinese remainder theorem implies that L ⊗ L ≅ ⨁_{σ∈G} L^{σ}. The lemma follows.

Lemma 4.2. Suppose B is an algebra defined over a field K, L is a Galois extension of K, and L′ ⊆ 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 \overline{σ} ∈ G/H, there is a natural isomorphism

Ind_{L′}^{L}(M\overline{σ}) ≅ (Ind_{L′}^{L} M)^{σ}.

Proof. The easiest way to see this is to calculate the adjoints. A map of B ⊗ Lmodules (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\overline{σ} ⟶ N is the same thing as a map of B ⊗ L′-modules M ⟶ (Res_{L′}^{L} N)\overline{σ^{−1}}. But there is an obvious isomorphism Res_{L′}^{L}(N^{σ^{−1}}) ≅ (Res_{L′}^{L} N)\overline{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, 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 generated B-modules and Galois invariant tensor-closed thick subcategories of finitely generated B ⊗ L-modules.

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_{K}^{L′} N′. Hence IR(D) is generated by the modules Ind_{L′}^{L} Ind_{K}^{L′} Res_{K}^{L′} N′. Using Lemmas 4.1 and 4.2, we find that Ind_{L′}^{L} Ind_{K}^{L′} Res_{K}^{L′} N′ 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 a field K of characteristic p if, for every element α of L, there is some n such that α^{p^{n}} ∈ 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 thick subcategories over B ⊗ L are determined by prime ideals, then so are 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). We have the commutative diagram

J(R) ⟶^{i_{∗}} J(R ⊗ L)
f_{B} ↓  ↓^{f_{B⊗L}}
L_{thick}^{⊗}(B) ⟶_{I} L_{thick}^{⊗}(B ⊗ L)

of Lemma 3.3. We know that f_{B} and I are injections, and we are assuming that f_{B⊗L} is an isomorphism. We claim that i_{∗} is an isomorphism. Given this, it is clear that f_{B} and I must also be isomorphisms.

Since the map i: R ⟶ R ⊗ L is an F -isomorphism, the map i^{∗}: Proj(R ⊗ L) ⟶ Proj(R) is a homeomorphism. This general fact is easy to see in this case. First of all, since R ⊗L is faithfully flat over R, i^{∗} is surjective. Furthermore, if 𝔮 is a prime ideal of R ⊗ L, then, because L is purely inseparable over K, we have 𝔮 = \sqrt{(i^{∗}𝔮)^{e}}, so i^{∗} is injective. It follows easily from this formula that i^{∗} is closed as well. Since J(R) is the collection of unions of closed sets in Proj(R), and similarly for J(R⊗L), it follows that i_{∗} is an isomorphism.

We would also like a theorem analogous to Theorem 4.3, asserting in the purely inseparable case that I: L_{thick}^{⊗}(B) ⟶ L_{thick}^{⊗}(B ⊗ L) is always an isomorphism, regardless of whether prime ideals determine thick subcategories over B⊗L. However, Lemma 4.1 does not hold in the purely inseparable case. In fact, if L is a finite purely inseparable extension of K, we can write L = K[x]/(x^{p^{n}} −t) for some t in K, where p is the characteristic of K. It follows that L ⊗ L ≅ L[y]/(y^{p^{n}} − t) ≅ L[y]/(y − x)^{p^{n}} as an (L, L)-bimodule, and this is an indecomposable bimodule.

References

[AM69] M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969.

[BS94] A. Bajer and H. Sadofsky, Cohomology of finite-dimensional connected cocommutative Hopf algebras, J. Pure Appl. Algebra 94 (1994), 115–126.

[BCR97] D. J. Benson, Jon F. Carlson, and Jeremy Rickard, Thick subcategories of the stable module category, Fund. Math. 153 (1997), 59–80.

[Ben98] D. J. Benson, Representations and cohomology. I, second ed., Cambridge University Press, Cambridge, 1998.

[FS97] E. M. Friedlander and A. Suslin, Cohomology of finite group schemes over a field, Invent. Math. 127 (1997), 209–270.

[HP99] M. Hovey and J. H. Palmieri, Stably thick subcategories of modules over Hopf algebras, Math. Proc. Cambridge Philos. Soc., to appear.

[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, 1975.

[Ver77] J.-L. Verdier, Catégories dérivées, Cohomologie Etale (SGA 4 \frac{1}{2} ) (P. Deligne, ed.), 1977, Springer Lecture Notes in Mathematics 569, pp. 262–311.

[Wil81] C. Wilkerson, The cohomology algebras of finite dimensional Hopf algebras, 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 98195

E-mail address: palmieri@member.ams.org