VARIETIES AND INDUCTION

Jon F. Carlson^{∗}

Department of Mathematics University of Georgia Athens, Georgia 30605

1. Introduction

In the last few years Dave Benson, Jeremy Rickard and the author have been involved in a project to extend the notions and methods of support varieties for finitely generated modules over group algebras to the category of all modules. For a finitely generated module, the definition of its variety is reasonably straightforward. If G is a finite group, k a field of characteristic p and M a finitely generated kGmodule, then the variety of M is the closed set in the maximal ideal spectrum of H^{∗}(G, k) corresponding to the ideal which is the annihilator of Ext_{kG}^{∗}(M, M). For an infinitely generated module things are much more complicated. For one thing the “variety” of an infinitely generated module is not really a variety but rather is a subset of the prime ideal spectrum of the cohomology ring.

The project was originally motivated by an attempt to recover the Krull-Schmidt Theorem in complexity quotient categories [3]. It has also led to the discovery of several unexpected results. In one exciting paper Rickard [9] has shown that the stable category of all kG-modules modulo projectives has idempotent objects which correspond to thick subcategories defined by varieties. These objects have played a role in the development of an extended definition of cohomological varieties for modules and an extension of the Tensor Product Theorem to the category of all kG-modules [4].

One of the biggest surprises has been that the results on infinitely generated modules can be used to prove new things about the category of finitely generated modules. For example, for certain types of groups we may use the methods to characterize the thick subcategories of the stable category of finitely generated kG-modules [5]. In another remarkable application, Benson [2] has settled several problems on the vanishing of the cohomology H^{∗}(G, M) for M a kG-module. One by-product of Benson’s study is a theorem which asserts that a periodic module whose variety has certain properties must be stably isomorphic to a module induced from a subgroup determined by the variety. Specific cases of this sort of behavior had been known before. However this seems to be the first result to pinpoint a general causal factor.

In this paper we generalize some of the ideas of Benson’s proof. For a variety V and a subgroup H we give necessary and sufficient conditions that a module,

Typeset by AMS-TEX

∗Partially supported by a grant from NSF.

whose variety is in V, is stably isomorphic to a module induced from a kH-module. The statement and proof of the main theorem are very technical and use almost all of the technology of varieties for infinitely generated modules. For this reason we have given a detailed outline of the necessary background. In the final section there are some examples which show how the theorem can be applied. One interesting aspect of the examples is that in all of the nontrivial cases, the subgroup H is not normal. There is probably good reason for this but the details seem fuzzy.

The reader should bear in mind that the conditions of the theorem are most easily verified and most meaningful in the case that the group is a p-group. Consequently the main theorem, corollary and examples should really be interpreted as results about the vertices and sources of modules over more general groups.

One of the ironies of this paper is that its original inspiration, which was Benson’s result in the one-dimensional case, could have been proved in the context of finitely generated modules. For suppose that ℓ is a line in V_{G}(k), and that ℓ is minimally supported (see section 4) on an elementary abelian p-subgroup E. Assume that ℓ is an orbit of maximal length under the action of N_{G}(E)/C_{G}(E). Then an easy generalization of the techniques of [6] show that, for some integers m and n, there is an exact sequence

0 → L → (Ω^{n}(k))^{m} → k_{D}^{↑G} → 0

where D = D_{G}(E) is the diagonalizer of E (the set of all y ∈ N_{G}(E) such that there exists a ∈ ℤ/pℤ with yxy^{−1} = x^{a} for all x ∈ E) and ℓ ∉ V_{G}(L). So if M is a finitely generated kG-module with V_{G}(M) = ℓ, then by the Tensor Product Theorem, L ⊗ M is projective. Hence

(Ω^{n}(k))^{m} ⊗ M ≅ (Ω^{n}(M))^{m} ⊕ proj ≅ k_{D}^{↑G} ⊗ M ⊕ proj
≅ M_{D}^{↑G} ⊕ proj .

Moreover because M is periodic, M^{m}⊕proj ≅ M_{D}^{↑G}⊕proj. The rest can be finished by a vertices and sources argument.

On the other hand, the arguments of [6] are based on the existence of certain transfer maps and are entirely inadequate for dealing with varieties which are not lines. Hence for varieties which are not lines it seems necessary to use some sort of generalization of Benson’s argument as we have given here.

2. Varieties and idempotent modules

In this section we review some of the background necessary for understanding the proof of the main theorem. Throughout the paper let k be an algebraically closed field of characteristic p > 0 and let K be an algebraically closed extension of k such that K has infinite transcendence degree over k. Let G be a finite group. The category of finitely generated kG-modules and kG-homomorphisms is denoted mod-kG. Let stmod-kG be the stable category of finitely generated kG-modules modulo projectives. That is, stmod-kG has the same objects as mod-kG, but for objects M and N, Hom_{stmod-kG}(M, N) = Hom_{kG}(M, N)/PHom_{kG}(M, N) where PHom_{kG} is the set of all morphisms which factor through a projective module. We use the symbols Mod-kG and StMod-kG to denote the corresponding categories of all kG-modules. We shall use the fact that the stable categories stmod-kG and StMod-kG are triangulated with triangles in the stable categories corresponding to exact sequences in the module categories. For a general reference on triangulated categories see [11].

The cohomology ring H^{∗}(G, k) is a finitely generated graded-commutative kalgebra. Let V_{G}(k) be its maximal ideal spectrum. Classically the variety of a module M in mod-kG is defined to be the closed subvariety V_{G}(M) = V_{G}(J(M)) where J(M) is the annihilator in H^{∗}(G, k) of the cohomology ring Ext_{kG}^{∗}(M, M) of M. For V a closed subvariety of V_{G}(M), the full subcategory M_{V} of all finitely generated kG-modules whose varieties are contained in V is a thick (triangulated) subcategory of stmod-kG. Basic information on varieties of finitely generated modules can be found in the books by Benson [1] and Evens [8].

In [4], the notion of “variety” was extended to the category of all kG-modules. The important thing to notice here is that the “variety” of an infinitely generated module may not be a variety in the usual sense. The definition goes as follows. Let V_{G}(k) be the collection of all homogeneous irreducible subvarieties of V_{G}(k). That is, V_{G}(k) is the projectivized prime ideal spectrum Proj H^{∗}(G, k), but without the topology. For a module M in StMod-kG, the variety of M, V_{G}(M), is the collection of all X in V_{G}(k) such that κ(X) ⊗ M is not a projective module, where κ(X) is a certain idempotent module associated to X. The definition of κ(X) is given later in this section. For now it is better that we emphasize that the technicality of the definition will not be important in the works of this paper. Instead we will measure the varieties using the concept of the rank variety which is outlined below.

Again we emphasize that V_{G}(M) is, in general, only a subset of the projectivized prime ideal spectrum. If M is in mod-kG then we have the special case that V_{G}(M) is closed under specialization. This means that if U and V are homogeneous irreducible subvarieties of V_{G}(k) with U ⊆ V and V ∈ V_{G}(M) then U ∈ V_{G}(M). In fact, for M finitely generated, V_{G}(M) = ℌ(V_{G}(M)) is the set of all irreducible homogeneous subvarieties of V_{G}(M). On the other hand if M is not finitely generated then no such assumptions can be made. The idempotent modules F (V) and κ(V) defined below are examples of modules whose varieties are not closed under specialization.

Now suppose that E = ⟨x_{1}, …, x_{n}⟩ is an elementary abelian p-group of order

p^{n}. For α = (α_{1}, …, α_{n}) ∈ k^{n}, let u_{α} = 1 + ∑_{i=1}^{n} α_{i}(x_{i} − 1). Then u_{α} is a unit of

order p in kE and kE is free as a k⟨u_{α}⟩-module. If α ≠ 0 then the subgroup ⟨u_{α}⟩ is called a cyclic shifted subgroup of kE. The classical rank variety of M ∈ mod-kE is given as

(2.1)  V_{E}^{r}(M) = {α ∈ k^{n} | M ↓_{⟨u_{α}⟩} is not free} ∪ {0} .

It is a closed homogeneous subvariety of V_{E}^{r}(k) = k^{n}. Moreover there is an isogeny τ : V_{E}(k) → V_{E}^{r}(k). If p = 2, this is actually an isomorphism, while if p > 2 then we must twist by the Frobenius map (see the proof of Theorem 7.2 of [6]). The point is that for any line ℓ (homogeneous irreducible dimension-one subvariety) of V_{E}(k), we have that ℓ ⊆ V_{E}(M) (or equivalently ℓ ∈ V_{E}(M)) if and only if the corresponding shifted subgroup ⟨u_{α}⟩, α = τ(γ) for γ ∈ ℓ, fails to act freely on M (i.e. α ∈ V_{E}^{r}(M)).

The equation defining V_{E}^{r}(M) in (2.1) makes sense also in the case that M is infinitely generated. However the result about the correspondence is no longer valid. Instead, to define a usable rank variety we must expand our horizons. Here is where the transcendental extension K of k comes into play. The new rank “variety” of any kG-module M is defined to be

V_{E}^{r}(K ⊗ M) = {α ∈ K^{n} | (K ⊗ M)_{⟨u_{α}⟩} is not free} ∪ {0}.

As with the cohomological variety defined before, this is not necessarily a closed subset of K^{n} if M is not finitely generated. However we still have a correspondence between cohomological varieties and rank varieties. If X ∈ V_{E}(k), then under the isogeny τ, τ(X) is a homogeneous irreducible subvariety of V_{E}^{r}(k). Let α ∈ K^{n} be an element which is generic for τ(X). Then X ∈ V_{E}(M) if and only if α ∈ V_{E}^{r}(K ⊗M).

If H is a subgroup of G then we have maps of varieties

res_{G,H}^{∗} : V_{H}(k) → V_{G}(k)

and of projectivized prime ideal spectra

res_{G,H}^{∗} : V_{H}(k) → V_{G}(k)

induced by the restriction map res_{G,H} : H^{∗}(G, k) → H^{∗}(H, k). As in the case of finite dimensional modules we have a Quillen-Alperin-Evens type of theorem saying that for any module M

(2.2)  V_{G}(M) = ⋃_{E∈A_{p}(G)} res_{G,E}^{∗}(V_{E}(M))

where A_{p}(G) is the collection of all elementary abelian p-subgroups of G [4]. In addition the fiber of all of the restriction maps is exactly what we expect. Specifically, suppose that X ∈ V_{H}(k) then

res_{G,H}^{∗}(X) ∈ V_{G}(M) if and only if X ∈ V_{H}(M) [4].

A lot of this can be summed up in the following result which will be a major tool in the proofs of the next two sections. For terminology, we say that a subvariety X ∈ V_{G}(k) is supported on a subgroup H if X ∈ res_{G,H}^{∗}(V_{H}(k)). We say that X is minimally supported on E if X is supported on E but not on any proper subgroup of E. Note that by (2.2) any X is minimally supported on some E ∈ A_{p}(G).

Lemma 2.3 ([4]). Suppose that X ∈ V_{G}(k) is supported on E ∈ A_{p}(G). Then there exists a shifted cyclic subgroup ⟨u⟩ in KE such that for any kG-module M, X ∈ V_{G}(M) if and only if (K ⊗ M)_{⟨u⟩} is not a free K⟨u⟩-module.

One property of the varieties which we shall require is expressed in the Tensor Product Theorem.

Theorem 2.4. [4]. Let M and N be kG-modules. Then

V_{G}(M ⊗ N) = V_{G}(M) ∩ V_{G}(N) .

One more tool is necessary for the proofs of this paper. In [9] Rickard showed that the stable category StMod-kG has modules T which are idempotent in the sense that T ⊗_{k} T is stably isomorphic to T, i.e. T ⊗_{k} T ≅ T ⊕ proj for some projective module proj. The modules can be defined for any subset V ⊆ V_{G}(k) which is closed under specialization. However for the purposes of this paper we stick to the case of a single closed subvariety V ⊆ V_{G}(k). The information which we need is expressed in the following theorem.

Theorem 2.5. [9]. Let V ⊆ V_{G}(k) be a closed subvariety. Then in StMod-kG there is a triangle of modules

(2.6)  E(V) ⟶^{σ} k ⟶ F (V) ⟶ Ω^{−1}(E(V))

with the properties:

1) E(V) is a filtered colimit of finitely generated modules in M_{V} (the full subcategory of stmod-kG of all finitely generated modules with varieties contained in V), and

2) F (V) is M_{V} -local, i.e. Hom_{StMod-kG}(M, F (V)) = 0 for all M ∈ M_{V}.

Moreover properties (1) and (2) characterize the triangle (2.6) up to isomorphism.

Notice that property (2) says that any homomorphism θ : M → F (V), for M ∈ M_{V}, must factor through a projective module. Similarly M ⊗ F (V) is a projective module and M ⊗ E(V) ≅ M ⊕ proj, whenever M ∈ M_{V}. The modules E(V) and F (V) are idempotent in the stable category and E(V) ⊗ F (V) is a projective module, hence isomorphic to {0} in StMod-kG.

For V ⊆ V_{G}(k) a closed subvariety we let ℌ(V) = {X ∈ V_{G}(k) | X ⊆ V }. In this notation we have that

V_{G}(E(V)) = ℌ(V) and V_{G}(F (V)) = V_{G}(k) ⧹ ℌ(V) .

It should be noted that the module κ(V) mentioned earlier can be defined as

κ(V) = E(V) ⊗ F (V′)

where V′ = ℌ(V) ⧹ {V } ⊆ V_{G}(k). So by the Tensor Product Theorem (2.4) V_{G}(κ(V)) = {V }.

Finally, we know by the Mackey formula that, for any subgroup H of G, V_{G}(k_{H}^{↑G}) = res_{G,H}^{∗}(V_{H}(k)). Hence we have that

V_{G}(k_{H}^{↑G}) = ℌ(res_{G,H}^{∗}(V_{H}(k))
= res_{G,H}^{∗}(V_{H}(k)) .

The next Lemma follows by a similar argument.

Lemma 2.7. If H is a subgroup of G and M is a kH-module then

V_{G}(M^{↑G}) = res_{G,H}^{∗}(V_{H}(M)) .

3. Varieties and Mackey decompositions

The goal of this section is to prove a technical result about the varieties of the pieces of the Mackey decomposition of the restriction of an induced module. For notation, suppose that M is a kH-module for H a subgroup of G. For x ∈ G, let

M̃_{x} = ((x ⊗ M) ↓_{H∩xHx^{−1}})^{↑H}.

The Mackey decomposition says that

M^{↑G} ↓_{H} ≅ ⨁_{HxH} M̃_{x}

where the sum is over a set of representations of the H-H double cosets in G.

Proposition 3.1. Suppose that H is a subgroup of G and let W ⊆ V_{H}(k) be a closed subvariety. For x ∈ G, let S_{x} be the set of all lines ℓ in V_{H}(k) having the property that there exists an elementary abelian p-subgroup E ⊆ H on which ℓ is supported, a line ℓ ⊆ V_{E}(k) and an element y ∈ H such that E^{yx} = (yx)^{−1}Eyx ⊆ H, res_{H,E}^{∗}(ℓ′) = ℓ and res_{H,E^{yx}}^{∗}(ℓ^{′yx}) ⊆ W. Suppose that M is a kH-module with V_{H}(M) = ℌ(W). Then V_{H}(M̃_{x}) = ℌ(U) where

U = ⋃_{ℓ∈S_{x}} ℓ .

Proof. First we should notice that from a philosophical standpoint we should expect that V_{H}(M̃_{x}) = ℌ(U) for some closed set U ⊆ V_{H}(k). This is because the variety of M̃_{x} should be the same regardless of whether M is finitely generated or infinitely generated. Hence we can assume that V_{H}(M̃_{x}) has the type of the variety of a finitely generated module. This will become even clearer in the argument to follow.

Suppose that Y ∈ V_{H}(k) is minimally supported on E. So Y = res_{H,E}^{∗}(X) for some homogeneous irreducible subvariety X ⊆ V_{E}(k). Let ⟨u⟩ ∈ KE be a cyclic shifted subgroup corresponding to X as in Lemma 2.3. Then Y ∈ V_{H}(M̃_{x}) if and only if K ⊗ M̃_{x} ↓_{⟨u⟩} is not free as a K⟨u⟩-module. But now

K ⊗ M̃_{x} ↓_{E} = x ⊗ (K ⊗ M) ↓_{H∩xHx^{−1}} ^{↑H} ↓_{E}

= ∑_{Ey(H∩xHx^{−1})} y ⊗ x ⊗ (K ⊗ M) ↓_{E∩y(H∩xHx^{−1})y^{−1}} ^{↑E}

= ∑ yx ⊗ (K ⊗ M) ↓_{E∩yxH(yx)^{−1}} ^{↑E}

where the sum is over a set of representatives of the E −(H ∩xHx^{−1}) double cosets in H. Now observe that by the minimality of the support, ⟨u⟩ is not in KE′ for any E′ proper in E and ⟨u⟩ acts freely on any KE-module induced from a proper subgroup of E. So we have that Lemma 3.2. Suppose that Y ∈ V_{H}(k) is minimally supported on E and Y = res_{G,E}^{∗}(X) for X ∈ V_{E}(k). Then Y ∈ V_{H}(M̃_{x}) if and only if there is an element y ∈ H such that E^{yx} ⊆ H and

X^{yx} ∈ V_{E^{yx}}(M) .

Now suppose that Y ∈ V_{H}(M̃_{x}) and ℓ is any (k-rational) line in Y. Then ℓ is supported on E and we may assume that ℓ = res_{H,E}^{∗}(j) for j ⊆ X also a k-rational line. By the lemma there exists y ∈ H such that X^{yx} ∈ V_{E^{yx}}(M). However V_{E^{yx}}(M) is closed under specialization by hypothesis. So j^{yx} ∈ V_{E^{yx}}(M) and res_{H,E^{yx}}^{∗}(j^{yx}) ∈ V_{H}(M) = ℌ(W) and ℓ ∈ S_{x}. It follows that Y = ⋃_{S_{y}} ℓ where

S_{y} = {ℓ ∈ S_{x} | ℓ ⊆ Y }. Hence also Y ⊆ U and Y ∈ ℌ(U).

On the other hand suppose that Y is any element of ℌ(W). Then Y = ⋃_{S_{y}} ℓ for

S_{y} as above. This is because all of the lines in U are also in S. Suppose that Y is minimally supported on E and Y = res_{G,E}(X). Let

Z = Y − ⋃_{E′⊆E}(Y ∩ res_{H,E′}^{∗}(V_{E′}(k))) .

Notice that Z is not empty and that closure of Z is Y. That is, Z is homogeneous and relatively open. Moreover

Z = res_{H,E}^{∗}(Z′) for
Z′ = X − ⋃_{E′⊆E}(X ∩ res_{E,E′}^{∗}(V_{E′}(k))) .

Now let ℓ be a line in Z. Then ℓ = res_{H,E}^{∗}(ℓ′) for some ℓ′ ∈ Z′. But in addition ℓ′ ∈ S, since ℓ ⊆ Y ⊆ U and both ℓ and ℓ′ are minimally supported on E. So there must exist y ∈ H such that E^{yx} ⊆ H (note that any elementary abelian p-subgroup which supports ℓ contains an H-conjugate of E), and

res_{H,E^{yx}}^{∗}(ℓ^{′yx}) ⊆ W.

Because Z′ is homogeneous it is the union of its lines. So

res_{H,E^{yx}}^{∗}(Z^{′xy}) ⊆ W.

Then by taking closures we get that

res_{H,E}^{∗}(X^{yx}) ⊆ W.

So by the Lemma, Y ∈ V_{H}(M̃_{x}). This proves the proposition.

Corollary 3.3. Suppose that M is a kH-module and that V_{H}(M) = ℌ(W) for W a closed set in V_{H}(k). Then V_{H}(M̃_{x}) has a nontrivial intersection with V_{H}(M̃_{y}) if and only if they share a k-rational line.

Proof. We know that V_{H}(M̃_{x}) = ℌ(U_{x}) and V_{H}(M̃_{y}) = ℌ(U_{y}) for U_{x} and U_{y} as in the theorem. But clearly

V_{H}(M̃_{x}) ∩ V_{H}(M̃_{y}) ≠ ∅

if and only if U_{x} ∩ U_{y} ≠ {0}.

4. The main theorem

In this section we give necessary and sufficient conditions on a subvariety V ⊆ V_{G}(k) and a subgroup H ⊆ G such that every module whose variety is in V must be induced from H. The case in which V is a line was originally done by Benson [2], who first applied the idempotent modules to this situation. The proof here is mostly an exercise in pushing past all of the technical difficulties of the extension. Indeed the theorem itself comes out as a very technical statement. The corollary that follows represents an attempt to make some sense of the technicalities. However the best view of what is happening might be coming from the examples of the next section.

For terminology, we say that a kG-module M is stably induced from a kHmodule if there is a kH-module L such that L^{↑G} ≅ M ⊕ proj.

Theorem. Suppose that H is a subgroup of G and that V ⊆ V_{G}(k) is a closed subvariety. Let M_{V} be the full subcategory of StMod-kG consisting of all kGmodules M with V_{G}(M) ⊆ ℌ(V). Then every object in M_{V} is stably induced from a kH-module if and only if there is a subvariety W ⊆ V_{H}(k) satisfying the following two conditions.

1) res_{G,H}^{∗}(W) = V.

2) Suppose that ℓ ∈ W is any line and that ℓ is minimally supported on E ⊆ H. Then for any x ∈ G, x ∉ H, if xEx^{−1} ⊆ H and ℓ = res_{H,E}^{∗}(ℓ′) then res_{H,xEx^{−1}}^{∗}(ℓ^{′x}) ∉ W.

Proof. First assume that every module in M_{V} is stably induced from H. Let E(V) be the idempotent module corresponding to M_{V}. Then E(V) = e^{↑G} for some kH-module e. Let

e^{↑G} ↓_{H}= ∑_{HxH} x ⊗ e ↓_{H∩xHx^{−1}} ^{↑H} ≅ ∑_{HxH} ẽ_{x}

be the Mackey decomposition in the notation of the proposition of the last section. Several things should be observed at this point.

First, it can be seen that

V_{H}(ẽ_{x}) ∩ V_{H}(ẽ_{y}) = ∅

if x and y are in different H-H double cosets. The proof of this goes as follows. Suppose that Y is a homogeneous irreducible subvariety in V_{H}(ẽ_{x}). Then there is an elementary abelian p-subgroup E on which Y is supported and a cyclic shifted subgroup ⟨u⟩ ⊆ KE corresponding to Y in the sense of Lemma 2.3. In particular K ⊗ ẽ_{x} ↓_{⟨u⟩} is not a free K⟨u⟩-module. From the triangle we obtain an exact sequence

0 ⟶ K ⊗ E(V) ⟶^{σ} K ⊕ proj ⟶ K ⊗ F (V) ⟶ 0

where K ⊗ F (V) ↓_{⟨u⟩} is free as a K⟨u⟩-module. So σ is stably (modulo projectives) an isomorphism of K⟨u⟩-modules, and

K ⊗ E(V) ↓_{⟨u⟩} ≅ K ⊕ proj.

We also know that

K ⊗ E(V) ↓_{⟨u⟩} ≅ ∑ K ⊗ ẽ_{x} ↓_{⟨u⟩} .

It follows that K ⊗ ẽ_{y} is not free as a K⟨u⟩-module if y ∉ HxH. Therefore if y ∉ HxH then Y ∉ V_{H}(ẽ_{y}) as claimed.

The second thing to observe is that V_{H}(tildee_{x}) is closed under specialization. For suppose that M is a finitely generated module with V_{G}(M) = V. Then V_{H}(M_{H}) = res_{G,H}^{∗}^{−1}(ℌ(V)) and contains V_{H}(ẽ_{x}) for all x. So by the Tensor Product Theorem (2.4)

V_{H}(M_{H} ⊗ ẽ_{x}) = V(ẽ_{x}).

But M ⊗ E(V) ≅ M ⊕ proj by Theorem 2.4. So M ⊗ E(V) ↓_{H} ≅ ∑ M ⊗ ẽ_{x} and for each x, M ⊗ ẽ_{x} ≅ L_{x} ⊕ proj for some finitely generated module L_{x}. Thus

V_{H}(ẽ_{x}) = V_{H}(M_{H} ⊗ ẽ_{x}) = V_{H}(L_{x}) = ℌ(V_{H}(L_{x}))

which is closed under specialization.

In particular, we have that V_{H}(e) = ℌ(W) for some closed subvariety W ⊆ V_{H}(k). Clearly we must have that res_{G,H}(W) = V (see Lemma 2.7). This proves (1). Now (2) follows by Proposition 3.1 (see also Corollary 3.3) and the disjunction of V_{H}(e) and V_{H}(ẽ_{x}) for x ∉ H. This proves the “only if” part of the theorem.

For the other half, assume that there exists W ⊆ V_{H}(k) satisfying conditions (1) and (2). Let e_{W} be the idempotent kH-module corresponding to W. By Theorem 2.5 there is a triangle

e_{W} ⟶^{σ} k ⟶ f_{W} ⟶ Ω^{−1}(e_{W}).

Let 1⊗σ : e_{W}^{↑G} = kG⊗_{kH} e_{W} → kG⊗_{kH} k ≅ k_{H}^{↑G} be the induced kG-homomorphism. Let ε : k_{H}^{↑G} → k (ε(∑ g ⊗ a_{g}) = ∑ a_{g}) be the relative augmentation. Then in StMod-kG there is a triangle

(4.2)  e^{↑G} ⟶^{ε(1⊗σ)} k ⟶ f′ ⟶ Ω^{−1}(e^{↑G}) .

Our objective is to show that the triangle (3.2) is isomorphic in StMod-kG to the triangle

(4.3)  E(V) ⟶^{γ} k ⟶^{θ} F (V) ⟶ Ω^{−1}(E(V)) .

For this we use the distinguishing characteristics of triangle (4.3) in Theorem 2.5. It is easy to see that e_{W}^{↑G} is a filtered colimit of modules in M_{V} since e_{W} is a filtered colimit of finitely generated modules whose varieties are contained in W. So to obtain an isomorphism of (4.2) and (4.3) we need only show that f′ is M_{V} -local. That is we need to prove that for any M in M_{V},

(4.4)  Hom_{StMod-kG}(M, f′) ≅ Hom_{StMod-kG}(k, M^{∗} ⊗ f′) = 0 .

Here the isomorphism on the Hom’s is a consequence of the finite dimensionality of M. To verify (4.4) it would be sufficient to prove that M^{∗} ⊗ f′ is projective or, because of the Tensor Product Theorem, that V_{G}(f′) is disjoint from ℌ(V).

So suppose that Y ∈ ℌ(V) and let X ∈ V_{H}(W) be an element such that res_{G,H}^{∗}(X) = Y. Assume that X is supported on an elementary abelian p-subgroup E and that ⟨u⟩ ∈ KE is a shifted cyclic subgroup corresponding to X in the sense of Lemma 2.3. Then because X is in W, the homomorphism σ′ : K⊗e_{W} → K⊗k = K induced by σ is stably a K⟨u⟩-isomorphism. By Proposition (3.1) and the hypothesis (ẽ_{W})_{x} is free as a K⟨u⟩-module for x ∉ H. Of course

K ⊗ e_{W}^{↑G} ↓_{H} ≅ (K ⊗ e_{W}) ⊕ ∑_{\begin{matrix}HxH \\ x∉H \end{matrix}} K ⊗ (ẽ_{W})_{x}.

Moreover, ε(1 ⊗ σ′) : K ⊗ e_{W}^{↑G} → K when restricted to K ⊗ e_{W} coincides with σ′. Hence ε(1 ⊗ σ′) is a stable isomorphism of k⟨u⟩-modules. Then from triangle (4.2) we see that K ⊗ f′ is free as a K⟨u⟩-module. Now by Lemma 2.2, we deduce that Y ∉ V_{G}(f′). Because this holds for all Y in ℌ(V), f′ is M_{V} -local, and triangles (4.2) and (4.3) are isomorphic.

Therefore e_{W}^{↑G} ≅ E(V) ⊕ proj. To finish the proof suppose that M is any object in M_{V}. Then

M ⊕ proj ≅ M ⊗ E(V) ⊕ proj
≅ M ⊗ e_{W}^{↑G} ≅ (M_{H} ⊗ e_{W})^{↑G} .

Corollary 4.5. Suppose that H ⊆ G is a subgroup and that W ⊆ V_{H}(k) is a closed homogeneous subvariety. Assume that the following conditions hold.

1) If g ∈ N_{G}(H), g ∉ H then W ∩ W^{g} = {0}.

2) If g ∉ N_{G}(H) and ℓ is any line in W then E^{g} ⊈ H for any elementary

abelian p-subgroup on which ℓ is supported. If M is a kG-module with V_{G}(M) ⊆ res_{G,H}^{∗}(ℌ(W)) then M ⊕ proj ≅ L^{↑G} for some kH-module L.

Proof. A direct check shows that W satisfies the conditions of the theorem.

5. Examples

In this final section we sketch a couple of examples to show how the main theorem can be used. The examples are in no way meant to be exhaustive, as the theorem will apply in many other situations. In the case that the variety V is a line in V_{G}(k), the result of Benson [2] comes close to saying it all. Simply choose an elementary abelian p-subgroup E on which V is minimally supported. Then if V = res_{G,E}^{∗}(ℓ′), we let H be the stabilizer of the line ℓ′ in N_{G}(E) and get that every kG-module with variety in ℓ is stably induced from H.

One way to construct more complicated examples is to consider root subgroups of Chevalley groups. Suppose that G = G(𝔽) is a group of Lie type defined over the field 𝔽 = GF (p^{m}) of p^{m} elements. Let Σ be a root system for the corresponding Lie algebra and let Σ^{+} be a set of positive roots. A Sylow p-subgroup of G is the group P generated the symbols x_{α}(t) for α ∈ Σ^{+} and t ∈ 𝔽. The relations (see [10]) on the generators are expressed by the equations

x_{α}(t)x_{α}(u) = x_{α}(t + u)

and the commutator relations

(5.1)  [x_{α}(t), x_{β}(u)] = ∏_{i,j} x_{iα+jβ}(c_{ij}t^{i}u^{j})

where c_{ij} are certain constants determined by α and β. The only property that we need to be concerned about here is that |c_{11}| = |r+1| where β−rα, …, β, …, β+qα is the α-string through β. Now fix a root γ ∈ Σ^{+} and let

Θ_{γ} = {α ∈ Σ^{+} | α + γ is not a root} .

Proposition 5.2. Assume that all roots in Σ have the same length. Suppose that γ ∈ Σ^{+} and that E = ⟨x_{γ}(t) | t ∈ 𝔽⟩. Let H = ⟨x_{β}(t) | t ∈ 𝔽, β ∈ Θ_{γ}⟩. Then every kP -module whose variety is supported on E is stably induced from a kH-module. Also every kG-module whose variety is supported on E has its vertex in H.

Proof. The first thing to notice is that E is in the center of H. This is because if β ∈ Θ_{γ} then no iγ + jβ is a root for i ≥ 1 and j ≥ 1. Hence [x_{γ}(t), x_{β}(u)] = 1 for all t, u ∈ 𝔽 by (5.1). So let W = res_{H,E}^{∗}(V_{E}(k)). Because E is in the center of H, no non-zero element of W is supported on any subgroup A with A ∩ E = {1}. That is any subgroup A which minimally supports a non-zero element of W must be H-conjugate to the subgroup of E which also minimally supports this element.

Now suppose that β ∈ Σ^{+} and β ∉ Θ_{γ}, then β +γ is a root and because all roots have the same length, β − γ and β + 2γ are not roots. Hence

[x_{β}, x_{γ}] = x_{β+γ}(±tu) ∈ H.

If x ∉ H then x = h(Πx_{β}(t_{β})) where the product is over all β ∈ Σ^{+}, β ∉ Θ_{γ}, and where t_{β} ∈ 𝔽 and h ∈ H. So E^{x} ∩ E = {1}. This proves that the set W ⊆ V_{H}(k) satisfies condition (2) of Theorem 4.1 (with P = G in the Theorem). If M is a kP -module with V_{P} (M) supported on E, then

V_{P} (M) ⊆ res_{P,E}^{∗}(ℌ(W)).

Hence M is stably induced from H. The second conclusion of the theorem is a consequence of the fact that any kG-module whose variety is supported on E is a direct summand of L^{↑G} for some kP -module L whose variety is also supported on E.

Remark 5.3. We should observe here that for p large enough the proposition holds without the restriction that all roots have the same length. The main thing is that in (5.1) for the commutator [x_{γ}(t), x_{β}(u)] the coefficient c_{11} should not be zero in 𝔽 whenver β ∈ Σ^{+} and β ∉ Θ_{γ}.

The second example is more specific though still a group of Lie type. Let G be the group of unipotent upper triangular n × n matrices with coefficients in 𝔽. So G is a Sylow p-subgroup for SL(n, 𝔽) and any element of G has the form

σ(γ_{ij}) = I + ∑_{i<j} γ_{ij}E_{ij}

where I is the n×n identity matrix, E_{ij} is the matrix with 1 in the (i, j) position and zero elsewhere and, of course, all γ_{ij} ∈ 𝔽. Now fix an index t such that 1 < t < n and let A = {σ(γ_{ij}) | γ_{ij} = 0 if j ≠ t} and H = {σ(γ_{ij}) | γ_{ij} = 0 if i = t}. Notice that A ⊆ H and A is an elementary abelian p-group. Also A is in the center of H. If x ∈ G and x ∉ H then A ∩ A^{x} = {1}. So the next proposition is true for exactly the same reasons as the last one.

Proposition 5.3. Suppose that M is a kG-module such that V_{G}(M) is supported on A. Then M is stably induced from a kH-module.

References

1. D. J. Benson, Representations and Cohomology II: Cohomology of Groups and Modules, Cambridge University Press, Cambridge, 1991.

2. D. J. Benson, Cohomology of modules in the principal block of a finite group, N.Y. J. Math. 1 (1995), 196-205.

3. D. J. Benson, J. F. Carlson and J. Rickard, Complexity and varieties for infinitely generated modules, Math. Proc. Cambridge Phil. Soc. 118 (1995), 223-243.

4. D. J. Benson, J. F. Carlson and J. Rickard, Complexity and varieties for infinitely generated modules,II, Math. Proc. Cambridge Phil. Soc. (to appear).

5. , Thick subcategories of the stable category, preprint.

6. J. F. Carlson, The varieties and the cohomology ring of a module, J. Algebra 85 (1983), 104-143.

7. J. F. Carlson, Decomposition of the trivial module in the complexity quotient category, J. Pure Appl. Algebra 106 (1996), 23-44.

8. L. Evens, The Cohomology of Groups, Oxford University Press, New York, 1991.

9. J. Rickard, Idempotent modules in the stable category, J. London Math. Soc. (to appear).

10. R. Steinberg, Lectures on Chevalley Groups, Yale University Lecture Notes, New Haven, 1968.

11. C. A. Weibel, An Introduction to Homological Algebra, Cambridge University Press, Cambridge, 1994.