THICK SUBCATEGORIES OF THE RELATIVE STABLE CATEGORY

JON F. CARLSON

Abstract. Let G be a finite group and k an algebraically closed field of of characteristic p > 0. Let H be a collection of p-subgroups of G. We investigate the relative stable category stmod_{H}(kG) of finitely generated modules modulo Hprojective modules. Triangles in this category correspond to H-split sequences. Hence, compared to the ordinary stable category there are fewer triangles and more thick subcategories. Our interest is in the spectrum of the category and its relationship to the induction functor. In some cases the spectrum is not noetherian.

1. Introduction

This is a start of an investigation into the nature of the thick subcategories of the relative stable category of modules over the modular group algebra of a finite group, relative to some collection of subgroups of that group. The study is partly motivate by the work of Benson and Wheeler [9] and most recently by of Wang and Zhang [25] proving that the Green correspondence can be viewed as an equivalence of relative stable or derived categories. It raises the question of what are these categories? What structure do they have? A popular way of addressing this question in other contexts has been to classify the thick subcategories, or the thick tensor ideal subcategories. This was accomplished by Hopkins and Neeman [20, 21] for the bounded derived category of a commutative Noetherian ring. For the ordinary stable category of a modular group algebra a classification was proved by the Benson, Rickard and the author [7] (see also [14]). More general theory is studied in papers such as [8].

Assume that k is an algebraically close field of characteristic p > 0 and that G is a finite group. Let H be a collection of nontrivial p-subgroups of G not containing a Sylow p-subgroup. The relative stable category stmod_{H}(kG) relative to H is the category of finitely generated kG-module whose morphisms are the quotient of the ordinary homomorphisms by those that factor through direct sums of modules induced from elements of H.

Date: January 11, 2017.

1991 Mathematics Subject Classification. 20C20 (primary), 20J06.

Research partially supported by NSA grant H98230-15-1-0007 and by Simons Foundation grant 054813-01.

In this paper we do not actually classify the thick subcategories or thick tensor ideal subcategories of the relative stable category. Instead, the aim is to show that the structure of the system of thick tensor ideal subcategories is richer than might otherwise have been thought. For example, when p = 2 and H contains a subgroup of 2-rank at least two, then the Balmer spectrum of prime thick tensor ideal subcategories is not noetherian. The hope is that the discoveries here may lead to a complete solution to the problem. In general the study has raised several questions for which answers do not seem to be currently available.

One motivation for this study is the hope of understanding the relation of tensor products and induction from subgroups. For example, one might ask when the tensor product of two modules is H-projective, i. e. is a direct summand of a module induced from the subgroup H. If H = {H} and M is a kG-module, let ⟨M⟩_{H} denote the thick tensor ideal generated by M. Then we know that ⟨M ⊗ N⟩_{H} = ⟨M⟩_{H} ∩ ⟨N⟩_{H}. Consequently, this tensor product is H-projective if and only if ⟨M⟩_{H} ∩ ⟨N⟩_{H} contains only the zero object. Another way of saying this is that the intersection of the spectra of the two subcategories is empty.

The next two sections of the paper present some preliminaries and definitions concerned with stable categories and annihilators of cohomology. In section 4, a primary method for the construction of thick tensor ideals is introduced. The method uses additive tensor ideals of the stable category of subgroups of members of the collection H of subgroups of G. In Section 5, then method is used to show that the spectrum of the relative stable category is not noetherian for many 2-groups and collections of subgroups. Other methods of creating tensor ideal thick subcategories are given Section 6 using idempotent modules and in Section 7 using inflations. In Section 8, we show that some thick subcategories defined by support varieties have no nonzero objects. Two examples, are explored in some detail in Section 9, and we consider a question of virtual projectivity in the last section.

2. Notation and preliminaries

Throughout the paper, we assume that G is a finite group and that k is an algebraically closed field of characteristic p. We assume that all modules are finitely generated unless otherwise stated. Our starting off point is the paper [15]. The relative stable category is defined there. See also [19] for general background. The essence of the material is the following.

Let H be a collection of p-subgroups of G. A kG-module is relatively H-projective if it is a direct sum of induced modules from the subgroups in H. The category stmod_{H}(kG) has finitely generated kG-modules as objects and morphisms given by

\underline{Hom}_{H}(M, N) = Hom_{stmod_{H}(kG)}(M, N) ≅ \frac{Hom_{kG}(M, N)}{PHom_{H}(M, N)}

where PHom_{H}(M, N) is the set of kG-homomorphisms from M to N that factor through H-projective modules. An exact sequence \xymatrix{0 \ar[r] & L \ar[r] & M \ar[r] & N \ar[r] & 0} is H split if it splits on restriction to every subgroup in H.

We recall that H-projectivity and H-split are the same as W -projective and W - split for W = ∑_{H∈H} k_{H}^{↑G}, the direct sum of the induced trivial modules from each of the elements of the collection H. See [15] for details.

In the relative stable category, the translation functor is the relative shift Ω_{H}^{−1}. That is, if M is a kG-module, its relative injective hull is the smallest H-projective module P (M) into which M injects by an H-split map. Thus, we have an H-split exact sequence

\xymatrix{0 \ar[r] & M \ar[r] & P (M) \ar[r] & Ω_{H}^{−1}(M) \ar[r] & 0.}

A triangle in the relative stable category

[figure: L; M; N; Ω_{H}^{−1}(L)]

corresponds to an H-split sequence

\xymatrix{0 \ar[r] & L \ar[r] & M \ar[r] & N \ar[r] & 0.}

It is an easy observation that the triangles behave well with respect to both tensor products and support varieties. See [16] for a full account of support varieties. Briefly, the main issues are as follows. The cohomology ring H^{∗}(G, k) = Ext_{kG}^{∗}(k, k) is a finitely generated, graded commutative k-algebra and for any finitely generated kG-modules M and N, Ext_{kG}^{∗}(M, N) is a finitely generated module over H^{∗}(G, k). Let V_{G}(k) = Proj H^{∗}(G, k), the spectrum of homogeneous prime ideals in the cohomology ring. The support variety of M is the set V_{G}(M) of all such prime ideal that contain the annihilator J(M) in H^{∗}(G, k) of the module Ext_{kG}^{∗}(M, M). Note that J(M) also annihilates Ext_{kG}^{∗}(M, N) and Ext_{kG}^{∗}(N, M) for any finitely generated module N.

The support variety has many useful properties. One, in particular, is the tensor product theorem that V_{G}(M ⊗ N) ≅ V_{G}(M) ∩ V_{G}(N). For a kG-module M, the operation − ⊗ M takes triangles to triangles. Also if V is a closed subset of V_{G}(k), and if two of the three objects in a triangle have their support varieties in V then so does the third. Hence, if V is a collection of closed subsets, closed under finite unions and specializations, then the full subcategory stmod_{H}(kG)_{V} consisting of all finitely generated modules whose varieties are contained in V is thick tensor ideal subcategory. If H consists only of the trivial subgroup, then we know that every thick tensor ideal has this form [7]. We see below that, in general, there are many more if H contains nontrivial p-subgroups.

An H-projective resolution of a module M is a resolution of M by H-projective modules that is H-split at every step. A H-injective resolution is defined similarly. The relative cohomology Ext_{H}^{∗}(M, N) and H_{H}^{∗}(G, M) is computed the usual way using these resolutions. It is well known that, in general, the relative cohomology ring is not finitely generated. Hence, it is does not seem possible to define relative support varieties. However, this does not appear to present problems and the structure of the relative stable category seems to have little to do with the relative cohomology rings.

There are several other ways in which the relative stable categories differ from the ordinary stable category. A cautionary example starts with G an elementary abelian group of order 4. Let H be a subgroup of order 2, and let H = {H}. Since k_{H}^{↑G} is a direct sum of two copies of k on restriction to H, we have an H-split exact sequence

\xymatrix{0 \ar[r] & k \ar[r] & k_{H}^{↑G} \ar[r] & k \ar[r] & 0}

It follows that Ω_{H}^{n}(k) ≅ k for all n. Consequently, the only nonzero indecomposable object in the thick subcategory ⟨k⟩ of stmod_{H}(kG) generated by k, is k itself. In addition, we point out that the subcategory ⟨k⟩ is generated by the module k in a finite number of steps. Consequently, there is no analog to the theorem of Oppermann and Stovicek [22] in the relative case. More generally, we have the following, the proof of which is left as an exercise.

Proposition 2.1. Suppose the G is an elementary abelian group of order p^{2} and that H ⊆ G is a subgroup of order p. Then for H = {H}, and all n, Ω_{H}^{2n+1}(k) ≅ Ω_{H}^{−1}(k) has dimension p − 1 and is the third object in the sequence

\xymatrix{0 \ar[r] & k \ar[r] & k_{H}^{↑G} \ar[r] & Ω_{H}^{−1}(k) \ar[r] & 0.}

In addition, Ω_{H}^{2n}(k) ≅ k for all n. The full subcategory of stmod_{H}(kG) consisting of any collection of objects that is close under the operation − ⊗ Ω_{H}(k) is a thick subcategory.

However, the reader should keep in mind that these subcategories are not thick tensor ideals.

3. Annihilators of cohomology

In preparation for one of the main theorems of this paper, it is necessary to establish some results on the annihilation of cohomology.

We say that ζ ∈ H^{d}(G, k) annihilates the cohomology of a module M, if it annihilates Ext_{kG}^{∗}(M, M). That is, ζ is in the kernel of the map H^{∗}(G, k) → Ext_{kG}^{∗}(M, M) given by cup product with the class of the identity element of M. It is equivalent to the statement that the map ζ ⊗ 1_{M} : Ω^{n}(k) ⊗ M → k ⊗ M ≅ M factors through a projective map. Note that ζ annihilates the cohomology of M if and only if it annihilates the cohomology of Ω^{t}(M) for any t, and annihilates Ext_{kG}^{∗}(M, N) and Ext_{kG}^{∗}(N, M) for any kG-module N.

A nonzero element ζ ∈ H^{d}(G, k) is represented by an element ζ : Ω^{d}(k) → k. If L_{ζ} is the kernel of this map then we have a triangle in the ordinary stable category:

[figure: L_{ζ}; Ω^{d}(k); ζ; k; Ω^{−1}(L_{ζ})]

In particular, there is an exact sequence

\xymatrix{E_{ζ} : 0 \ar[r] & Ω(k) \ar[r] & L_{ζ} ⊕ (proj) \ar[r] & Ω^{d}(k) \ar[r] & 0}

that represents ζ as an element of Ext_{kG}^{1}(Ω^{d}(k), Ω(k)) ≅ H^{d}(G, k). The projective summand, labeled (proj) is easily seen to be a single copy of the projective cover of the trivial module.

Lemma 3.1. For any kG-module M and ζ ∈ H^{d}(G, k), ζ annihilates the cohomology of M if and only if

L_{ζ} ⊗ M ≅ Ω(M) ⊕ Ω^{d}(M) ⊕ (proj)

Proof. Because E_{ζ} represents the cohomology element ζ, it must be that ζ annihilates the cohomology of M if and only if E_{ζ} ⊗M splits. But this happens if and only if the middle term of E_{ζ} ⊗M is isomorphic to the right hand term in the lemma. □

With regards to the annihilation of the cohomology of one of the modules L_{ζ}, there is the following.

Theorem 3.2. If p > 2 and d is even, then ζ annihilates the cohomology of L_{ζ} for any ζ ∈ H^{d}(G, k). The same is true if p = 2, for elements ζ that are in the subalgebra generated by images of the squares of homogeneous element in H^{∗}(G, 𝔽_{p}) under the inclusion H^{∗}(G, 𝔽_{p}) → H^{∗}(G, k).

Proof. The first statement is one of the main theorems of [11]. The second statement follows from [26, Corollary 1.4] using the Cartan formula for Steenrod operations (see Chapter 7 of [16]). □

Proposition 3.3. Assume that G has p-rank at least 2. Suppose that ζ is a nonnilpotent element of H^{d}(G, k), such that ζ annihilates the cohomology of L_{ζ}. Then for any n ≥ 1, ζ^{n} does not annihilate the cohomology of L_{ζ^{n+1}}.

Proof. By Quillen’s theorem (see Theorem 8.3.1 of [16]) there is an elementary abelian p-subgroup E of G such that the restriction of ζ to E is not nilpotent.

Consequently, it is sufficient to prove the theorem with the assumption that G = E is elementary abelian, and we make this assumption.

Assume that ζ^{n} annihilates the cohomology of L_{ζ^{n+1}}. Then by Lemma 3.1,

L_{ζ^{n}} ⊗ L_{ζ^{n+1}} ≅ Ω(L_{ζ^{n}}) ⊕ Ω^{(n+1)d}(L_{ζ^{n}}) ⊕ (proj) ≅ Ω(L_{ζ^{n+1}}) ⊕ Ω^{nd}(L_{ζ^{n+1}}) ⊕ (proj) .

By a dimension argument, the above is not possible. For this we restrict to the rank-two case. That is, suppose that E is a subgroup of G of p-rank 2 or a shifted subgroup of kE of rank 2. By a shifted subgroup we mean a subgroup of the group of units of kG generated by elements of the form 1 + a for a ∈ Rad(kG). Thus, if E is a shifted subgroup of rank 2, then E is generated by x = 1 + a and y = 1 + b where a and b are linearly independent modulo Rad^{2}(kG), and kE ≅ k[a, b]/(a^{p}, b^{p}) is isomorphic to the group algebra of an elementary abelian group of order p^{2}. The inclusion kE ↪ kG is a flat embedding and kG is a projective module over kE. See Section 4.6 of [16] for more details on shifted subgroup.

The point is that there must exist some E, a subgroup or shifted subgroup of rank 2, such that the restriction of ζ to H^{∗}(E, k) is not zero. Then the restriction of a module of the form Ω^{m}(L_{ζ^{t}}) are the not projective and are isomorphic in the stable category, i. e. modulo projective summands, to L_{res_{G,E}(ζ^{t})}. As an aside, we note that if E is only a shifted subgroup then the tensor product operation on kG-modules may not commute with restriction to kE-module. However, this does not matter as we are only comparing dimensions in the display above.

We are assuming that G = E an elementary abelian p-group of order p^{2}. Then an easy calculation show that for d = 2c, Dim(Ω^{2c}(k)) = cp^{2}+1. Hence, the dimension of L_{ζ^{n}} is ncp^{2}. Note that because ζ is not nilpotent L_{ζ^{n}} is a periodic module of period 2 and Dim(L_{ζ^{n}}/ Rad(L_{ζ^{n}}) = Dim(H^{nd}(E, k)) − 1 = nc. Hence, Dim(Ω(L_{ζ^{n}})) = ncp^{2} = Dim(Ω^{(n+1)d}(L_{ζ^{n}})). while, Dim(Ω(L_{ζ^{n+1}})) = (n + 1)cp^{2} = Dim(Ω^{nd}(L_{ζ^{n+1}})). The proposition now follows by comparing the dimensions of the modules in the expression for L_{ζ^{n}} ⊗ L_{ζ^{n+1}}. □

Theorem 3.4. Suppose that ζ ∈ H^{d}(G, k) is a nonzero cohomology element with the property that ζ^{n} annihilates the cohomology of L_{ζ^{n}} for all n > 0. Let M be a kG-module with V_{G}(M) ⊆ V_{G}(ζ). Let m be the least positive integer such that ζ^{m} annihilates the cohomology of M. Let n be an integer with 1 ≤ n ≤ m. Then ζ^{n−1} does not annihilate the cohomology of L_{ζ^{n}} ⊗ M.

Proof. In the stable category stmod(kG), there is a commutative diagram

\xymatrix{… \ar[r] & Ω^{nd}(M) \ar[r]^{ζ^{n}} \ar[d]^{ζ^{n−1}} & M \ar[r] \ar[d]^{ζ^{n−1}} & L_{ζ^{n}} ⊗ M \ar[r] \ar[d]^{ζ^{n−1}} & … \\ … \ar[r] & Ω^{d}(M) \ar[r]^{ζ^{n}} & Ω^{(1−n)d}(M) \ar[r]^{γ} & Ω^{(1−n)d}(L_{ζ^{n}} ⊗ M) \ar[r] & …}

Here, the maps marked ζ^{n} should be read as multiplication by ζ^{n} or the class in the stable category of the map ζ^{n} ⊗ 1_{U} : Ω^{nd}(k) ⊗ U → U for U as appropriate. The two rows in the diagram are triangles in the stable category.

Assume that ζ^{n−1} annihilates the cohomology of L_{ζ^{n}} ⊗ M. Then the right hand down arrow is the zero map. This implies that γζ^{n−1} : M → Ω^{(1−n)d}(L_{ζ^{n}} ⊗ M) is the zero map. So there exists a map θ : M → Ω^{d}(M) such that ζ^{n}θ = ζ^{n−1} : M → Ω^{(1−n)d}(M). This leads to a contradiction because, then, ζ^{m}θ = ζ^{m−1} is the zero map on M, and ζ^{m−1} annihilates the cohomology of M, thus contradicting the hypothesis. This completes the proof. □

One further result on the nature of L_{ζ} module is required later in the paper. We begin with a couple of helpful lemmas.

Lemma 3.5. Suppose that ζ ∈ H^{d}(G, k) be a nonzero element. If V_{G}(ζ) = W_{1} ∪ ⋯ ∪ W_{n}, where W_{1}, …, W_{n} are mutually-disjoint connected closed sets in V_{G}(k), then the module L_{ζ} decomposes as a direct sum L_{ζ} ≅ L_{1} ⊕ ⋯ ⊕ L_{n} where each L_{i} is an indecomposable module with V_{G}(L_{i}) = W_{i}.

Proof. See Lemma 4.1 of [10]. □

Lemma 3.6. Suppose that G is a p-group, and that E and F are two elementary abelian subgroups of G, both having rank at least three. Then there exist elementary abelian subgroup E_{1} and E_{2}, with rank at least three, such that the intersections E ∩ E_{1}, E_{1} ∩ E_{2} and E_{2} ∩ F all have rank at least two. As a consequence, if W ⊆ V_{G}(k) is the union of the closed set res_{G,E}^{∗}(V_{E}(k)) for E elementary abelian with rank at least three, and ζ ∈ H^{d}(G, k) for some d > 0, then W ∩ V_{G}(ζ) is connected.

Proof. The first statement is Alperin’s trick in the proof of Theorem 4 in [1]. The second is a consequence of the first and the fact that if E has rank two then res_{G,E}^{∗}(V_{E}(k)) ∩ V_{G}(ζ) can not be empty. □

Theorem 3.7. Suppose that ζ ∈ H^{d}(G, k), and assume that d > 1 if p = 2. Let H be a collection of p-subgroups of G none of which is a Sylow p-subgroup. If G has p-rank at least 3, then L_{ζ} is not H-projective.

Proof. First note that it is sufficient to prove the proposition in the case that G is a p-group since we may assume that all of the elements of H lie in a chosen Sylow p-subgroup S. Also if L_{ζ} is H-projective, then so also is its restriction to S. By Quillen’s Theorem, the variety V_{G}(k) is the union of the images res_{G,E}^{∗}(V_{E}(k)) where E runs through a set of representatives of the maximal elementary abelian p-subgroups of G. These images are the components of the variety. At least one of the maximal elementary abelian subgroups (call it F) has rank at least 3 and hence one of the components res_{G,F}^{∗} (V_{F} (k)) has dimension at least two. Its intersection with V_{G}(ζ) = V_{G}(L_{ζ}) has dimension at least one and must be connected.

By Lemma 3.5, if V_{G}(L_{ζ}) = W_{1} ∪ ⋯ ∪ W_{r} is a decomposition into disjoint closed sets, then L_{ζ} = L_{1}⊕⋯⊕L_{r} where V_{G}(L_{i}) = W_{i}. Consequently, L_{ζ} has a direct summand, say L_{1}, whose variety W_{1} contains res_{G,F}^{∗} (V_{F} (k)) ∩ V_{G}(L_{ζ}) and has dimension at least one.

If L_{ζ} is H-projective, then by transitivity, we may assume that L_{1} is a direct summand of N^{↑G} for N an indecomposable kH-module where H is a maximal subgroup of G. By Green’s Indecomposability Theorem (Theorem 8 of [18]), N^{↑G} is indecomposable and hence it is equal to L_{1}. Because, the varieties of N and N^{↑G} have the same dimension, the variety of the kH-module N must have dimension at least one. Because H is maximal, it is normal and for some g ∉ H, (L_{1})_{↓H} = ∑_{i=0}^{p−1} g ⊗ N is a direct sum of G-conjugates of N. On the other hand, (L_{1})_{↓H} is a direct summand of

(L_{ζ})_{↓H} ≅ L_{res_{G,H}(ζ)} ⊕ Q

where Q is projective. By some easy analysis we see that Q must be zero.

There are two possibilities. The first is that res_{G,H}(ζ) = 0, in which case (L_{ζ})_{↓H} ≅ Ω(k_{H}) ⊕ Ω^{d}(k_{H}). If p = 2, then because d > 1, and H has p-rank at least 2, Ω^{d}(k_{H}) is not a G-conjugate of Ω(k_{H}). If p > 2, then (N^{↑G})_{↓H} has at least p indecomposable summands. Hence, regardless of the characteristic, this case is not possible.

For the second case, we assume that res_{G,H}(ζ) ≠ 0. We claim that, in such circumstances, the subgroup H must have rank at least three. The reason is that if H has p-rank two then the components of V_{H}(k) have dimension one, and the closed subvarieties V_{H}(g^{i} ⊗ N), having dimension also equal to one, must each contain one of these components. However, these components all have a common intersection which is res_{H,Z}(V_{Z}(k)) where Z is the unique elementary abelian subgroup in the center of G. Hence the closed subvarieties V_{H}(g^{i} ⊗ N) are not disjoint if H has p-rank two. The contradiction shows that H has p-rank at least three.

Now note that the elementary abelian subgroup F which was chosen earlier can be assumed to be in H. Hence, for some i

res_{H,F}^{∗} (V_{F} (k)) ∩ V_{H}((L_{1})_{↓H}) ⊆ V_{H}(g^{i} ⊗ N),

and we can assume that i = 0. But then, for any j,

V_{H}(g^{j} ⊗ N) = V_{H}(N)^{g^{j}}

contains res_{G,E}^{∗}(V_{E}) ∩ V_{H}((L_{1})_{↓H}) for E = F^{g_{j}}. But by Lemma 3.6, these varieties are all connected. Hence, this situation is also impossible. □

Theorem 3.8. Suppose that ζ ∈ H^{d}(G, k), and assume that d > 1 if p = 2. Let H be a collection of p-subgroups of G none of which is a Sylow p-subgroup. If the center of a Sylow p-subgroup of G has p-rank at least 2, then L_{ζ} is not H-projective.

Proof. The proof follow by a very similar argument as that of Theorem 3.7. If G has p-rank 3 then the proof follows from that theorem. As in that proof, we may assume that G is a p-group. Let Z be the center of G. It is assumed that Z ≅ (ℤ/pℤ)^{2} and is the unique maximal elementary abelian p-subgroup of G. A consequence of this fact and the fact that Z is central is that the restriction map res_{G,E}^{∗} is an isomorphism from V_{E}(k) and V_{G}(k). Suppose that L is any direct summand of L_{ζ}. Then either V_{G}(L) = V_{G}(k), or V_{G}(L) is a closed point in V_{G}(k) ≅ ℙ^{1} not shared by the variety of any other direct summand of L_{ζ}. In neither case is it possible for there to exist a maximal subgroup H such that L is a direct summand of N^{↑H} for some kHmodule N. First, by Green’s Indecomposability Theorem, N^{↑H} is indecomposable and hence equal to L. But additionally, the restriction of N^{↑H} to E is a direct sum of p modules all having the same variety. Because the indecomposable summands of (L_{ζ})_{↓E} = L_{res_{G,E}(ζ)} have disjoint varieties, this is not possible. □

4. Additive tensor ideals

In this section we demonstrate a method of constructing thick subcategories of stmod_{H}(kG) from additive tensor ideals in stmod(kH) for any subgroup H in the collection H. From this construction we can get infinite chains of thick subcategories in stmod_{H}(kG), all consisting of objects with the same ordinary support variety. The indication is that the spectrum of the category stmod_{H}(kG) is not noetherian, and this is proved for many cases in Section 5.

The main definition is the following.

Definition 4.1. Let G be a finite group. An additive tensor ideal in stmod(kG) is a full subcategory that it closed under finite directs sums, direct summands and tensor product by arbitrary objects in stmod(kG). By the last statement we mean that if D is a additive tensor ideal and if M ∈ D, N any kG-module, then M ⊗ N is in D. Such a subcategory D is called an ATI for short.

Notice that we are not requiring that an ATI be a triangulated subcategory, though any triangulated thick tensor ideal is very naturally an ATI. Also it is clear that an ATI is invariant under the translation functor Ω^{−1} because this functor is equivalent to tensoring with Ω^{−1}(k).

The next two examples have some importance in other discussions in this paper.

Proposition 4.2. Assume that the field k is algebraically closed. Let D be the full subcategory of all modules that are direct sums of indecomposable module having dimension divisible by p. Then D is an additive tensor ideal.

Proof. By [5], an absolutely indecomposable module M has the property that there exists a module N such that k is a direct summand of M ⊗ N, if and only if the dimension of M is not divisible by p. This implies that if the dimension of either one of two indecomposable modules M and N is divisible by p, then the dimension of any direct summand of M ⊗ N is divisible by p. Otherwise, M ⊗ N ⊗ L would have a direct summand isomorphic to k for some module L. Thus D is an ATI as asserted. Note that the assumption that k is algebraically closed is important for invoking the results of [5]. □

For the second example, suppose that W is a closed subset of V_{G}(k), and suppose that I is an ideal in H^{∗}(G, k) such that W ⊆ V_{G}(I), the closed set of homogeneous prime ideals that contain I. Recall that if M is a kG-module with V_{G}(M) ⊆ W, then I^{n} annihilates the cohomology of M for some n, sufficiently large. Then we have the following.

Proposition 4.3. Let D = D_{I} be the full subcategory of stmod(kG) consisting of all modules M such that I annihilates the cohomology of M. Then D_{I} is an ATI.

Proof. It is clear that D is closed under direct sums and summands. Also if N is any kG-module, then I also annihilates the cohomology of M ⊗ N. □

Now let D_{1} = D_{I}. defined as above, and let D_{n} = D_{I^{n}}, the full subcategory of all kG-module M with V_{G}(M) ⊆ W and such that I^{n} annihilates the cohomology of M. Then we have a nested collection of ATIs

D_{1} ⊆ D_{2} ⊆ D_{3} ⊆ ⋯ ⊆ stmod(kG)_{W} .

There are certainly cases in which we know that these are all distinct. There has been some work on proving the distinction in general.

The point of the definition is the following.

Theorem 4.4. Suppose that H is a subgroup of some H′ ∈ H and that D is an ATI in stmod(kH). Let C = C(D) be the full subcategory of stmod_{H}(kG) consisting of all kG-modules M with the property that M_{↓H} is in D. Then C is a thick tensor ideal subcategory of stmod_{H}(kG).

Proof. The tensor property follows directly from the definition of an ATI. That is, if M is in C and N is any kG-module, then (M ⊗ N)_{↓H} ≅ M_{↓H} ⊗ N_{↓H} is in D. So M ⊗ N is in C. Clearly, direct sums and summands of elements of C are in C. The subcategory C is invariant under the translation functor since this functor is equivalent to tensoring with Ω_{H}^{−1}. Consequently, the only question is triangles.

Suppose that L → M → N → Ω_{H}^{−1}(L) is a triangle in stmod_{H}(kG) and that two of the three objects, say, L and N, are in C. Then there is an H-split sequence

\xymatrix{0 \ar[r] & L \ar[r] & M ⊕ Q \ar[r] & N \ar[r] & 0.}

for some H-projective module Q. Because the sequence is H-split, it also splits on restriction to the subgroup H, and the middle term has the form M_{↓H} ≅ L_{↓H} ⊕ N_{↓H} which is in D. Therefore M is in C and we have that M is a thick tensor ideal. □

From Proposition 4.3 we immediately have the following.

Theorem 4.5. Suppose that H is a subgroup of some H′ in H. Let I be an ideal in H^{∗}(H, k), and let 𝔛 = 𝔛_{I} be full subcategory of all kG-modules M with the property that the cohomology of the restriction of M to H is annihilated by I. Then 𝔛 is a thick tensor ideal in stmod_{H}(kG).

Following the ideas of Balmer [2] we define primes.

Definition 4.6. An additive tensor ideal D in stmod(kH) is said to be prime if for any two kH-modules M and N, the tensor product M ⊗ N is in D, then one of the two modules M or N is in D.

With the construction in Theorem 4.4 we can show the following.

Theorem 4.7. Suppose that D is an additive tensor ideal in stmod(kH) for some subgroup H which is contained in some element of H. If in addition, D is prime then the thick tensor ideal C = C(D), defined as in Theorem 4.4, is a prime thick tensor ideal in stmod_{H}(kG).

Proof. Suppose that M and N are two kG-modules neither of which is in C. Then each of M_{↓H} and N_{↓H} has a direct summand that is not in D. The tensor product of these two direct summands is a direct summand of (M ⊗ N)_{↓H} that is not in D. Consequently, M ⊗ N is not in C. □

Proposition 4.2 leads to the following.

Theorem 4.8. Suppose that H is a subgroup of some H′ in H. Let 𝔜 = 𝔜_{H} be the full subcategory of all kG-modules M such that every indecomposable summand of M_{↓H} has dimension divisible by p. Then 𝔜 is a prime thick tensor ideal in stmod_{H}(kG).

Proof. That 𝔜 is a thick tensor ideal follows from Proposition 4.2 and Theorem 4.4. It is prime because the dimension of the tensor product of modules whose dimension is not divisible by p, is not divisible by p. □

Remark 4.9. It is not clear when the thick ideal defined in Theorem 4.5 are primes. .

5. Non-noetherian spectra

In this section we show that if p = 2 and H is a collection of subgroup of G that contains a subgroup H ⊆ G having 2-rank at least two, then the spectrum of the relative stable category stmod_{H}(kG) is not noetherian. The proof that we give is constructible in the sense that we show explicitly how to find an infinite ascending chain of prime thick subcategories if we assume the hypothesis. The proof depends heavily on the results in the last two section.

Throughout the section, let H be a collection of p-subgroups of the finite group G. As usual it is assumed that no element of H is a Sylow p-subgroup of G.

Theorem 5.1. Let G denote a finite group such that either the 2-rank of G is at least three or the center of a Sylow 2-subgroup of G has rank two. Suppose that p = 2 and that the collection H contains a subgroup of 2-rank two or more. Then the spectrum of the relative stable category stmod_{H}(kG) is not noetherian.

Proof. Suppose that E ⊂ H is an elementary abelian subgroup of order 4 for H some element of H. Our objective is to show that there is an infinite increasing chain of prime additive tensor ideals in stmod(kE). Then Theorem 4.7 gives a nondecreasing chain of prime tensor ideal thick subcategories of stmod_{H}(kG). It then remains only to show that that an infinite subset of the elements of the chain are distinct.

We begin with a quick review of some of what is known about kE-modules and their cohomology. Recall that the field k is algebraically closed. All of the modules for kE are known and their tensor products are also known [17]. The indecomposable modules come in three types. First there is the free indecomposable module kE. Second comes the modules of odd dimension. These are the modules Ω^{n}(k) for n ∈ ℤ. If M is an indecomposable kE-module with V_{G}(M) = V_{G}(k), then M ≅ Ω^{n}(k) for some n and the dimension of n is odd (not divisible by two). The third class consists of periodic indecomposable modules. Every one is isomorphic to L_{ζ^{n}} for some ζ ∈ H^{1}(G, k) and some n > 0. Recall that H^{∗}(G, k) ≅ k[ζ_{1}, ζ_{2}], the polynomial ring generated by ζ_{1} and ζ_{2}, both having degree one. Then any element of H^{d}(E, k) is a product of d homogeneous polynomials of degree one. If ζ ∈ H^{d}(E, k) and ζ = γ_{1}^{a_{1}} ⋯ γ_{t}^{a_{t}} where each γ_{i} ∈ H^{1}(E, k), γ_{i} ≠ γ_{j} for i ≠ j and a_{1} + … + a_{t} = d, then L_{ζ} ≅ ⊕_{j=1}^{t}L_{γ_{j}^{a_{j}}}.

It can be seen from 3.1 and the tensor product formulas in [17] that for ζ ∈ H^{1}(E, k)), ζ^{n} annihilates the cohomology of L_{ζ^{n}} whenever n > 1. Let ζ be any nonzero element in H^{1}(E, k) of degree 1. For i > 1, let S_{i} = S_{i,ζ} be the collection of all indecomposable periodic modules except those isomorphic to L_{ζ^{j}} for j > i. This collection is precisely the class of all finitely generated indecomposable kEmodule M such that either V_{E}(M) does not contain V_{E}(ζ) or that ζ^{i} annihilates the cohomology of M. Let C_{i} be the full additive subcategory generated by S_{i}. That is, an object in C_{i} is a direct sums of copies of modules in S_{i}. We claim that C_{i} is a prime additive tensor ideal in stmod(kE).

The closure of C_{i} under direct sums and summands is clear from the definition. The tensor property follows from the tensor product formula on varieties (V_{E}(M ⊗ N) = V_{E}(M) ⊕ V_{E}(N) for modules M and N) and the fact that if γ ∈ H^{m}(E, k) annihilates the cohomology of a module M, then it annihilates the cohomology of M ⊗ N for any kE-module M. If M is a kE-module that it not in C_{i} then either M has a direct summand isomorphic to Ω^{n}(k) for some n or isomorphic to L_{ζ^{j}} for j > i. The tensor product of two such module is also not in C_{i}. This can be seen from the multiplication table in [17] or using Theorem 3.4. Thus C_{i} is a prime ATI.

We have that there is a chain C_{2} ⊆ C_{3} ⊆ … of prime ATI’s. For each i, let B_{i} be the full subcategory of stmod_{H}(kG) consisting of all modules M whose restriction, M_{↓E} to E is in C_{i}. Then by Theorem 4.7, there is a nondecreasing sequence B_{2} ⊆ B_{3} ⊆ … of prime thick tensor ideals in stmod_{H}(kG). It remains only to show that an infinite number of the elements of this chain are distinct. This proves that there is an infinite ascending chain of primes and that the spectrum is not noetherian.

For any nonzero element γ ∈ H^{d}(G, k), the restriction has the property that res_{G,E}(γ) = ∏_{i=1}^{d} φ_{i} where each φ_{i} is a linear combination of ζ_{1} and ζ_{2}. We claim that for some degree d, there is an element γ ∈ H^{d}(G, K) with a decomposition as a product of elements of degree 1 as above, such that at least one of the elements φ_{i} is equal to ζ. The reason is that for E = ⟨g_{1}, g_{2}⟩, there is a shifted cyclic subgroup ⟨u⟩ for u = 1 + a(g_{1} − 1) + b(g_{2} − 1) ∈ kE, for some a, b ∈ k, such that ζ and its scalar multiples are the only elements in H^{1}(G, k) that vanish on restriction to ⟨u⟩. That is, the inclusion of k⟨u⟩ into kE is a flat embedding and the restriction res_{E,⟨u⟩} : H^{∗}(E, k) → H^{∗}(⟨u⟩, k) is a homomorphism from a polynomial ring in two variable to a polynomial ring in one variable. The kernel is the ideal generated by ζ. Because the ring H^{∗}(E, k) is finitely generated as a module over the restriction to E of H^{∗}(G, k), it is necessary that some nonzero element in the image of that restriction be in the kernel of the restriction to k⟨u⟩.

At this point we have shown that there is an element γ ∈ H^{d}(G, k) and an integer t ≥ 1 such that res_{G,E}(γ) = ζ^{t}θ ≠ 0 where θ ∈ H^{d−t}(E, k) is not in the ideal generated by ζ. Hence, for any j > 0, we have that the restriction of L_{γ^{j}} has the form

(L_{γ^{j}})_{↓E} ≅ L_{ζ^{tj}} ⊕ L_{θ^{j}} ⊕ (proj) .

The proof is complete as we observe that this module is in C_{tj} but not in C_{tj+ℓ} for any ℓ > 0, and, in addition it is not H-projective by Theorem 3.7 or Theorem 3.8. □

Remark 5.2. It is natural to expect that the theorem also holds in the case that G has 2-rank two and the center of a Sylow 2-subgroup fails to have rank two. The problem is showing that there is an appropriate L_{ζ} module that is not H-projective. Certainly, an analog of Theorem 3.8 should not be expected. But as seen in the case of the dihedral group in Section 9 it might be possible to find some desired L_{ζ} modules that are not H-projective, even though others are relatively projective.

A natural question to ask is what happens if p > 2 and H has a subgroup of p-rank at least two. A similar proof might work. If E is an elementary abelian subgroup of order p^{2}, then the cohomology ring of E has the form H^{∗}(E, k) ≅ k[ζ_{1}, ζ_{2}]⊗Λ(η_{1}, η_{2}). Here Λ is the exterior algebra generate by the two degree one elements and for i = 1, 2, ζ_{i} is the Bockstein of η_{i} and has degree 2. The element ζ should be chosen in degree 2, to be a linear combination of ζ_{1} and ζ_{2}. All of the arguments of the above proof go through with only minor changes, except one! That argument is that we have no proof that the additive tensor ideal C_{i} is prime. This is the additive subcategory generated by all indecomposable module whose varieties don’t contain V_{G}(ζ), and all indecomposable modules modules M with V_{G}(M) = V_{G}(ζ) and with cohomology annihilated by ζ^{i}. It is certainly true here that ζ^{i} annihilates the cohomology of L_{ζ^{i}}. In light of Theorem 3.4, C_{i} would be prime provided the following had an affirmative answer.

Question 5.3. Suppose that M and N are kE-modules such that V_{G}(M) = V_{G}(N) = V_{G}(ζ) for some nonzero ζ chosen as above. Suppose that for some i > 0, ζ^{i} does not annihilate the cohomology of either M or N. Is it the case that ζ^{i} does not annihilate the cohomology of M ⊗ N.

Or we might ask a weaker question.

Question 5.4. In above notation, does there always exist a prime additive tensor ideal D_{i} that contains C_{i} but not L_{ζ^{i+1}}.

An affirmative answer here would also fill in the missing step in a proof of Theorem 5.1 for odd primes.

6. Idempotent modules

In this section we consider the role that idempotent modules play in the structure of the relative stable category. One simple principle is the following. As before assume that H is a collection of p-subgroups of G. The proof is left to the reader.

Lemma 6.1. Suppose that X is any kG-module, even perhaps infinitely generated. The full subcategory of all finitely generated kG-modules M such that M ⊗ X is relatively H-projective is a thick tensor ideal in stmod_{H}(kG).

But more than that is true. For every thick tensor ideal, there exist such a module. For notation let StMod_{H}(kG) be the relative stable category of all kGmodule, not just finitely generated. Let stmod_{H}(kG)^{⊕} be the smallest triangulated full subcategory of StMod_{H}(kG) that contains stmod_{H}(kG) and is closed under arbitrary coproducts (direct sums). The following was proved by Rickard for the ordinary stable category [23]. In glorious generality, much more than we need here, it is proved in [3].

Proposition 6.2. Suppose that C is a tensor ideal thick subcategory of stmod_{H}(kG). There exist kG-modules E = E^{C} and F = F^{C} in stmod_{H}(kG)^{⊕} such that there is a triangle (called an idempotent triangle)

[figure: Δ_{C} : E; γ; k; λ; F]

in stmod_{H}(kG)^{⊕} having the following properties.

(1) γ ⊗ 1 : E ⊗ E → E is an isomorphism.

(2) E ⊗ F = {0}.

(3) 1 ⊗ λ : F → F ⊗ F is an isomorphism.

(4) For any finitely generated kG-module M, M ⊗ E ∈ C^{⊕}. The map 1 ⊗ λ : M → M ⊗ F is C-local, meaning that if φ : X → M is a map with X in C then (1 ⊗ λ) ∘ φ = 0.

(5) For any object M in stmod_{H}(kG), M ⊗ F ≅ {0} if and only if M is in C.

The modules E and F are called idempotent modules because of their behavior in properties (1) and (3). Condition (2) says that E ⊗ F is an H-projective module. We note that the category stmod(kG) has no nonzero idempotent modules other than the trivial module. If J is a collection of p-subgroups of G, such that for every J ∈ J there is an H in H with J ⊆ H, then any relatively J -projective module is also relatively H-projective. Consequently if C is a thick tensor ideal in stmod_{J} (kG), then the associated idempotent module F^{J} defines a thick tensor ideal D in stmod_{H}(kG). That it, D consist of all finitely generated module whose tensor with F^{J} is relatively H projective. In general, D will contain modules that are not in C. We give two possibilities from the standpoint of modules defined by varieties.

First there are some cases where the idempotent modules for the ordinary stable category and the relative stable category coincide.

Proposition 6.3. Let V_{H} = ∪_{H∈H} res_{G,H}^{∗}(V_{H}(k)), and let V_{H} be the set of all closed subsets of V_{H}. Let W be a collection of closed subsets of V_{G}(k) that is closed under finite unions and specializations. Assume that V_{H} ∩W = {∅}. Then the subcategories stmod_{H}(kG)_{W} and stmod(kG)_{W} are naturally equivalent, meaning that they have exactly the same objects, morphisms and triangles. Moreover F_{W} = F_{W}^{H}, where these are the idempotent F-modules corresponding to the subcategories stmod(kG)_{W} of stmod(kG) and stmod_{H}(kG)_{W} of stmod_{H}(kG), respectively. Also, the subcategory stmod_{H}(kG) consisting of all finitely generated module whose tensor with F_{W} is zero coincides with stmod_{H}(kG)_{W}.

Proof. If M is a finitely generated module with V_{G}(M) ∈ W, then V_{G}(M) ∩ V_{H} = ∅. Thus, the restriction of M to H is a projective modules for every H in H. If N is another such module and φ : M → N is a homomorphism that factors through an H-projective module, then φ factors through a projective module. Hence Hom_{stmod_{H}(kG)}(M, N) = Hom_{stmod(kG)}(M, N). Likewise, a triangle in stmod(kG)_{W} L → M → N → Ω^{−1}(L) must come from an exact sequence

\xymatrix{0 \ar[r] & L \ar[r] & M \ar[r] & N \ar[r] & 0}

which is H-split, because, when restricted to any H in H, N_{↓H} is projective. Hence the triangle is also a triangle in stmod_{H}(kG)_{V}. This proves the first statement.

The second statement is a consequence of the standard construction of the triangle of idempotent modules for the subcategory. That is both triangles are constructed in exactly the same way (see [23]). The third statement follows from the second. □

The state of the idempotents and the subcategories that they determine in other situations seems more confused. We offer the following as an example. For notation, for a kG-module X, perhaps infinitely generate, let U^{H}(X) be the thick tensor ideal subcategory of stmod_{H}(kG) consisting of all modules M such that X ⊗ M is Hprojective.

Proposition 6.4. Let W be a collection of closed subsets of a fixed closed subset W ⊆ V_{G}(k). Suppose that H ∈ H is a subgroup such that W ∩ V_{H} ≠ V_{H}, where V_{H} = res_{G,H}^{∗}(V_{H}(k)). Let M be an indecomposable kG-module with the properties that M is not H-projective and that M is an extension having either the form

\xymatrix{0 \ar[r] & L \ar[r] & M \ar[r] & N^{↑G} \ar[r] & 0,}  (6.5)

or the form

\xymatrix{0 \ar[r] & N^{↑G} \ar[r] & M \ar[r] & L \ar[r] & 0,}  (6.6)

where L is a kG-module such that V_{G}(L) ⊆ W and N is a kH-module such that res_{G,H}^{∗}(V_{H}(N)) ∉ W. Then, M is in U^{H}(F_{W}) but is not in U^{H}(F_{W}^{H}).

Proof. Note that neither of the displayed extensions can be H-split since one of the end terms is H-projective and M is indecomposable. We know that M is not in U^{H}(F_{W}^{H}) because its variety is not contained in W. On the other hand if we tensor the sequence 6.5 with F_{W}, we get an exact sequence

\xymatrix{0 \ar[r] & L ⊗ F_{W} \ar[r] & M ⊗ F_{W} \ar[r] & N^{↑G} ⊗ F_{W} \ar[r] & 0.}

Now notice that the first term L ⊗ F_{W} is projective, by the varieties condition. Hence, this term is projective and injective and the above sequence splits. At the other end, N^{↑G} ⊗ F_{W} ≅ (N ⊗ (F_{W})_{H})^{↑G}, by Frobenius reciprocity. Therefore this module is H-projective and M is in U^{H}(F_{W}) as asserted. □

We present one example that demonstrates the conditions in the proposition are not vacuous. Indeed, such examples are ubiquitous.

Example 6.7. Let G = ⟨x, y, z⟩ be an elementary abelian group of order 8. Write X = x − 1, Y = y − 1 and Z = z − 1 so that kG is the truncated polynomial ring kG = k[X, Y, Z]/(X^{2}, Y^{2}, Z^{2}). Let J = ⟨x⟩ and W = res_{G,J}^{∗}(V_{J}(k)). Assume that H = {H} where H = ⟨x, y⟩. Thus the variety of a kG-module M contains W if and only if the restriction of M to J is not projective. Let F be the free kG-module on the symbols a, b, c, and let M = F/I where I is the ideal generated by

Xa, Xb − ZY a, Y c, Xc − ZY b

It can be checked that M has a k-basis consisting of the classes modulo I of the elements in the set {a, Y a, Za, Y Za, b, Y b, Zb, Y Zb, c, Zc}, so its dimension is 10. It can be checked that it is indecomposable. It is not H-projective, because it is not a direct summand of (M_{↓H})^{G}, which can be seen by noting that M_{↓H} has a direct summand of dimension 2 spanned by the classes of a and Y a.

Let L denote the submodule of M, generated by the classes modulo I of a and b. Then L is an extension

\xymatrix{0 \ar[r] & L′ \ar[r] & L \ar[r] & L′′ \ar[r] & 0}

where L′ is generated by the class of a. Both L′ and L′′ are isomorphic to k_{J}^{↑G}. However, as before, it can be checked that L is indecomposable and not H-projective. Moreover, its variety is precisely {W }. On the other hand, M/L ≅ k_{H}^{↑G}, which can be seen by the fact that H acts trivially on this 2-dimensional module.

From all of this we see that M is an extension of the form described in 6.5. The dual of M is an extension of the form 6.6.

7. Thick subcategories by inflation

In this section we show how thick subcategories of the relative stable category can be defined by inflating idempotent modules from quotient groups. As before let H be a collection of p-subgroups of G.

The basic principle is the following.

Theorem 7.1. Suppose that N is a normal p-subgroup of G and that N ⊆ H for some H ∈ H. Let J = {HN/N| H ∈ H}. Suppose that V is a collection of subvarieties of V_{G/N}(k) that is closed under finite unions and specializations. Let 𝔘_{V} be the thick tensor ideal subcategory of stmod_{H}(kG) generated by the inflations to kG of all k(G/N)-modules M with V_{G/N}(M) ∈ V. If W is another such collection and if stmod_{J} (k(G/N))_{W} is not contained in stmod_{J} (k(G/N))_{V}, then 𝔘_{W} is not contained in 𝔘_{V}.

Proof. Assume that E_{V}^{H} → k → F_{V}^{H} → Ω_{H}^{−1}(E_{V}^{H}) is the idempotent triangle in StMod_{J} (k(G/N)) corresponding to the thick subcategory stmod_{J} (k(G/N))_{V}. Let F_{V}^{H} denote also the inflation to a kG-module. By hypothesis, there is a module M in stmod_{J} (k(G/N))_{W} that is not also contained in stmod_{J} (k(G/N))_{V}. Consequently, M ⊗ F_{V}^{H} is not J -projective, and its inflation to a kG-module (which we also call M) has the property that M ⊗ F_{V}^{H} is not in U_{(F_{V})}^{H}, the thick tensor ideal of modules whose tensor with F_{V} is H-projective. For this we should observe that an inflated module that is H-projective is also J -projective. It follows that M is not in 𝔘_{V}, though it is clearly in 𝔘_{W}. □

Remark 7.2. While it is tempting to try to replace the condition “stmod_{J} (k(G/N))_{W} is not contained in stmod_{J} (k(G/N))_{V}” by the condition “W is not in V”, the resulting statement would not in general be correct. The reason is that stmod_{J} (k(G/N))_{W} might fail to contain any nonzero objects even though W is not empty and not in V. See Section 8 for an explanation as to how such a thing could happen.

Remark 7.3. As in the above theorem, suppose that N is a normal p-subgroup of G and that N ⊆ H for some H ∈ H. Let J = {HN/N| H ∈ H}. The inflation map induces a functor from stmod_{J} (k(G/N)) to stmod_{H}(kG) that is full and faithful and preserves triangles. The image of a thick subcategory C of stmod_{J} (k(G/N)) is a thick subcategory. However the image C under the inflation map is not, in general, a thick tensor ideal even when C is a thick tensor ideal.

The next example shows again that there are tensor ideal thick subcategories as defined above that do not come from the spectrum of the cohomology of G.

Example 7.4. Suppose that p = 2 and the G = ⟨g, h|g^{2} = h^{2} = (gh)^{4}⟩ is a dihedral group of order 8. Let N = ⟨(gh)^{2}⟩ be the center of G and let H = {N}. Then G/N is an elementary abelian group of order 4, V_{G/N}(k) ≅ ℙ^{1}, a projective line, and H^{∗}(G/N, k) ≅ k[ζ_{1}, ζ_{2}], where res_{G/N,⟨h⟩}(ζ_{1}) = 0 and res_{G/N,⟨g⟩}(ζ_{2}) = 0. If V is a point on the line other than [0, 1] or [1, 0] (where [a, b] ∈ ℙ^{1} corresponds to the prime ideal generated by bζ_{1} − aζ_{2}), then the subcategory 𝔘_{V} (V = {V }) is a proper subcategory of stmod_{H}(kG)_{W} for W = {res_{G,N}^{∗}(V_{N}(k))}.

8. Empty thick tensor ideals

The purpose of this section is to remind the reader that it is possible to have thick tensor ideals which are defined by nonempty closed sets of the spectrum V_{G}(k) and which have no nonzero objects. The point is that for some groups G and collections of p-subgroups H, there exist closed subsets V in V_{G}(k) having the property that every kG-module M with V_{G}(M) ⊆ V is H-projective.

An example is contained in the following theorem due to Dave Benson [4]. For reference, we recall the theorem of Quillen that V_{G}(k) = ∪_{E⊆G} res_{G,E}^{∗}(V_{E}(k)), where E runs through the elementary abelian p-subgroups of G. This says that if ℓ is a point in V_{G}(k), then ℓ is in the image of some map on varieties V_{E}(k) → V_{G}(k) induced by restriction to an some elementary abelian p-subgroup E of G. It is equivalent to saying that every homogeneous prime ideal of H^{∗}(G, k) contains the kernel of the restriction map H^{∗}(G, k) → H^{∗}(E, k) to some elementary abelian p-subgroup of G.

Theorem 8.1. [4] Suppose that ℓ is a point in V_{G}(k). Let E be an elementary abelian p-subgroup of G, that is minimal with respect to the property that ℓ is contained in the image of the restriction map res_{G,E}^{∗} : V_{E}(k) → V_{G}(k). Let ℓ̂ be a point in V_{E}(k) such that res_{G,E}^{∗}(ℓ̂) = ℓ. Let D denote the subgroup of N_{G}(E) of all elements that stabilize ℓ̂. If M is a kG-module with V_{G}(M) = {ℓ}, then M is stably induced from D, meaning that the direct sum of M with some projective module is an induced module from D.

This leads to the following theorem concerning thick subcategories.

Corollary 8.2. Suppose that ℓ, E and D are as in the theorem. Let V = {ℓ} be the collection of closed sets whose only nonempty member is the closed point ℓ. Suppose that a Sylow p-subgroup of D is conjugate to a subgroup of H for some H in the collection H. Then every object in the thick tensor ideal stmod_{H}(kG)_{V} is isomorphict to the zero module.

Proof. By the theorem, if M is a nonprojective object in stmod_{H}(kG)_{V}, then V_{G}(M) = ℓ and there exists a kD-module L such that L^{↑G} ≅ M ⊕ P for P some projective module. Thus, M is relatively kD-projective and hence relatively kSprojective for S a Sylow p-subgroup of D. By hypothesis, S is conjugate to a subgroup of some H ∈ H and hence M is relatively kH-projective and M ≅ {0} in the relative stable category. □

We should emphasize here that there is an analogue of Theorem 8.1 for connected closed sets ℓ of dimension greater than zero [12]. The principle is still the same, meaning, basically, that ℓ = res_{G,E}(ℓ̂), where ℓ̂ ∈ V_{E}(k) has the property that its conjugates under the action of N_{G}(E)/D are disjoint (D being the stabilizer as in the Theorem). Then the conclusion is that if D is conjugate to a subgroup of H for some H in H, then stmod_{H}(kG)_{V} has only zero objects. In addition, as occurs in the example of the dihedral group in the next section, there are other circumstances under which the thick subcategories associated to a variety consists only of the zero module.

9. Examples

In this section we present two examples. In the case that G is a Klein four group and H is any nonempty collection of proper subgroups, all thick tensor ideals in the relative stable category are determined by the support varieties of their objects. On the other hand if G is a dihedral group of order 8 and H is the set of the two elementary abelian subgroups of order 4, then there are many other thick tensor ideals and only a finite number are determined by support varieties. In both of these examples there is a complete classification of the finite dimensional modules.

First suppose that G = ⟨g, h⟩ is an elementary abelian group of order 4. We assume that k is algebraically closed and has characteristic 2. The indecomposable kG modules are classified in [17] and described in the second paragraph of the proof of Theorem 5.1. We prove the following.

Proposition 9.1. Suppose that H is any nonempty collection of proper subgroups of G. If C is a thick tensor ideal in stmod_{H}(kG) then C = stmod_{H}(kG)_{W}, where W is the collection of support varieties of all of the objects in C.

Proof. The reader is referred to the proof of Theorem 5.1 for a discussion of the cohomology ring and the structure of the indecomposable modules in this case. For the proof, it is only necessary to show that if M is in C, and M not zero, then any module N with V_{G}(N) ⊆ V_{G}(M) is also in C. And this statement need only be proved for indecomposable modules M. If M has odd dimension, then M ≅ Ω^{n}(k) for some n. In this case, M ⊗ M^{∗} ≅ k ⊕ P for some projective module P. Thus, k is in C and C = stmod_{H}(kG).

Suppose that M is not H-projective and has even dimension. Then M ≅ L_{ζ^{r}} for some ζ ∈ H^{1}(H, k) and some r. We know that the almost split sequence for L_{ζ^{r}} has the form

\xymatrix{0 \ar[r] & L_{ζ^{r}} \ar[r] & L_{ζ^{r−1}} ⊕ L_{ζ^{r+1}} \ar[r] & L_{ζ^{r}} \ar[r] & 0}  (9.2)

if r > 1 and

\xymatrix{0 \ar[r] & L_{ζ} \ar[r] & L_{ζ^{2}} \ar[r] & L_{ζ} \ar[r] & 0}  (9.3)

for r = 1. Moreover, this sequence splits on restriction to every proper subgroup of G provided M is not isomorphic to k_{J}^{↑G} for J a subgroup of order 2. In this last case r = 1 and M = L_{ζ} for ζ = ζ_{1}, ζ_{2} or ζ_{1} + ζ_{2}. The sequence 9.3 splits on restriction to every proper subgroup of G except J, and because M is not H-projective, J ∉ H. Hence in this case the sequence 9.3 is H-split.

So by the above and by the H-splitting of sequence 9.2, we have that if L_{ζ^{r}} is in C, the so is L_{ζ^{r+1}} and (if r > 1) L_{ζ^{r−1}}. Thus if if L_{ζ^{r}} is in C then every module whose support variety is the point V_{G}(ζ) is in C. □

Our next example is the dihedral group of order 8,

G = ⟨g, h⟩/(g^{2} = h^{2} = (gh)^{4} = 1)

Assume that H = {E_{1}, E_{2}}, the two elementary abelian subgroups of order 4. We let E_{1} = ⟨g, (gh)^{2}⟩ and E_{2} = ⟨h, (gh)^{2}⟩. The cohomology ring of G has the form

H^{∗}(G, k) = k[η_{1}, η_{2}, ζ]/(η_{1}η_{2})

(see [13]). Here, η_{1} and η_{2} have degree 1, while ζ has degree 2. The cohomology ring of E_{1} and E_{2} are polynomial rings, H^{∗}(E_{i}, k) = k[α_{i}, β_{i}], generated by elements in degree 1. With proper choice of the generators, the restriction maps from G can be taken to be

res_{G,E_{1}}(η_{1}) = α_{1}, res_{G,E_{1}}(η_{2}) = 0, res_{G,E_{1}}(ζ) = α_{1}β_{1} − β_{1}^{2},
res_{G,E_{2}}(η_{1}) = 0, res_{G,E_{2}}(η_{2}) = α_{2}, res_{G,E_{2}}(ζ) = α_{2}β_{2} − β_{2}^{2},

The spectrum V_{G}(k) consists of two copies of ℙ_{k}^{1} joined at a point. The two copies are the images of the map induced by restriction onto each of E_{1} nd E_{2}. The reader should be careful here, because the map induced by restriction res_{G,E_{i}}^{∗} : V_{E_{i}}(k) → V_{G}(k) is generically two-to-one. So it folds V_{E_{i}}(k) ≅ ℙ_{k}^{1} over on itself. The point of intersection of the two components is res_{G,Z}^{∗}(V_{Z}(k)), the maximal ideal containing the kernel of the restriction to the center of G.

This leads us to our first conclusion,

Proposition 9.4. Let V be any nontrivial irreducible closed subvariety of V_{G}(k). Then stmod_{H}(kG)_{V} contains only the zero module, unless V is one of res_{G,Z}^{∗}(V_{Z}(k)), res_{G,E_{1}}^{∗}(V_{E_{1}}(k)) or res_{G,E_{2}}^{∗}(V_{E_{2}}(k)).

Proof. The second two subvarieties listed in the proposition are the components of V_{G}(k) of dimension one. All other irreducible subvarieties are closed points. Choose a point 𝔭 = res_{G,E_{1}}^{∗}([a, b]) for [a, b] ∈ V_{E_{1}}(k) ≅ ℙ^{1}. Here, [a, b] is the zero set of γ = bα_{1} + aβ_{1} ∈ H^{1}(E_{1}, k). But notice that the action of G on H^{∗}(E_{1}, k) fixes α_{1}, which is in the image of the restriction from G and interchanges β_{1} with α_{1} + β_{1}. Consequently, the point [a, b] is conjugated to the point [a, a + b]. The only fixed point under this action is the point [0, b] = res_{E_{1},Z}^{∗}(V_{Z}(k)). This point is excluded by the hypothesis since then 𝔭 = res_{G,Z}^{∗}(V_{Z}(k)) by transitivity.

By Theorem 8.1 for 𝔭 ≠ [1, 0] or [1, 1] which correspond to cyclic subgroups of E_{1} we have that every module whose variety is in 𝔭 is induced from a kE_{1}-module. As E_{1} is in H, we have that all objects in stmod_{H}(kG)_{{𝔭}} are isomorphic to the zero object by Theorem 8.1. Hence we are done except for the two cases that V = res_{G,U}^{∗}(V_{U}(k)) for U = ⟨g⟩ or U = ⟨h⟩. These two cases are not exactly covered by Benson’s Theorem though the same principles are at work and the same result (that every module having support variety V is H-projective) could be proved using Benson’s methods. For the sake of completeness, we present a more direct proof after the discussion of the structure of the modules given below. □

The indecomposable finitely generated kG-modules are all known [24]. See also Chapter 7 of [13] for a more thorough discussion. For notation, let x = g − 1 and y = h − 1, so that the central element z = (gh)^{2} − 1 = xy + yx + xyx + yxy + xyxy. Note that x^{2} = 0 = y^{2} and xyxy = yxyx. The indecomposable kG-modules come in bands and strings. A string module can best be described by a word in the alphabet x, x^{−1}, y, y^{−1}. The requirement is that each x or x^{−1} must be followed by a y or y^{−1} and vice versa. So if a_{1}a_{2} … a_{n} is a string in the given alphabet, the module that it represents has a basis m_{1}, m_{2}, …, m_{n+1}. The actions of x and y on this basis is read off of the string by the rule xm_{i} = m_{i+1} if a_{i} = x, xm_{i+1} = m_{i} if a_{i} = x^{−1}, ym_{i} = m_{i+1} if a_{i} = y and ym_{i+1} = m_{i} if a_{i} = y^{−1}. On the ends, the rule is that nothing happens that is not indicated. So, for example if a_{1} = x or x^{−1}, then ym_{1} = 0. As an example, the module Ω^{2}(k) is represented by the string xyxy^{−1}xy^{−1}x^{−1}y^{−1} which can be seen as a diagram in the form

\xymatrix{• \ar[d]^{x} &  &  &  & • \ar[d]^{y} \\ • \ar[d]^{y} &  &  &  & • \ar[d]^{x} \\ • \ar[dr]^{x} &  & • \ar[dl]^{y} \ar[dr]^{x} &  & • \ar[dl]^{y} \\  & • &  & •}

Indeed, it can be shown that the string for Ω^{2n}(k) is (xyxy^{−1})^{n}(xy^{−1}x^{−1}y^{−1})^{n}.

Every band module also has an associated string as before. The difference is that the string must have even length, each node is replaced by a vector space V_{i}, having the same dimension for all nodes, and a letter in the string indicates an action by x or y mapping V_{i} to V_{i+1} or V_{i+1} to V_{i} by the identity map as indicated in the string modules. The other difference is that the copy of V on the right hand end is identified by a formula with the other copies of V in the socle so that the total dimension comes out to be the dimension of V times the length of the string. There are rules for the identification, but they need not concern us here.

It is worth mentioning that if M is a module of even dimension, then V_{G}(M) a proper subset of V_{G}(k). This is because in every string of odd length, there are either an even number of letters that are in {x, x^{−1}} or there are an even number that are in {y, y^{−1}}, and hence the module is either free when restricted to ⟨g⟩ or when restricted to ⟨h⟩.

To finish the proof of Proposition 9.4, we now sketch of a proof that a module M whose support is the point res_{G,U}^{∗}(V_{U}(k)) for U = ⟨g⟩ or U = ⟨h⟩ is H-projective. Specifically we show, assuming that U = ⟨g⟩, that M is E_{1}-projective. The proof for U = ⟨h⟩ is almost the same. First note that

V_{G}(ζ) = res_{G,⟨g⟩}^{∗}(V_{i⟨g⟩}(k)) ∪ res_{G,⟨h⟩}^{∗}(V_{⟨h⟩}(k)).

It follows that if M is a module with support variety equal to V, then M ⊗ L_{ζ^{n}} ≅ Ω(M) ⊕ Ω^{2n}(M) ⊕ (proj) for n sufficiently large. That is, for n sufficiently large, the element ζ^{n} annihilates the cohomology Ext_{kG}^{∗}(M, M) and the given tensor product formula holds. Thus M is a direct summand of Ω^{−1}(M) ⊗ L_{ζ}, and if L_{ζ^{n}} is Hprojective then so is M. Now L_{ζ^{n}} = L_{1} ⊕ L_{2} is a direct sum of two string modules given by the strings (xyxy^{−1})^{n−1}xyx and (yxy)^{−1}(x(yxy)^{−1})^{n−1} respectively. The support variety of L_{1} is res_{G,⟨h⟩}^{∗}(V_{⟨h⟩}(k)) while V_{G}(L_{2}) = res_{G,⟨g⟩}^{∗}(V_{⟨g⟩}(k)). We claim that L_{1} is induced from a kE_{2}-module while L_{2} is induced from a kE_{1}-module. The proof of the claim is a straightforward, though tedious, exercise. Note that kE_{2} ≅ k[y, z]/(y^{2}, z^{2}) where z = (gh)^{2} + 1 is central in kG. The kE_{2}-string-module N with string (zx^{−1})^{n−1}z has the property that N^{↑G} ≅ L_{1}. Here is the tedious part, which we leave mostly as an exercise. The process is to take a set of generators for N. The obvious ones are v_{1} at the end of the string and then v_{i} = (zy^{−1})^{i−1}v_{1} so that xv_{1} = 0, and yv_{i+1} = zv_{i} for i = 1, …, n − 1. Then one shows that N^{↑G} ≅ kG ⊗_{kE_{2}} N ≅ L_{1}, by producing a basis on which the generators x and y act according to the specified string. One can start constructing a collection of generators by letting w_{1} = 1 ⊗ v_{1}, with yw_{1} = 0 and xyxw_{1} = x ⊗ zv_{1}. Then the other element of a set of generator can be constructed using identities such as that y(1 ⊗ v_{j+1}) + ∑_{i=1}^{j}(1 + x) ⊗ v_{i} = x ⊗ zv_{j}.

Knowing the modules we can prove some statements about the other support varieties.

Proposition 9.5. Let D be one of the subcategories stmod_{H}(kG)_{V} where V is the set of of all closed sets in either res_{G,E_{1}}^{∗}(V_{E_{1}}(k)) or res_{G,E_{2}}^{∗}(V_{E_{2}}(k)). Then D contains an infinite ascending chain of thick tensor ideals that are prime in stmod_{H}(kG).

Proof. Assume that V = res_{G,E_{1}}^{∗}(V_{E_{1}}(k)). Then the variety of any module M in D when restricted to E_{2} has precisely the point [0, 1] as this is the inverse image of the intersection

res_{G,E_{1}}^{∗}(V_{E_{1}}(k)) ∩ res_{G,E_{2}}^{∗}(V_{E_{2}}(k)) = res_{G,Z}^{∗}(V_{Z}(k)).

This is the zero set of α_{2}. Recall that res_{G,E_{2}}(η_{2}) = α_{2}, and we see that V = V_{G}(η_{2}). In particular, for any i > 0, L_{η_{2}^{i}} is in stmod_{H}(kG)_{V} and the restriction of L_{η_{2}^{i}} to E_{2} is L_{α_{2}^{i}}.

Exactly as in the proof of Theorem 5.1, let C_{i} be the additive subcategory consisting all indecomposable kE_{2}-modules M such that either V_{E_{2}}(M) does not contain V_{E_{2}}(α_{2}) or that α_{2}^{i} annihilates the cohomology of M. Then C_{i} is a prime additive tensor ideal (ATI). Let B_{i} be the full subcategory of stmod_{H}(kG) consisting of all modules whose restriction to E_{2} is in C_{i}. Then, B_{i} is a thick tensor ideal by Theorem 4.5. Note that L_{η_{2}^{i}} is in B_{i} but not in B_{i+1}.

Finally, note that L_{η_{2}^{i}} is not H-projective. The reason is that it is indecomposable. By Green’s Indecomposability Theorem [18], which says that the induction of an indecomposable module is indecomposable, it would have to be induced from an indecomposable module on E_{1} or E_{2}. But then by the Mackey Formula, its restriction to E_{2} would have at least two nonprojective summands. □

Proposition 9.6. Suppose that D the subcategory stmod_{H}(kG)_{V} where V is the set of of all closed sets in res_{G,Z}^{∗}(V_{Z}(k)). Then D contains two infinite ascending chains of thick tensor ideals that are prime in stmod_{H}(kG).

Proof. The variety of any nonzero module M in D when restricted to either E_{1} or E_{2} is precisely the closed point [0, 1] because this is the inverse image of res_{G,Z}^{∗}(V_{Z}(k)). Hence, for j = 1, 2 and i > 0, there are prime additive tensor ideals C_{i}^{j}, and prime thick tensor ideals B_{i}^{j}, defined exactly as in the previous proof. That is, B_{i}^{j} is the full subcategory generated all indecomposable kG-modules M such that the restriction of M to E_{j} either has variety V_{E_{j}}(M_{↓E_{j}}) not containing V_{E_{j}}(α_{j}) or has cohomology annihilated by α_{j}^{i}. □

The thick tensor ideal B_{r,s} = B_{1}^{r} ∩ B_{2}^{s} is not prime simply because the tensor product of a module in B_{r+1,s} and a module in B_{r,s+1} is in B_{r,s}. They are however distinct. The reader may check that the that the module obtained from the string module defined by (x^{−1}y^{−1}x^{−1}y)^{s}(x^{−1}yxy)^{r}, and identifying the two ends is in B_{r,s} but not in B_{r−1,s} or B_{r,s−1}.

Other thick subcategories can be derived from inflations as in Section 7. Suppose that Z = Z(G) is the center of G. The cohomology ring of G/Z is H(G/Z) = k[γ_{1}, γ_{2}], where γ restricts to zero on the subgroup generated by the class of h and γ_{2} is zero on restriction to the subgroup generated by the class of g. The spectrum of the cohomology ring is V_{G/Z}(k) ≅ ℙ^{1}. For any collection of closed subsets W of V_{G/Z}(k) there is a thick tensor ideal 𝔘_{W}. generated by the inflations of the kG/Zmodules whose varieties are contained in W.

Proposition 9.7. If W is a subset of S = ℙ^{1} \ {[1, 0], [0, 1]}, then the thick tensor ideals 𝔘_{W} defined as above are subcategories of B_{1,1} and are distinct for different choices of W.

Proof. The distinction follows from Theorem 7.1. The reason for the exclusion of the points [1, 0] and [0, 1], is to insure that any module in the category has variety contained in res_{G,Z}(V_{Z}(k)). □

We remark that as in Section 6, the idempotent F-module in the idempotent triangle of stmodĤ(k(G/Z))_{W} (where Ĥ = {E_{1}/Z, E_{2}/Z} is not the same as that of 𝔘_{W}. So there is yet another thick tensor ideal associated to the inflation. Indeed, we are left with the question.

Question 9.8. Does every thick tensor ideals of stmod_{H}(kG) have the form of one of the subcategories discussed so far?

10. One more question

Finally, we remind the reader that there is one other construction of thick tensor ideals in the relative stable category given in [15]. As before let H be a collection of p-subgroups of G, not containing a Sylow p-subgroup. A module M is said to be virtually H-projective provided that for n sufficiently large every cocyle ζ : Ω^{n}(M) → M factors through an H-projective module. In Theorem 6.4 of [15] is is shown that the full subcategory of stmod_{H}(kG) consisting of virtually H-projective modules is a thick tensor ideal. Moreover, it is shown that if M is a kG-module whose support variety intersects trivially the variety of the ideal of images of transfers Tr_{H}^{G} : H(H, k) → H(G, k) for H ∈ H, then M is virtually H-projective. The question is: is there some sort of converse to this result?

Question 10.1. Does there exist a collection of closed subsets W of V_{G}(k), closed under specialization, such that stmod_{H}(kG)_{W} is the subcategory of all virtually H-projective modules?

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Department of Mathematics, University of Georgia, Athens, GA 30602, USA

E-mail address: jfc@math.uga.edu