TRANSFER MAPS AND VIRTUAL PROJECTIVITY

Jon F. Carlson^{∗}, Chuang Peng, and Wayne W. Wheeler^{∗}

Abstract. When G is a finite group and H is a subgroup of G, the concept of relative Hprojectivity is of fundamental importance in modular representation theory. Unfortunately, in some ways the properties of relatively H-projective modules are not as closely related to other ideas of representation theory as one might hope. For example, the variety of a module bears only a weak connection to relative projectivity. The purpose of this paper is to introduce and study a related concept known as virtual H-projectivity that seems to be much more closely related to the theory of varieties. The study is actually carried out in the more general setting of virtual projectivity with respect to a finitely generated module W, and the main tool is a transfer map Tr_{W} that generalizes the usual transfer map Tr_{H}^{G} with respect to a subgroup H.

1. Introduction

Relative homological algebra and relative projectivity have long been a rich source of problems and inspiration in the modular representation theory of finite groups, and it does not seem likely that the stream of interesting results on the subject will soon come to an end. This paper introduces a new variant of the concept of relative projectivity, called virtual projectivity because it has the same spirit as other virtual notions in group cohomology. In this case, however, the term virtual refers to properties of modules and representations rather than properties of groups.

The concept of virtual projectivity arises naturally from previous work that connects relative projectivity and varieties for modules. Let k be an algebraically closed field of characteristic p > 0, and assume for convenience that G is a finite p-group. Let H be a subgroup of G, and let I = Tr_{H}^{G}(H^{∗}(H, k)) be the image of the transfer map. With some effort Benson’s result in Corollary 3.2 of [4] can be restated to show that if M is a finitely generated periodic kG-module whose variety V_{G}(M) intersects V_{G}(I) trivially, then M is stably induced from a kH-module; in other words, there exist a finitely generated kH-module L and a projective kG-module P such that M ⊕ P ≅ L↑^{G}. A more general theorem for modules that are not periodic has been proven by the first author [11]. In fact, Benson’s original result is also somewhat more general than that stated here. By viewing it as a theorem about transfers, however, we obtain a shorter statement and a remarkably easy proof (see Corollary 4.5).

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 20C20, 20J06.

∗Partially supported by the NSF.

Because the variety of an indecomposable periodic module M is a line through the origin in the affine variety V_{G}(k), we can regard it as a point in the corresponding projective variety V̄_{G}(k). Then the assumption of Benson’s theorem is that V̄_{G}(M) is contained in the open set V̄_{G}(k) − V̄_{G}(I). For the generalization in [11], however, this open condition on varieties is not sufficient; instead, a very technical condition on V̄_{G}(M) is necessary to insure that M is stably induced. Nevertheless, the open condition is quite natural, and it provides the motivation for our definition of virtual projectivity.

The approach taken here, as in [12], is to fix an arbitrary finite group G and to consider projectivity relative to a finitely generated kG-module W. A module M is W -projective if it is a summand of W ⊗ N for some kG-module N. If H is a subgroup of G, then a kGmodule M is relatively H-projective if and only if it is relatively W -projective for W = k_{H}^{↑G}. Section 2 summarizes the basic facts about homological algebra relative to a module W. The major tool used throughout the paper is the transfer map Tr_{W} : Ext_{kG}^{∗}(W, W) → H^{∗}(G, k) defined and studied in Section 3. When H is a subgroup of G and W = k_{H}^{↑G}, the map Tr_{W} has the same image as the usual transfer map Tr_{H}^{G} : H^{∗}(H, k) → H^{∗}(G, k). Section 4 is devoted to studying other properties of the image of Tr_{W} and the variety defined by this image.

Virtual projectivity, defined in Section 5, shares many of the same formal properties as relative projectivity. For example, it is closed under direct sums and under tensor products with arbitrary finitely generated modules. It has one interesting property, however, that is not shared by relative projectivity: if M is a finitely generated kG-module such that the variety defined by the image of Tr_{W} intersects the variety V_{G}(M) trivially, then M must be virtually W -projective (see Proposition 5.3). Example 7.1 shows that such a module need not be W -projective. In fact, this example describes a module W such that all finitely generated kG-modules are virtually W -projective, but not all are relatively W -projective. Section 7 also includes an example of a subgroup H ⊆ G and a kG-module that is virtually H-projective but not H-projective.

Section 6 presents one other interesting relationship between relative projectivity and virtual projectivity. One can define a W -stable category \underline{mod}^{W} (kG) that is an analogue of the usual stable category \underline{mod}(kG). The objects of \underline{mod}^{W} (kG) consist of finitely generated kG-modules, and the morphisms are equivalence classes of kG-homomorphisms; two homomorphisms are equivalent if their difference factors through a W -projective module. The main results of Section 6 show that \underline{mod}^{W} (kG) is a triangulated category and that the virtually W -projective modules form a thick subcategory. It would be quite interesting to determine what the other thick subcategories are, but that task probably requires a much better understanding of relative cohomology.

2. Relative projectivity with respect to a module

The main purpose of this section is to establish some notation and basic facts concerning projectivity relative to a module W. Much of this material already appears in [12], but it is summarized here for the sake of completeness. The reader should be warned that the definition of relative W -projectivity used here is not the same as that in [12], but the equivalence of the two definitions is proven in Proposition 2.4 below. For most of the results concerning a relatively W -projective module M it is necessary to assume that W is finitely generated. On the other hand, many results remain valid even if M is infinitely generated, and we have chosen to work in that generality.

Let Mod(kG) denote the category of right kG-modules and mod(kG) the subcategory of finitely generated modules. If M and N are kG-modules such that M is isomorphic to a direct summand of N, then we write M | N. If W is a kG-module and

E : 0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0

is a short exact sequence such that E ⊗ W splits, then E is said to be W -split. A monomorphism f : M′ → M is said to be W -split if the short exact sequence

0 ⟶ M′ ⟶^{f} M ⟶ Coker f ⟶ 0

is W -split, and W -split epimorphisms are defined similarly.

Definition 2.1. Let W and P be kG-modules. Assume that if f : M → M′′ is any W -split epimorphism and g ∈ Hom_{kG}(P, M′′), then there is a map h ∈ Hom_{kG}(P, M) such that fh = g. Then P is said to be W -projective (or relatively W -projective). The idea of a W -injective module is defined similarly.

If W is a kG-module, then the evaluation map α_{W} : W^{∗}⊗W → k is the homomorphism defined by α_{W} (f ⊗ w) = f(w). This map will play an important part in the following work, and we begin by recording one of its basic properties.

Lemma 2.2. If W is a finitely generated kG-module, then the evaluation map is W -split.

Proof. Let {w_{1}, …, w_{n}} be a basis for W, and let {w_{1}^{∗}, …, w_{n}^{∗}} denote the dual basis for W^{∗}. Then the map 1_{W} ⊗α_{W} : W ⊗W^{∗}⊗W → W has a right inverse β : W → W ⊗W^{∗}⊗W given by β(w) = ∑_{i=1}^{n} w_{i} ⊗ w_{i}^{∗} ⊗ w. Hence α_{W} is W -split. □

Suppose that W is a finitely generated kG-module. It is easy to see that if X | W, then any monomorphism, epimorphism, or short exact sequence that is W -split must also be X-split. Lemma 2.2 implies that W | W ⊗W^{∗}⊗W and by duality that W^{∗} | W^{∗}⊗W ⊗W^{∗}. Thus we obtain the following result.

Proposition 2.3. Let W be a finitely generated kG-module. Then a monomorphism, an epimorphism, or a short exact sequence is W -split if and only if it is W^{∗}-split.

Proposition 2.4. Let W and M be kG-modules, and assume that W is finitely generated. Then the following statements are equivalent:

(1) M is W -projective;

(2) M | W ⊗ W^{∗} ⊗ M;

(3) there is a kG-module N such that M | W ⊗ N.

Proof. To prove that (1) implies (2), suppose that M is W -projective. If α_{W} : W^{∗} ⊗W → k is the evaluation map, then α_{W} ⊗ 1_{M} : W^{∗} ⊗ W ⊗ M → M is a W -split epimorphism. Hence there is a map f : M → W^{∗} ⊗ W ⊗ M such that (α_{W} ⊗ 1_{M})f = 1_{M}, and M | W^{∗} ⊗ W ⊗ M.

It is trivial that (2) implies (3), so suppose that (3) holds. Let f : U′ → U be a W -split epimorphism. If g : M → U is a kG-homomorphism, then we must show that there is a kG-homomorphism h : M → U′ such that fh = g. Because M | W ⊗ N, it suffices to prove this result when M = W ⊗ N. But in this case there is a commutative diagram

Hom_{kG}(W ⊗ N, U′) ⟶^{f_{∗}} Hom_{kG}(W ⊗ N, U)
↓  ↓
Hom_{kG}(N, W^{∗} ⊗ U′) ⟶_{(1_{W^{∗}} ⊗f)_{∗}} Hom_{kG}(N, W^{∗} ⊗ U)

in which the vertical maps are isomorphisms. Moreover, f : U′ → U is a W^{∗}-split epimorphism, so the map (1_{W^{∗}} ⊗f)_{∗} is an epimorphism and hence so is f_{∗}. This completes the proof. □

An argument dual to that given in Proposition 2.4 shows that a module M is W - injective if and only if there is a kG-module N such that M | W ⊗ N. Thus we obtain the following result.

Proposition 2.5. Assume that W is finitely generated. Then a kG-module is relatively W -projective if and only if it is relatively W -injective.

Let W and M be kG-modules, and assume that W is finitely generated. A W -split epimorphism ε : P → M is called a W -projective cover of M if P is W -projective and Ker ε has no W -projective summands. The W -injective hull of M is defined similarly. The following fundamental result is essentially a restatement of Theorem 3.5 of [12].

Proposition 2.6. Let W and M be kG-modules, and assume that W is finitely generated. Then there is a W -projective module P and a W -split epimorphism ε : P → M. Similarly, there is a W -injective module I and a W -split monomorphism δ : M → I. If M is finitely generated, then M has a W -projective cover and a W -injective hull, both of which are unique up to isomorphism.

Proof. Because the evaluation map α_{W} : W^{∗} ⊗ W → k is W -split, there is a W -split epimorphism α_{W} ⊗ 1_{M} : W^{∗} ⊗ W ⊗ M → M from a W -projective module onto M. Moreover, if M is finitely generated, then it is easy to see that M has a W -projective cover that is unique up to isomorphism. The corresponding statements for W -injective modules follow by a dual argument. □

Let W be a finitely generated module. If M is any right kG-module and α_{W} : W^{∗} ⊗ W → k is the evaluation map, then we define Ω_{W} (M) to be the kernel of the W -split epimorphism α_{W} ⊗1_{M} : W^{∗}⊗W ⊗M → M. Taking duals gives a W -split monomorphism α_{W}^{∗} : k → W ⊗ W^{∗}, and we define Ω_{W}^{−1}(M) to be the cokernel of α_{W}^{∗} ⊗ 1_{M} : M → W ⊗ W^{∗} ⊗ M.

Let C be a full subcategory of mod(kG) that is closed under taking finite direct sums and summands. Suppose that for any finitely generated module M there is a module U in C and a map f : U → M such that if g : W → M is any homomorphism with W in C, then g = fh for some h : W → U. Then C is said to be contravariantly finite, and the module U together with the map f : U → M is called a right C-approximation of M (see [1]). It is known that M always has a minimal right C-approximation that is a direct summand of any other right C-approximation. The concept of a covariantly finite subcategory of mod(kG) is dual to that of a contravariantly finite subcategory, and C is said to be functorially finite if it is both covariantly and contravariantly finite. It is easy to see that if W is any finitely generated kG-module, then the W -projective modules in mod(kG) form a functorially finite subcategory.

Proposition 2.7. Suppose that C is a contravariantly finite subcategory of mod(kG) such that if M is in C and N is any finitely generated kG-module, then M ⊗ N is in C. Let f : W → k be the minimal right C-approximation of k. Then C is the subcategory of all W -projective modules in mod(kG). In particular, C is functorially finite.

Proof. Let C be any object of C. Because there is a natural isomorphism Hom_{kG}(L ⊗ M^{∗}, N) ≅ Hom_{kG}(L, M ⊗ N) whenever M is finitely generated, we get a commutative diagram

Hom_{kG}(C ⊗ C^{∗}, W) ⟶^{f_{∗}} Hom_{kG}(C ⊗ C^{∗}, k)
↓  ↓
Hom_{kG}(C, C ⊗ W) ⟶_{(1_{C}⊗f)_{∗}} Hom_{kG}(C, C)

in which the vertical maps are isomorphisms. The module C ⊗ C^{∗} is in C, so f_{∗} is onto and hence so is (1_{C} ⊗ f)_{∗}. Thus there is a map g : C → C ⊗ W such that (1_{C} ⊗ f)g = 1_{C}. It follows that C | C ⊗ W so that C is relatively W -projective.

Conversely, we know that W is in C and hence by assumption that W ⊗ M is in C for any finitely generated module M. But C is closed under taking direct summands, so every finitely generated W -projective module is in C. Hence C is the category of all finitely generated W -projective modules, as desired. □

3. The transfer map

It is well known that the transfer map is closely related to the usual concept of relative projectivity. The objective of this section is to define and study the basic properties of a more general transfer map associated to projectivity with respect to a module. We begin with some definitions and notation.

If M and N are kG-modules, let PHom_{kG}(M, N) be the subspace of Hom_{kG}(M, N) consisting of all homomorphisms that factor through a projective module; these maps are called projective homomorphisms. The stable category \underline{Mod}(kG) has the same objects as Mod(kG), but the morphisms from M to N are given by the k-vector space

\underline{Hom}_{kG}(M, N) = Hom_{kG}(M, N)/ PHom_{kG}(M, N).

The full subcategory of finitely generated modules is denoted by \underline{mod}(kG).

For each kG-module M let ε_{M} : P (M) → M denote a fixed projective cover of M, and set Ω(M) = Ker ε. If n > 1, then Ω^{n}(M) is defined inductively by setting Ω^{n}(M) = Ω(Ω^{n−1}(M)). The modules Ω^{−n}(M) are defined for n ≥ 1 by a dual construction using injective hulls. Note that the definition of Ω(M) is essentially a special case of the definition of Ω_{W} (M) given in the previous section: if W = kG, then Ω_{W} (M) and Ω(M) are isomorphic in \underline{Mod}(kG). We will often use the fact that for n > 0 there are isomorphisms

Ext^{n}(M, N) ≅ \underline{Hom}_{kG}(Ω^{n}(M), N) ≅ \underline{Hom}_{kG}(M, Ω^{−n}(N)).

Suppose that M, N, and W are kG-modules and that W is finitely generated. We define the transfer map Tr_{W} : Hom_{kG}(M ⊗ W, N ⊗ W) → Hom_{kG}(M, N) to be the composition

Hom_{kG}(M ⊗ W, N ⊗ W) ≅ Hom_{kG}(M, N ⊗ W^{∗} ⊗ W) ⟶^{(1_{N} ⊗α_{W})_{∗}} Hom_{kG}(M, N),

where α_{W} : W^{∗} ⊗ W → k is the evaluation map. If w_{1}, …, w_{n} is a basis of W and w_{1}^{∗}, …, w_{n}^{∗} is the dual basis of W^{∗}, then there is a kG-homomorphism σ_{W} : k → W ⊗ W^{∗} satisfying σ_{W} (1) = ∑_{i=1}^{n} w_{i} ⊗ w_{i}^{∗}. Moreover, σ_{W} is independent of the chosen basis. The transfer Tr_{W} can also be described as the composition

Hom_{kG}(M ⊗ W, N ⊗ W) ≅ Hom_{kG}(M ⊗ W ⊗ W^{∗}, N) ⟶^{(1_{M} ⊗σ_{W})^{∗}} Hom_{kG}(M, N).

The isomorphism F : Hom_{kG}(M ⊗ W, N ⊗ W) → Hom_{kG}(M ⊗ W ⊗ W^{∗}, N) has the form (F γ)(m ⊗ w ⊗ w^{∗}) = (1_{N} ⊗ w^{∗})γ(m ⊗ w), so an explicit formula for the transfer is given by

Tr_{W} (γ)(m) = ∑_{i=1}^{n}(1_{N} ⊗ w_{i}^{∗})γ(m ⊗ w_{i}).

If f ∈ Hom_{kG}(M′, M) and g ∈ Hom_{kG}(N, N′), then it follows that

g ∘ Tr_{W} (γ) ∘ f = Tr_{W} ((g ⊗ 1_{W})γ(f ⊗ 1_{W})).

Note that if γ ∈ Hom_{kG}(M ⊗ W, N ⊗ W) is a projective homomorphism, then so is the corresponding map F γ ∈ Hom_{kG}(M ⊗ W ⊗ W^{∗}, N). Hence Tr_{W} (γ) is projective, and it follows that Tr_{W} induces a map Tr_{W} : Ext_{kG}^{n}(M ⊗ W, N ⊗ W) → Ext_{kG}^{n}(M, N) for all n ≥ 0. In particular, the image of the transfer map Tr_{W} : Ext_{kG}^{∗}(W, W) → Ext_{kG}^{∗}(k, k) = H^{∗}(G, k) is a homogeneous ideal.

Suppose that H is a subgroup of G and W = k_{H}^{↑G}, and let g_{1}, …, g_{n} be a set of representatives for the right cosets of H in G. Let ϕ : Hom_{kH}(M, N) → Hom_{kG}(M ⊗ W, N ⊗ W) be the homomorphism satisfying ϕ(f)(m ⊗ 1 ⊗ g_{i}) = f(mg_{i}^{−1})g_{i} ⊗ 1 ⊗ g_{i} for 1 ≤ i ≤ n. Then the map ϕ gives the following relationship between Tr_{W} and Tr_{H}^{G}.

Proposition 3.1. Let H be a subgroup of G, and set W = k_{H}^{↑G}. Then Tr_{W} ∘ϕ = Tr_{H}^{G}, and Tr_{H}^{G}(H^{∗}(H, k)) = Tr_{W} (Ext_{kG}^{∗}(W, W)).

Proof. It is easy to see that Tr_{W} ∘ϕ = Tr_{H}^{G}, and this fact implies that Tr_{H}^{G}(H^{∗}(H, k)) ⊆ Tr_{W} (Ext_{kG}^{∗}(W, W)). To prove the opposite inclusion, let 1 = g_{1}, …, g_{n} be a set of representatives for the right cosets of H in G, and set w_{i} = 1 ⊗ g_{i} ∈ k_{H}^{↑G} for 1 ≤ i ≤ n. Then w_{1}, …, w_{n} is a basis for W = k_{H}^{↑G}. Let w_{1}^{∗}, …, w_{n}^{∗} be the dual basis of W^{∗}, and consider the transfer map Tr_{W} : \underline{Hom}_{kG}(Ω^{m}(k) ⊗ W, W) → \underline{Hom}_{kG}(Ω^{m}(k), k) for m ≥ 0. If η : Ω^{m}(k) ⊗ W → W is a kG-homomorphism, define a kH-homomorphism η′ : Ω^{m}(k) → k by η′(x) = w_{1}^{∗}(η(x ⊗ w_{1})). Then

Tr_{H}^{G}(η′)(x) = ∑_{i} η′(xg_{i}^{−1})

= ∑_{i} w_{1}^{∗}(η(xg_{i}^{−1} ⊗ w_{1}))

= ∑_{i} w_{1}^{∗}(η(x ⊗ w_{i})g_{i}^{−1})

= ∑_{i} w_{i}^{∗}(η(x ⊗ w_{i}))

= Tr_{W} (η)(x).

Hence Tr_{W} (Ext_{kG}^{∗}(W, W)) ⊆ Tr_{H}^{G}(H^{∗}(H, k)), and this completes the proof. □

Proposition 3.2. Let f : M → N be a kG-homomorphism, and let W be a finitely generated kG-module. Then f is in the image of Tr_{W} : Hom_{kG}(M ⊗ W, N ⊗ W) → Hom_{kG}(M, N) if and only if f factors through a relatively W -projective module. In particular, the module M is relatively W -projective if and only if there is a γ ∈ Hom_{kG}(M ⊗ W, M ⊗ W) with Tr_{W} (γ) = 1_{M}.

Proof. If f = Tr_{W} (β), let β′ : M ⊗W ⊗W^{∗} → N be the kG-homomorphism corresponding to β under the isomorphism Hom_{kG}(M ⊗ W, N ⊗ W) ≅ Hom_{kG}(M ⊗ W ⊗ W^{∗}, N). Then f factors as

M ⟶^{1_{M} ⊗σ_{W}} M ⊗ W ⊗ W^{∗} ⟶^{β′} N,

and M ⊗ W ⊗ W^{∗} is W -projective.

Conversely, let P be a W -projective module, and suppose that there are maps g : M → P and h : P → N such that f = hg. We may assume without loss of generality that P = U ⊗ W for some kG-module U. If we can prove that 1_{P} = Tr_{W} (γ) for some γ ∈ Hom_{kG}(P ⊗ W, P ⊗ W), then it follows that

f = h ∘ Tr_{W} (γ) ∘ g = Tr_{W} ((h ⊗ 1_{W})γ(g ⊗ 1_{W})).

Let α_{W} : W^{∗}⊗W → k denote the evaluation map, and let π : W ⊗W ⊗W^{∗} → W^{∗}⊗W ⊗W be the isomorphism that interchanges the tensor factors W ⊗ W and W^{∗}. Then (1_{U} ⊗ α_{W} ⊗ 1_{W})(1_{U} ⊗ π)(1_{U} ⊗ 1_{W} ⊗ σ_{W}) = 1_{U⊗W}. If γ ∈ Hom_{kG}(U ⊗ W ⊗ W, U ⊗ W ⊗ W) denotes the map corresponding to (1_{U} ⊗ α_{W} ⊗ 1_{W})(1_{U} ⊗ π) under the isomorphism Hom_{kG}(U ⊗ W ⊗ W, U ⊗ W ⊗ W) ≅ Hom_{kG}(U ⊗ W ⊗ W ⊗ W^{∗}, U ⊗ W), then it follows that 1_{U⊗W} = Tr_{W} (γ), as desired.

The final conclusion of the proposition is an easy consequence of the first statement. □

Proposition 3.3. Let M, N, W, and X be kG-modules with W and X finitely generated. Then the composition of transfer maps

Hom_{kG}(M ⊗ W ⊗ X, N ⊗ W ⊗ X) ⟶ Hom_{kG}(M ⊗ W, N ⊗ W) ⟶ Hom_{kG}(M, N) satisfies Tr_{W} ∘ Tr_{X} = Tr_{W ⊗X}.

Proof. Let w_{1}, …, w_{s} be a basis for W and x_{1}, …, x_{t} a basis for X, and let w_{1}^{∗}, …, w_{s}^{∗} and x_{1}^{∗}, …, x_{t}^{∗} be the dual bases of W^{∗} and of X^{∗}. If γ : M ⊗ W ⊗ X → N ⊗ W ⊗ X is a kG-homomorphism, then

(Tr_{W} ∘ Tr_{X})(γ)(m) = ∑_{i}(1_{N} ⊗ w_{i}^{∗}) Tr_{X}(γ)(m ⊗ w_{i})

= ∑_{i,j}(1_{N} ⊗ w_{i}^{∗})(1_{N} ⊗ 1_{W} ⊗ x_{j}^{∗})γ(m ⊗ w_{i} ⊗ x_{j})

= ∑_{i,j}(1_{N} ⊗ w_{i}^{∗} ⊗ x_{j}^{∗})γ(m ⊗ w_{i} ⊗ x_{j})

= Tr_{W ⊗X}(γ)(m),

as desired. □

Let M, M′, N, and N′ be kG-modules. If ζ ∈ Ext_{kG}^{m}(M, M′) and η ∈ Ext_{kG}^{n}(N, N′), then ζ and η are represented by homomorphisms ζ̂: Ω^{m}(M) → M′ and η̂: Ω^{n}(N) → N′. We write ζ ∪ η for the element of Ext_{kG}^{m+n}(M ⊗ N, M′ ⊗ N′) represented by ζ̂ ⊗ η̂ : Ω^{m}(M) ⊗ Ω^{n}(N) → M′ ⊗ N′.

Proposition 3.4. Suppose that W and X are finitely generated kG-modules. If ζ ∈ Ext_{kG}^{m}(W, W) and η ∈ Ext_{kG}^{n}(X, X), then

Tr_{W} (ζ) · Tr_{X}(η) = Tr_{W ⊗X}(ζ ∪ η).

Proof. Let w_{1}, …, w_{s} be a basis for W and x_{1}, …, x_{t} a basis for X, and let w_{1}^{∗}, …, w_{s}^{∗} and x_{1}^{∗}, …, x_{t}^{∗} be the dual bases of W^{∗} and of X^{∗}. Let ζ′ : Ω^{m}(k) ⊗ W → W be a kG-homomorphism representing ζ, and let η′ : Ω^{n}(k) ⊗ X → X be a kG-homomorphism representing η. Then ζ′ ⊗ η′ : Ω^{m}(k) ⊗ W ⊗ Ω^{n}(k) ⊗ X → W ⊗ X represents ζ ∪ η ∈ Ext_{kG}^{m+n}(W ⊗ X, W ⊗ X), and for any u ∈ Ω^{m}(k) and v ∈ Ω^{n}(k) we have

(Tr_{W} (ζ′) ⊗ Tr_{X}(η′))(u ⊗ v) = ∑_{i,j} w_{i}^{∗}(ζ′(u ⊗ w_{i})) ⊗ x_{j}^{∗}(η′(v ⊗ x_{j}))

= ∑_{i,j}(w_{i}^{∗} ⊗ x_{j}^{∗})(ζ′ ⊗ η′)(u ⊗ w_{i} ⊗ v ⊗ x_{j})

= Tr_{W ⊗X}(ζ′ ⊗ η′)(u ⊗ v).

Hence Tr_{W} (ζ) · Tr_{X}(η) = Tr_{W ⊗X}(ζ ∪ η), as desired. □

Suppose that H and K are subgroups of G. Let ζ ∈ H^{m}(H, k) be represented by a kH-homomorphism ζ′ : Ω^{m}(k) → k, and let η ∈ H^{n}(K, k) be represented by a kKhomomorphism η′ : Ω^{n}(k) → k. Let g_{1}, …, g_{s} be a set of representatives for the right cosets of H in G, and define ζ̂: Ω^{m}(k)⊗k_{H}^{↑G} → k_{H}^{↑G} by setting ζ̂(x⊗1⊗g_{i}) = ζ′(xg_{i}^{−1})⊗g_{i} for 1 ≤ i ≤ s. Define η̂ : Ω^{n}(k) ⊗ k_{K}^{↑G} → k_{K}^{↑G} similarly, and set W = k_{H}^{↑G} and X = k_{K}^{↑G}. Then there is an isomorphism

W ⊗ X = k_{H}^{↑G} ⊗ k_{K}^{↑G} ≅ ⨁_{g∈H\G/K} k_{g^{−1}Hg∩K}^{↑G},

and the previous proposition gives the Mackey formula

Tr_{H}^{G}(ζ) · Tr_{K}^{G}(η) = Tr_{W} (ζ̂) · Tr_{X}(η̂)

= Tr_{W ⊗X}(ζ̂ ∪ η̂)

= ∑_{g∈H\G/K} Tr_{g^{−1}Hg∩K}^{G}(ζ^{g} · η).

If W is a kG-module and f : M → N is a kG-homomorphism that factors through a W -projective module, then we say that f is W -projective. We end this section with two results concerning relatively projective homomorphisms.

Proposition 3.5. Let M and N be kG-modules, and assume that M is finitely generated. If f : M → N is a kG-homomorphism, then f is (M ⊗ N)-projective.

Proof. Let α_{M} : M^{∗} ⊗ M → k denote the evaluation map. Fix a basis {m_{1}, …, m_{s}} of M, and let {m_{1}^{∗}, …, m_{s}^{∗}} be the dual basis of M^{∗}. Define β : M → M ⊗ M^{∗} ⊗ M by β(m) = ∑_{i=1}^{s} m_{i} ⊗ m_{i}^{∗} ⊗ m. Then the composition

M ⟶^{β} M ⊗ M^{∗} ⊗ M ⟶^{f⊗1⊗1} N ⊗ M^{∗} ⊗ M ⟶^{1_{N} ⊗α_{M}} N

is equal to f, and f factors through the (M ⊗ N)-projective module N ⊗ M^{∗} ⊗ M, as desired. □

Corollary 3.6. Let M, W, and X be finitely generated kG-modules. If f : L → M is W -projective and g : M → N is X-projective, then gf : L → N is (W ⊗ X)-projective.

Proof. Because M is finitely generated, it has a W -projective cover P and an X-injective hull I, and both P and I are finitely generated. Moreover, f factors through P and g factors through I, so Proposition 3.5 implies that gf factors through a (P ⊗ I)-projective module. Hence gf also factors through a (W ⊗ X)-projective module, as desired. □

4. Varieties and the image of the transfer map

In this section we use the theory of varieties to study some of the properties of the ideal Tr_{W} (Ext_{kG}^{∗}(W, W)) in H^{∗}(G, k) when W is a finitely generated kG-module. Let V_{G}(k) denote the maximal ideal spectrum of the cohomology ring H^{∗}(G, k). If ℑ is any ideal in H^{∗}(G, k), then V_{G}(ℑ) denotes the closed subvariety consisting of all maximal ideals containing ℑ. For any kG-module M let 𝔍(M) denote the annihilator of Ext_{kG}^{∗}(M, M) in H^{∗}(G, k); if M is finitely generated, set V_{G}(M) = V_{G}(𝔍(M)). These varieties have a number of interesting properties that are treated in detail in the standard texts [2, 13].

Proposition 4.1. Let W and M be kG-modules, and assume that W is finitely generated. Then

𝔍(W) · Tr_{W} (Ext_{kG}^{∗}(M ⊗ W, M ⊗ W)) = 0.

Proof. Suppose that ζ ∈ 𝔍(W) and f ∈ Ext_{kG}^{∗}(M ⊗ W, M ⊗ W) are homogeneous. Let ζ′ : k → Ω^{−n}(k) be the kG-homomorphism representing ζ, and let f′ : Ω^{m}(k)⊗M ⊗W → M ⊗ W be a kG-homomorphism representing f. Then ζ′ ⊗ 1_{W} represents ζ · 1_{W} = 0, so ζ′ ⊗ 1_{W} is a projective homomorphism and hence so is ζ′ ⊗ 1_{M} ⊗ 1_{W}. It follows that

(ζ′ ⊗ 1_{M}) ∘ Tr_{W} (f′) = Tr_{W} ((ζ′ ⊗ 1_{M} ⊗ 1_{W})f′)

is also projective, so ζ · Tr_{W} (f) = (ζ · 1_{M}) Tr_{W} (f) = 0, as desired. □

Proposition 4.2. Let P be a Sylow p-subgroup of G, and let W be a finitely generated kG-module. If res_{G,Z(P)}^{∗}(V_{Z(P)}(k)) ⊈ V_{G}(W), then Tr_{W} (Ext_{kG}^{∗}(W, W)) = 0.

Proof. Suppose that res_{G,Z(P)}(ζ) is nilpotent for every homogeneous ζ ∈ 𝔍(W). Then 𝔍(W) ⊆ res_{G,Z(P)}^{∗}(𝔪) for all 𝔪 ∈ V_{Z(P)}(k), so res_{G,Z(P)}^{∗}(V_{Z(P)}(k)) ⊆ V_{G}(W). But this contradicts the assumptions of the proposition, so we conclude that there is a homogeneous element ζ ∈ 𝔍(W) such that res_{G,Z(P)}(ζ) is not nilpotent. Then the techniques of [7] imply that ζ is a regular element in H^{∗}(G, k), and Tr_{W} (Ext_{kG}^{∗}(W, W)) = 0 by Proposition 4.1. This completes the proof. □

The following result is an immediate consequence of Proposition 4.2.

Corollary 4.3. Suppose that G is an abelian p-group and W is a finitely generated module such that V_{G}(W) ≠ V_{G}(k). Then Tr_{W} (Ext_{kG}^{∗}(W, W)) = 0.

Proposition 4.4. Let W and M be finitely generated modules, and suppose that ζ ∈ Tr_{W} (Ext_{kG}^{n}(W, W)) is an element such that V_{G}(M) ∩ V_{G}(ζ) = 0. Then M is relatively W -projective.

Proof. By abuse of notation we also write ζ for the unique kG-homomorphism from Ω^{n}(k) to k representing ζ ∈ H^{n}(G, k). Recall that L_{ζ} denotes the kernel of this homomorphism so that there is a short exact sequence

0 ⟶ L_{ζ} ⟶ Ω^{n}(k) ⟶^{ζ} k ⟶ 0.

Because V_{G}(L_{ζ} ⊗ M) = V_{G}(ζ) ∩ V_{G}(M) = 0, it follows that L_{ζ} ⊗ M is projective. Hence the map ζ ⊗ 1_{M} : Ω^{n}(k) ⊗ M → M is a split epimorphism. Let γ : M → Ω^{n}(k) ⊗ M be a map such that (ζ ⊗ 1_{M})γ = 1_{M}. Because ζ ∈ Tr_{W} (Ext_{kG}^{n}(W, W)), it follows that ζ ⊗ 1_{M} factors through a W -projective module and hence so does 1_{M}. Thus M is relatively W -projective, as desired. □

Corollary 4.5. Assume that W is finitely generated. If M is a finitely generated periodic module such that

V_{G}(M) ∩ V_{G}(Tr_{W} (Ext_{kG}^{∗}(W, W))) = 0,

then M is relatively W -projective.

Proof. We may assume that M is indecomposable. Because it is periodic, V_{G}(M) is a line, and there is an element ζ ∈ Tr_{W} (Ext_{kG}^{∗}(W, W)) such that V_{G}(M) ∩ V_{G}(ζ) = 0. Hence M is W -projective by Proposition 4.4, as desired. □

The reader may find it interesting to compare the results of this section with the standard theorems on the image of the transfer map in [9] or in the generalization [3].

For example, suppose that W = ⨁ k_{H}^{↑G}, where the sum runs over all subgroups H ⊆ G such that p divides |G : H|. If P is a Sylow p-subgroup of G, then [9] shows that

V_{G}(Tr_{W} (Ext_{kG}^{∗}(W, W))) = res_{G,Z(P)}^{∗}(V_{Z(P)}(k)).

5. Virtual projectivity

In the last section Corollary 4.5 gave a condition on varieties that forces a finitely generated periodic module M to be W -projective, and it is natural to try to find an analogue of this corollary when M is not periodic. As we will see in Theorem 5.1 below, the condition used in Corollary 4.5 gives a weaker conclusion when M is not periodic. We have referred to the resulting property as virtual W -projectivity, and this concept is the main subject of study in this section.

Theorem 5.1. Let W and M be finitely generated modules such that

V_{G}(M) ∩ V_{G}(Tr_{W} (Ext_{kG}^{∗}(W, W))) = 0.

If N is any finitely generated kG-module, then

Ext_{kG}^{n}(M, N) = Tr_{W} (Ext_{kG}^{n}(M ⊗ W, N ⊗ W))

for sufficiently large values of n.

Proof. Suppose that ζ_{1}, …, ζ_{m} ∈ H^{∗}(G, k) are homogeneous generators of the ideal Tr_{W} (Ext_{kG}^{∗}(W, W)). For 1 ≤ i ≤ m let C^{(i)} be the complex

C^{(i)} :  ⋯ ⟶ 0 ⟶ Ω^{n_{i}}(k) ⟶^{ζ_{i}} k ⟶ 0 ⟶ ⋯

with Ω^{n_{i}}(k) in degree one and k in degree zero, and set C = C^{(1)} ⊗ ⋯ ⊗ C^{(m)}. Then C is exact except in degree m, and H_{m}(C) = ⊗_{i=1}^{m}L_{ζ_{i}} so that there is an exact sequence

0 → ⨂_{i=1}^{m} L_{ζ_{i}} → C_{m} → ⋯ → C_{0} → 0.

Because

V_{G}(⨂_{i=1}^{m} L_{ζ_{i}} ⊗ M) = ⋂_{i=1}^{m} V_{G}(ζ_{i}) ∩ V_{G}(M) = V_{G}(ζ_{1}, …, ζ_{m}) ∩ V_{G}(M) = 0,

we conclude that ⊗_{i=1}^{m}L_{ζ_{i}} ⊗ M is projective. It follows that the cokernel C′_{m} of the map ⊗_{i=1}^{m}L_{ζ_{i}} ⊗ M → C_{m} ⊗ M is a summand of C_{m} ⊗ M, and there is an exact sequence

C′_{∗} : 0 → C′_{m} → C_{m−1} ⊗ M → ⋯ → C_{0} ⊗ M → 0.

Now let N → I^{∗} be an injective coresolution of N. Then there is a spectral sequence with E_{0}^{st} = Hom_{kG}(C′_{s}, I^{t}). The E_{1} page is given by taking homology with respect to the differential Hom_{kG}(1_{C′}, ∂), so E_{1}^{st} ≅ Ext_{kG}^{t}(C′_{s}, N). If t > 0, then it is easy to see that Ext_{kG}^{t}(C′_{∗}, N) is isomorphic to a summand of the Koszul complex of the H^{∗}(G, k)-module Ext_{kG}^{∗}(M, N) with respect to the elements ζ_{1}, …, ζ_{m}. It follows from Theorem 2.3 of [10] that E_{1}^{∗,t} ≅ Ext_{kG}^{t}(C′_{∗}, N) is exact for sufficiently large values of t. In particular, the map

⨁_{i=1}^{m} Ext_{kG}^{t−n_{i}}(M, N) ⟶^{(ζ_{1},...,ζ_{m})} Ext_{kG}^{t}(M, N)

is onto. If f ∈ Ext_{kG}^{t}(M, N), then there are elements g_{i} ∈ Ext_{kG}^{t−n_{i}}(M, N) for 1 ≤ i ≤ m such that f = g_{1}ζ_{1} + ⋯ + g_{m}ζ_{m}. Let γ_{i} ∈ Ext_{kG}^{∗}(W, W) be an element such that ζ_{i} = Tr_{W} (γ_{i}) for 1 ≤ i ≤ m. Then 1_{M} · ζ_{i} = 1_{M} · Tr_{W} (γ_{i}) = Tr_{W} (1_{M} ∪ γ_{i}), so

f = g_{1}ζ_{1} + ⋯ + g_{m}ζ_{m}
= g_{1} Tr_{W} (1_{M} ∪ γ_{1}) + ⋯ + g_{m} Tr_{W} (1_{M} ∪ γ_{m})
= Tr_{W} ((g_{1} ∪ γ_{1}) + ⋯ + (g_{m} ∪ γ_{m})).

Hence Ext_{kG}^{t}(M, N) = Tr_{W} (Ext_{kG}^{t}(M ⊗ W, N ⊗ W)) for sufficiently large values of t, and this completes the proof. □

Theorem 5.1 suggests the following definition.

Definition 5.2. Let W and M be kG-modules, and assume that W is finitely generated. We will say that M is virtually W -projective if

Ext_{kG}^{n}(M, M) = Tr_{W} (Ext_{kG}^{n}(M ⊗ W, M ⊗ W))

for all sufficiently large values of n. If H is a subgroup of G such that M is virtually W -projective for W = k_{H}^{↑G}, then we also say that M is virtually H-projective.

If we set Im Tr_{W} = Tr_{W} (Ext_{kG}^{∗}(W, W)), then we can use this definition to rephrase Theorem 5.1 as follows.

Proposition 5.3. Let U be the open set defined by setting U = V_{G}(k) − V_{G}(Im Tr_{W}). If M is a finitely generated kG-module with V_{G}(M) ⊆ U ∪ {0}, then M is virtually W - projective.

In Section 7 we give examples of virtually W -projective modules that are not W - projective. It is easy to see, therefore, that the class of W -projective modules is strictly smaller than the class of virtually W -projectives. One would expect that some of the basic properties of W -projective modules should extend to the larger class of modules, and the next result shows that this expectation is correct.

Proposition 5.4. Let W and M be finitely generated kG-modules, and suppose that M is virtually W -projective. Then V_{G}(M) ⊆ V_{G}(W). Moreover, if N is any finitely generated kG-module, then M ⊗ N is virtually W -projective.

Proof. Because M is virtually W -projective, Proposition 4.1 shows that 𝔍(W) annihilates Ext_{kG}^{n}(M, M) for any sufficiently large integer n. Then 𝔍(W)^{n+1} ⊆ 𝔍(M) so that V_{G}(M) ⊆ V_{G}(𝔍(W)^{n+1}) = V_{G}(W).

To prove the second statement, note that Ext_{kG}^{∗}(M ⊗N, M ⊗N) is a finitely generated right module over Ext_{kG}^{∗}(M, M). Let f_{1}, …, f_{s} be homogeneous generators, and set d = max{deg f_{i} | 1 ≤ i ≤ s}. Let N_{0} be an integer such that if n ≥ N_{0}, then Ext_{kG}^{n}(M, M) = Tr_{W} (Ext_{kG}^{n}(M ⊗ W, M ⊗ W)). If n ≥ N_{0} + d and g ∈ Ext_{kG}^{n}(M ⊗ N, M ⊗ N), then there are elements ξ_{i} ∈ Ext_{kG}^{n_{i}}(M, M) for 1 ≤ i ≤ s with

g = f_{1} · (ξ_{1} ∪ 1_{N}) + ⋯ + f_{s} · (ξ_{s} ∪ 1_{N}).

Because n_{i} = n − deg f_{i} ≥ n − d ≥ N_{0}, it follows that ξ_{i} = Tr_{W} (ζ_{i}) for some ζ_{i} ∈ Ext_{kG}^{n_{i}}(M ⊗ W, M ⊗ W), and

g = ∑_{i=1}^{s} f_{i} · (Tr_{W} (ζ_{i}) ∪ 1_{N}) = Tr_{W} (∑_{i=1}^{s}(f_{i} ∪ 1_{W}) · (ζ_{i} ∪ 1_{N})).

Thus M ⊗ N is virtually W -projective, as desired. □

Proposition 5.5. Let W and M be finitely generated kG-modules. Then the following statements are equivalent:

(1) M is virtually W -projective;

(2) if N is any finitely generated kG-module, then

Ext_{kG}^{n}(M, N) = Tr_{W} (Ext_{kG}^{n}(M ⊗ W, N ⊗ W))

for sufficiently large values of n;

(3) if N is any finitely generated kG-module, then

Ext_{kG}^{n}(N, M) = Tr_{W} (Ext_{kG}^{n}(N ⊗ W, M ⊗ W))

for sufficiently large values of n.

Proof. It is clear that either (2) or (3) implies (1). To show that (1) implies (2), let N be a finitely generated kG-module. Suppose that N_{0} is an integer such that Ext_{kG}^{n}(M, M) = Tr_{W} (Ext_{kG}^{n}(M ⊗ W, M ⊗ W)) for all n ≥ N_{0}. The right module Ext_{kG}^{∗}(M, N) is finitely generated over the ring Ext_{kG}^{∗}(M, M), so we can choose homogeneous generators f_{1}, …, f_{s} ∈ Ext_{kG}^{∗}(M, N). Set d = max{deg f_{i} | 1 ≤ i ≤ s}. If n ≥ N_{0} + d and g ∈ Ext_{kG}^{n}(M, N), then there are elements ξ_{i} ∈ Ext_{kG}^{n_{i}}(M, M) for 1 ≤ i ≤ s with g = f_{1}ξ_{1} + ⋯ + f_{s}ξ_{s}. Moreover, n_{i} = n − deg f_{i} ≥ n − d ≥ N_{0}, so ξ_{i} = Tr_{W} (ζ_{i}) for some ζ_{i} ∈ Ext_{kG}^{n_{i}}(M ⊗ W, M ⊗ W). Then

g = f_{1} · Tr_{W} (ζ_{1}) + ⋯ + f_{s} · Tr_{W} (ζ_{s}) = Tr_{W} ((f_{1} ∪ 1_{W})ζ_{1} + ⋯ + (f_{s} ∪ 1_{W})ζ_{s})

so that Ext_{kG}^{n}(M, N) = Tr_{W} (Ext_{kG}^{n}(M ⊗ W, N ⊗ W)), and (2) holds. The proof that (1) implies (3) is similar and is omitted. □

Proposition 5.6. Let W, X, M, and N be finitely generated kG-modules. (1) If X is W -projective, then

Tr_{X}(Ext_{kG}^{n}(M ⊗ X, N ⊗ X)) ⊆ Tr_{W} (Ext_{kG}^{n}(M ⊗ W, N ⊗ W))

for n ≥ 0.

(2) If X is virtually W -projective, then

Tr_{X}(Ext_{kG}^{n}(M ⊗ X, N ⊗ X)) ⊆ Tr_{W} (Ext_{kG}^{n}(M ⊗ W, N ⊗ W))

for sufficiently large values of n.

Proof. To prove (1), let ξ ∈ Ext_{kG}^{n}(M ⊗ X, N ⊗ X). If f : Ω^{n}(k) ⊗ M → N is a kGhomomorphism representing Tr_{X}(ξ), then f factors through a relatively X-projective module by Proposition 3.2. But if X is W -projective, then any X-projective module is also W -projective. Thus Tr_{X}(ξ) ∈ Tr_{W} (Ext_{kG}^{n}(M ⊗ W, N ⊗ W)), as desired.

To prove (2), suppose that X is virtually W -projective. Then N ⊗ X is virtually W -projective, so the map

Ext_{kG}^{n}(M ⊗ X ⊗ W, N ⊗ X ⊗ W) ⟶^{Tr_{W}} Ext_{kG}^{n}(M ⊗ X, N ⊗ X)

is an epimorphism for sufficiently large values of n by Proposition 5.5. Hence

Tr_{X}(Ext_{kG}^{n}(M ⊗ X, N ⊗ X)) = Tr_{X} Tr_{W} (Ext_{kG}^{n}(M ⊗ X ⊗ W, N ⊗ X ⊗ W))
= Tr_{X⊗W} (Ext_{kG}^{n}(M ⊗ X ⊗ W, N ⊗ X ⊗ W))
= Tr_{W} Tr_{X}(Ext_{kG}^{n}(M ⊗ W ⊗ X, N ⊗ W ⊗ X))
⊆ Tr_{W} (Ext_{kG}^{n}(M ⊗ W, N ⊗ W)),

and this completes the proof. □

Suppose that W, X, and M are finitely generated kG-modules. If M | W ⊗ W^{∗} ⊗ M and M | X ⊗ X^{∗} ⊗ M, then certainly M | W ⊗ X ⊗ W^{∗} ⊗ X^{∗} ⊗ M. In other words, if M is both W -projective and X-projective, then it is (W ⊗X)-projective. The next proposition shows that the analogous result also holds for virtually projective modules.

Proposition 5.7. Let W, X, and M be finitely generated kG-modules. If M is virtually W -projective and virtually X-projective, then it is virtually (W ⊗ X)-projective.

Proof. Choose homogeneous generators ξ_{1}, …, ξ_{s} for the k-algebra Ext_{kG}^{∗}(M, M), and set d = max{deg ξ_{i} | 1 ≤ i ≤ s}. Let N_{0} ≥ 0 be an integer such that if n ≥ N_{0}, then Ext_{kG}^{n}(M, M) = Tr_{W} (Ext_{kG}^{n}(M ⊗ W, M ⊗ W)) = Tr_{X}(Ext_{kG}^{n}(M ⊗ X, M ⊗ X)). Suppose that n ≥ 2N_{0}+d and ζ ∈ Ext_{kG}^{n}(M, M). Then ζ is a linear combination of elements of the form ξ_{i_{1}} ⋯ ξ_{i_{l}}. For some t with 1 ≤ t ≤ l the elements α = ξ_{i_{1}} ⋯ ξ_{i_{t}} and β = ξ_{i_{t+1}} ⋯ ξ_{i_{l}} satisfy deg α ≥ \frac{n−d}{2} and deg β ≥ \frac{n−d}{2}. Then deg α ≥ N_{0} and deg β ≥ N_{0}, so there are homogeneous elements α′ ∈ Ext_{kG}^{∗}(M ⊗ X, M ⊗ X) and β′ ∈ Ext_{kG}^{∗}(M ⊗ W, M ⊗ W) such that α = Tr_{X}(α′) and β = Tr_{W} (β′). Proposition 3.2 and Corollary 3.6 now imply that ξ_{i_{1}} ⋯ ξ_{i_{l}} = αβ ∈ Tr_{W ⊗X}(Ext_{kG}^{n}(M ⊗W ⊗X, M ⊗W ⊗X)). Hence Ext_{kG}^{n}(M, M) = Tr_{W ⊗X}(Ext_{kG}^{n}(M ⊗ W ⊗ X, M ⊗ W ⊗ X)), and M is virtually (W ⊗ X)-projective, as desired. □

6. Triangulating the relative stable category

If W is a finitely generated kG-module, then one can define an analogue of the stable category in which all W -projective modules are isomorphic to zero. The purpose of this section is to show that the resulting W -stable category is triangulated and that the finitely generated virtually W -projective modules form a thick subcategory. Although group theorists have studied cohomology relative to a subgroup H for many years, the observation that relative stable categories are triangulated does not seem to have appeared in print even for the H-stable category.

Lemma 6.1. Suppose that W is a finitely generated kG-module. Consider a commutative diagram of modules

0 ⟶ M′ ⟶ M ⟶^{g} M′′ ⟶ 0
↓  ↓^{γ}  ∥
0 ⟶ N′ ⟶ N ⟶_{h} M′′ ⟶ 0

in which both rows are short exact sequences. If the top row is W -split, then the bottom row is also W -split.

Proof. If the top row is W -split, then there is a homomorphism β : W ⊗ M′′ → W ⊗ M such that (1_{W} ⊗ g)β = 1_{W ⊗M′′}. Then (1_{W} ⊗ h)(1_{W} ⊗ γ)β = (1_{W} ⊗ g)β = 1_{W ⊗M′′}, so the bottom row is also W -split. □

Let W be a finitely generated module. If M and N are kG-modules, let PHom_{kG}^{W}(M, N) be the subspace of Hom_{kG}(M, N) consisting of all homomorphisms that factor through a W -projective module. Define the W -stable category of kG to be the category \underline{Mod}^{W} (kG) in which the objects consist of right kG-modules, and the morphisms from M to N are defined by setting

\underline{Hom}_{kG}^{W}(M, N) = Hom_{kG}(M, N)/ PHom_{kG}^{W}(M, N).

The full subcategory of \underline{Mod}^{W} (kG) consisting of finitely generated kG-modules is denoted \underline{mod}^{W} (kG).

Let α_{W} : W^{∗} ⊗ W → k denote the evaluation map. Because α_{W} is W -split, the dual map α_{W}^{∗} : k → W ⊗ W^{∗} is a W -split monomorphism. For any module M set I_{W} (M) = W ⊗ W^{∗} ⊗ M and ι_{M} = α_{W}^{∗} ⊗ 1_{M} : M → I_{W} (M). If f : M → N is a kG-homomorphism, then there is a commutative diagram of short exact sequences

0 ⟶ M ⟶^{ι_{M}} I_{W} (M) ⟶ Ω_{W}^{−1}(M) ⟶ 0
f↓  ↓  ∥
0 ⟶ N ⟶_{g} L ⟶_{h} Ω_{W}^{−1}(M) ⟶ 0

in Mod(kG). Letting \underline{f} denote the image of f in \underline{Hom}_{kG}^{W}(M, N), we obtain a sequence of modules and maps

M ⟶^{\underline{f}} N ⟶^{\underline{g}} L ⟶^{\underline{h}} Ω_{W}^{−1}(M)

in \underline{Mod}^{W} (kG), called a standard triangle. This standard triangle can also be written as a sextuple (M, N, L, \underline{f}, \underline{g}, \underline{h}). More generally, any sextuple isomorphic to a standard triangle will be called a distinguished triangle, or usually just a triangle for simplicity.

Let T denote the collection of all triangles in \underline{Mod}^{W} (kG). By using Lemma 6.1 and essentially the same argument given by Happel for the usual stable category (Theorem I.2.6 of [14]), one obtains the following result.

Theorem 6.2. The collection of sextuples in T is a triangulation of \underline{Mod}^{W} (kG), and \underline{mod}^{W} (kG) is a triangulated subcategory of \underline{Mod}^{W} (kG).

The W -stable category \underline{Mod}^{W} (kG) behaves largely like the stable category \underline{Mod}(kG). For example, the relationship between triangles in \underline{Mod}^{W} (kG) and short exact sequences in Mod(kG) generalizes that for the usual stable category. The proof of the following result is left to the reader; it is a simple matter of combining Lemma 6.1 above with the argument used to prove the analogous result for the stable category (see Lemma I.2.7 of [14]).

Proposition 6.3. Let W be a finitely generated kG-module, and suppose that

0 ⟶ M′ ⟶^{f} M ⟶^{g} M′′ ⟶ 0

is a W -split short exact sequence in Mod(kG). Then there is a triangle in \underline{Mod}^{W} (kG) of the form

M′ ⟶^{\underline{f}} M ⟶^{\underline{g}} M′′ ⟶ Ω_{W}^{−1}(M′).

Moreover, every triangle in \underline{Mod}^{W} (kG) is isomorphic to one that arises from a W -split short exact sequence in this way.

Recall that a subcategory C of a triangulated category D is said to be thick if C is a triangulated subcategory and is closed under taking direct summands of objects.

Theorem 6.4. Let W be a finitely generated kG-module. Then the finitely generated virtually W -projective modules form a thick subcategory of \underline{mod}^{W} (kG).

Proof. It is clear that any summand of a finitely generated virtually W -projective module is again virtually W -projective, so it suffices to prove that the virtually W -projective modules form a triangulated subcategory of \underline{mod}^{W} (kG). Let M′ and M′′ be virtually W -projective modules in \underline{mod}^{W} (kG), and suppose that there is a W -split short exact sequence

E :  0 ⟶ M′ ⟶^{f} M ⟶^{g} M′′ ⟶ 0.

By Proposition 6.3 it is enough to show that M is also virtually W -projective.

Let N be any finitely generated kG-module. Then the exact sequence E gives rise to a commutative diagram

Ext^{n}(M′′ ⊗ W, N ⊗ W) ⟶^{(g⊗1)^{∗}} Ext^{n}(M ⊗ W, N ⊗ W) ⟶^{(f⊗1)^{∗}} Ext^{n}(M′ ⊗ W, N ⊗ W)
Tr_{W} ↓  ^{Tr}_{W} ↓  ↓^{Tr}_{W}
Ext_{kG}^{n}(M′′, N)  ⟶_{g^{∗}}  Ext_{kG}^{n}(M, N)  ⟶_{f^{∗}}  Ext_{kG}^{n}(M′, N).

If n is sufficiently large, then Tr_{W} (Ext_{kG}^{n}(M′′ ⊗ W, N ⊗ W)) = Ext_{kG}^{n}(M′′, N) and Tr_{W} (Ext_{kG}^{n}(M′ ⊗ W, N ⊗ W)) = Ext_{kG}^{n}(M′, N). If α ∈ Ext_{kG}^{n}(M, N), then there is a β ∈ Ext_{kG}^{n}(M′ ⊗ W, N ⊗ W) such that Tr_{W} (β) = f^{∗}(α). Because E is W -split, there is a map h : M ⊗ W → M′ ⊗ W such that h(f ⊗ 1_{W}) = 1_{M′⊗W}. Then

f^{∗}(Tr_{W} (h^{∗}(β))) = Tr_{W} ((f ⊗ 1_{W})^{∗}h^{∗}(β)) = Tr_{W} (β) = f^{∗}(α).

Hence f^{∗}(α − Tr_{W} (h^{∗}(β))) = 0, and there is an element γ ∈ Ext_{kG}^{n}(M′′ ⊗ W, N ⊗ W) such that

α − Tr_{W} (h^{∗}(β)) = g^{∗}(Tr_{W} (γ)) = Tr_{W} ((g ⊗ 1_{W})^{∗}(γ)).

Thus α = Tr_{W} (h^{∗}(β)+(g⊗1_{W})^{∗}(γ)) so that Tr_{W} (Ext_{kG}^{n}(M⊗W, N⊗W)) = Ext_{kG}^{n}(M, N), and this completes the proof. □

In [15] Rickard showed that if C is any thick subcategory of \underline{mod}(kG), then there are corresponding idempotent functors E_{C} and F_{C} from \underline{Mod}(kG) to \underline{Mod}(kG). These functors are given by tensoring with modules E_{C}(k) and F_{C}(k) that are idempotent in the sense that E_{C}(k) ⊗ E_{C}(k) ≅ E_{C}(k) and F_{C}(k) ⊗ F_{C}(k) ≅ F_{C}(k) in \underline{Mod}(kG). Although some modifications are necessary, much of Rickard’s theory developed in Sections 4 and 5 of [15] generalizes to the W -stable category. In particular, one can prove the following result.

Proposition 6.5. Let W be a finitely generated kG-module, and let C be a thick subcategory of \underline{mod}^{W} (kG). If M is any kG-module, then there is a triangle

T^{W} (M) : E_{C}^{W} (M) ⟶ M ⟶ F_{C}^{W} (M) ⟶ Ω_{W}^{−1}(E_{C}^{W} (M))

in \underline{Mod}^{W} (kG) satisfying the following conditions:

(1) The module E_{C}^{W} (M) lies in the smallest full triangulated subcategory of \underline{Mod}^{W} (kG) that contains C and is closed under taking arbitrary direct sums.

(2) The module F_{C}^{W} (M) lies in the subcategory of \underline{Mod}^{W} (kG) consisting of those modules N such that \underline{Hom}_{kG}^{W}(C, N) = 0 for all C in C.

Moreover, E_{C}^{W} : \underline{Mod}^{W} (kG) → \underline{Mod}^{W} (kG) and F_{C}^{W} : \underline{Mod}^{W} (kG) → \underline{Mod}^{W} (kG) are idempotent functors.

In the case of Proposition 6.5 it is not clear whether the functors E_{C} and F_{C} are always given by tensoring with a kG-module. This may be the subject of further investigation.

7. Examples

In this section we present some examples and related results that have served as motivation for parts of this study and might be of further interest. In particular, we describe a finitely generated module W such that k is virtually W -projective but not W -projective. Another example shows that even for a subgroup H of G virtual H-projectivity is a weaker condition than relative H-projectivity. In both situations many examples are possible in addition to those we have chosen to present.

Example 7.1. There exist a group G and a finitely generated kG-module W such that k is virtually W -projective but not W -projective.

Let p = 3, and let G = ⟨g_{1}, g_{2}⟩ be an elementary abelian 3-group of rank 2. Then Ω^{2}(k) is a 10-dimensional kG-module with generators u, v_{1}, v_{2} satisfying (g_{1} −1)v_{1} = (g_{2} −1)^{2}u and (g_{2} − 1)v_{2} = (g_{1} − 1)^{2}u. The element u generates a submodule of dimension 8, and both v_{1} and v_{2} generate submodules of dimension 3.

Let W be the quotient of Ω^{2}(k) by the 1-dimensional submodule spanned by (g_{1}−1)^{2}v_{1}. Then W is a 9-dimensional indecomposable kG-module, so k is not a summand of W ⊗W^{∗} by Theorem 2.1 of [5] and hence k is not W -projective by Proposition 2.4. On the other hand, every homomorphism from Ω^{2}(k) to k is W -projective. Moreover, if n ≥ 2, then H^{n}(G, k) = H^{2}(G, k) · H^{n−2}(G, k) so that every homomorphism from Ω^{n}(k) to k is W - projective. Thus k is virtually W -projective but not W -projective. □

Example 7.1 is a special case of the general fact described in the next proposition. The proof is essentially the same as that given in the discussion above.

Proposition 7.2. Suppose that ζ_{1}, …, ζ_{n} ∈ H^{t}(G, k) are elements such that H^{∗}(G, k) is a finitely generated module over the subring generated by ζ_{1}, …, ζ_{n}. Let M be a submodule of rad Ω^{t}(k) such that W = Ω^{t}(k)/M is indecomposable and has dimension divisible by p. Then every finitely generated kG-module is virtually W -projective. Moreover, if N is a finitely generated W -projective module, then p divides dim N.

It is interesting to note that Example 7.1 has no analogue for projectivity relative to a subgroup. Indeed, it is easy to prove the following result.

Proposition 7.3. Let G be a p-group, and let 𝔛 be any collection of proper subgroups of G. Then k is not virtually 𝔛-projective.

Proof. It suffices to prove the result when 𝔛 is the collection of all proper subgroups of G. Set

I = ∑_{X∈𝔛} Tr_{X}(Ext_{kG}^{∗}(X, X)).

Then I is a graded ideal of H^{∗}(G, k), and V_{G}(I) = res_{G,Z(G)}^{∗}(V_{Z(G)}(k)) by Theorem A of [9]. In particular, V_{G}(I) ≠ {0} so that I does not contain all elements of H^{∗}(G, k) of sufficiently high degree. Hence k is not virtually 𝔛-projective, as desired. □

Although Theorem 6.4 shows that the virtually W -projective modules form a thick subcategory of \underline{mod}^{W} (kG), the previous result shows that they do not generally form a thick subcategory of \underline{mod}(kG). Indeed, suppose that G is a p-group that is not elementary abelian, E is the collection of all maximal elementary abelian p-subgroups of G, and

W = ⨁_{E∈E} k_{E}^{↑G}.

Then V_{G}(W) = V_{G}(k), and Proposition 3.2 of [6] implies that the smallest thick subcategory of \underline{mod}(kG) containing W is all of \underline{mod}(kG). But Proposition 7.3 shows that k is not virtually W -projective, so the virtually W -projective modules do not form a thick subcategory of \underline{mod}(kG).

Example 7.4. Let G be a Sylow 2-subgroup of SL(3, 4). Then there exist a subgroup H of G and a finitely generated kG-module that is virtually H-projective but not H-projective.

To describe such an example, we write the field 𝔽_{4} of four elements as 𝔽_{2}(α) for some α ∈ 𝔽_{4} with α^{2} + α + 1 = 0. We may assume that G ⊆ SL(3, 4) is generated by the matrices

x_{1} = [ \begin{matrix}1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ] , x_{2} = [ \begin{matrix}1 & α & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ] ,

y_{1} = [ \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix} ] , y_{2} = [ \begin{matrix}1 & 0 & 0 \\ 0 & 1 & α \\ 0 & 0 & 1 \end{matrix} ] ,

z_{1} = [ \begin{matrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ] , z_{2} = [ \begin{matrix}1 & 0 & α \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} ] .

The elements z_{1} and z_{2} generate the center of G, and the other elements satisfy the relations

y_{1}x_{1}y_{1} = x_{1}z_{1}, y_{1}x_{2}y_{1} = x_{2}z_{2}, y_{2}x_{1}y_{2} = x_{1}z_{2}, y_{2}x_{2}y_{2} = x_{2}z_{1}z_{2}.

Let H = ⟨x_{1}, x_{2}, z_{1}, z_{2}⟩ so that H is an elementary abelian 2-group of order 16. Then H^{∗}(H, k) = k[ζ_{1}, ζ_{2}, ζ_{3}, ζ_{4}] is a polynomial ring in four generators of degree one. Set x_{3} = z_{1} and x_{4} = z_{2}. If we identify H^{1}(H, k) with Hom(H, k), then we may assume that ζ_{i}(x_{j}) = 1 if i = j and ζ_{i}(x_{j}) = 0 if i ≠ j. The action of G on the cohomology ring of H fixes ζ_{1} and ζ_{2}, and

\begin{aligned}ζ_{3}^{y_{1}} & = ζ_{1} + ζ_{3}, \\ ζ_{3}^{y_{2}} & = ζ_{2} + ζ_{3}, \end{aligned}  \begin{aligned}ζ_{4}^{y_{1}} & = ζ_{2} + ζ_{4}, \\ ζ_{4}^{y_{2}} & = ζ_{1} + ζ_{2} + ζ_{4}. \end{aligned}

Now identify H with the additive group of 𝔽_{2}^{4} and V_{H}(k) with the affine space k^{4} ≅ k⊗_{𝔽_{2}}H so that the actions of y_{1} and y_{2} on V_{H}(k) are given by

(a, b, c, d)^{y_{1}} = (a, b, a + c, b + d),
(a, b, c, d)^{y_{2}} = (a, b, b + c, a + b + d).

Set I = Tr_{H}^{G}(H^{∗}(H, k)). Using the fact that res_{G,H}(Tr_{H}^{G}(ζ_{3}^{3})) = ζ_{1}^{2}ζ_{2} + ζ_{1}ζ_{2}^{2} and res_{G,H}(Tr_{H}^{G}(ζ_{3}ζ_{4})) = ζ_{1}^{2} + ζ_{1}ζ_{2} + ζ_{2}^{2}, one can show that V_{G}(I) = res_{G,H}^{∗}({(0, 0, c, d) | c, d ∈ k}).

Let E_{1} = ⟨x_{1}, x_{2}⟩ and E_{2} = ⟨x_{1}z_{1}, x_{2}⟩, and set M_{1} = k_{E_{1}}^{↑G} and M_{2} = k_{E_{2}}^{↑G}. If V_{1} and V_{2} are the subvarieties of V_{H}(k) given by V_{1} = {(a, b, 0, 0) | a, b ∈ k} and V_{2} = {(a, b, a, 0) | a, b ∈ k}, then V_{G}(M_{1}) = res_{G,H}^{∗}(V_{1}) and V_{G}(M_{2}) = res_{G,H}^{∗}(V_{2}). Using computer calculations, one can construct a finitely generated kG-module N that arises as an extension

0 ⟶ M_{1} ⟶ N ⟶ M_{2} ⟶ 0.

For one such calculation the resulting module N has the property that the restriction N↓_{H} is indecomposable. In this case, then, N cannot be relatively H-projective. On the other hand, V_{G}(N) ⊆ V_{G}(M_{1}) ∪ V_{G}(M_{2}) = res_{G,H}^{∗}(V_{1} ∪ V_{2}) so that

V_{G}(N) ∩ V_{G}(I) ⊆ res_{G,H}^{∗}(V_{1} ∪ V_{2}) ∩ res_{G,H}^{∗}({(0, 0, c, d) | c, d ∈ k}) = {0}.

Thus Theorem 5.1 implies that N is virtually H-projective, as desired. □

In [12] a nonzero homogeneous element ζ ∈ H^{∗}(G, k) is said to be productive if it annihilates Ext_{kG}^{∗}(L_{ζ}, L_{ζ}). We end with a result showing that when ζ is productive, L_{ζ^{2}} is not usually L_{ζ}-projective.

Proposition 7.5. Let ζ be a homogeneous element of H^{∗}(G, k) that is not nilpotent. If ζ is productive and L_{ζ^{2}} is L_{ζ}-projective, then G has p-rank one.

Proof. Let n = deg ζ. We will prove by induction on t that L_{ζ^{t}} is L_{ζ}-projective. There is nothing to prove when t = 1, so assume that t > 1 and that L_{ζ^{t−1}} is L_{ζ}-projective. By the Octahedral Axiom there is a commutative diagram

Ω^{nt+1}(k) ⟶^{Ω^{n+1}(ζ^{t−1})} Ω^{n+1}(k) ⟶ Ω^{n}(L_{ζ^{t−1}}) ⟶ Ω^{nt}(k)
∥  ^{Ω(ζ)}↓  ↓  ∥
Ω^{nt+1}(k) ⟶^{Ω(ζ^{t})}  Ω(k) ⟶  L_{ζ^{t}}  ⟶ Ω^{nt}(k)
↓  ↓
L_{ζ}  L_{ζ}
↓  ↓
Ω^{n}(k) ⟶ Ω^{n−1}(L_{ζ^{t−1}})

in which all rows and columns are triangles in \underline{mod}(kG). Because L_{ζ^{t−1}} is L_{ζ}-projective and ζ is productive, ζ annihilates the cohomology of L_{ζ^{t−1}}. But ζ also annihilates the cohomology of L_{ζ}, so the existence of the triangle

Ω^{n}(L_{ζ^{t−1}}) ⟶ L_{ζ^{t}} ⟶ L_{ζ} ⟶ Ω^{n−1}(L_{ζ^{t−1}})

implies that ζ^{2} annihilates the cohomology of L_{ζ^{t}}. In particular, ζ^{2} annihilates the cohomology of L_{ζ^{2}} so that ζ^{2} is productive. Thus L_{ζ^{t}} is L_{ζ^{2}}-projective by Proposition 4.7 of [12], so L_{ζ^{t}} is also L_{ζ}-projective, as desired. Proposition 5.9.5 of [2] now implies that there are stable isomorphisms

Ω(L_{ζ}) ⊕ Ω^{nt}(L_{ζ}) ≅ L_{ζ^{t}} ⊗ L_{ζ} ≅ Ω(L_{ζ^{t}}) ⊕ Ω^{n}(L_{ζ^{t}}).

Write V_{G}(ζ) = V_{1} ∪ ⋯ ∪ V_{s}, where V_{i} ∩ V_{j} = {0} if i ≠ j and each V_{i} is a nonzero closed homogeneous subvariety of V_{G}(ζ) such that the corresponding projective variety V̄_{i} is connected. Because ζ is not nilpotent, V_{G}(ζ) is a proper subvariety of V_{G}(k). Thus by Lemma 4.1 of [8] there are indecomposable modules M_{1}, …, M_{s} such that L_{ζ} = M_{1} ⊕⋯⊕M_{s} and V_{G}(M_{i}) = V_{i}. Similarly, there are indecomposable modules N_{1}, …, N_{s} such that L_{ζ^{t}} = N_{1}⊕⋯⊕N_{s} and V_{G}(N_{i}) = V_{i}. Then the isomorphism Ω(L_{ζ})⊕Ω^{nt}(L_{ζ}) ≅ Ω(L_{ζ^{t}}) ⊕ Ω^{n}(L_{ζ^{t}}) implies that

Ω(M_{i}) ⊕ Ω^{nt}(M_{i}) ≅ Ω(N_{i}) ⊕ Ω^{n}(N_{i})

for 1 ≤ i ≤ s. Hence either M_{i} ≅ N_{i} ≅ Ω^{n(t−1)}(M_{i}) or M_{i} ≅ Ω^{n−1}(N_{i}) ≅ Ω^{n(t+1)−2}(M_{i}), and M_{i} and N_{i} are periodic for all i. Hence L_{ζ} is periodic. Moreover, the isomorphism L_{ζ} ⊕ Ω^{nt−1}(L_{ζ}) ≅ L_{ζ^{t}} ⊕ Ω^{n−1}(L_{ζ^{t}}) implies that

dim H^{nt}(G, k) ≤ dim Ω^{nt}(k)
= dim L_{ζ^{t}} + 1
= dim Ω^{nt−1}(L_{ζ}) + dim L_{ζ} − dim Ω^{n−1}(L_{ζ^{t}}) + 1
≤ dim Ω^{nt−1}(L_{ζ}) + dim L_{ζ} + 1.

Because L_{ζ} is periodic, it follows that dim H^{nt}(G, k) is bounded and hence G has p-rank one by Quillen’s Dimension Theorem. This completes the proof. □

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

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

Department of Mathematics, Morehouse College, Atlanta, Georgia 30314

E-mail address: peng@math.morehouse.edu

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

E-mail address: www@sloth.math.uga.edu