VIRTUAL PROJECTIVITY AND THE STABLE MODULE CATEGORY

DAVID J. BENSON AND JON F. CARLSON

Abstract. We introduce the concept of a nearly null map in the stable module category, and relate it to the notion of virtual projectivity. We show that the trivial module k is virtually M-projective if and only if the map f: N → k in the triangle N ⟶^{f} k → M ⊗ M^{∗} is nearly null. If these conditions hold, then M generates the stable module category in fewer than ℓ steps, where ℓ is the radical length of the group algebra. We give examples to show that if M generates the stable module category then k is not necessarily virtually M-projective. We also give examples to show that for a given group, the degree of virtual projectivity of k with respect to a module M is not bounded. Finally, we develop a theory of virtual M-variety of a module X. This is a subset of the spectrum of the cohomology ring which controls virtual projectivity of X with respect to M.

1. Introduction

The stable category of modules over the modular group algebra of a finite group is a tensor triangulated category that is the source of many mysteries in spite of the numerous efforts to understand its structure. While the thick tensor-ideal subcategories are classified by specialization closed subsets of the spectrum of the cohomology ring, much less is known about additive tensor ideals of objects or tensor closed ideals of morphisms. In even some of the simplest cases, it is not known how many steps it takes to (tensor) generate the category starting with an arbitrary generating object (see [3]).

In this paper we investigate two concepts related to relative projectivity. Let G be a finite group and k a field of characteristic p > 0. The projectivity of a kG-module relative to a subgroup of G is an often used concept in modular representation theory. The generalization to projectivity relative to a module was formulated in an unpublished manuscript of Okuyama and was further studied by the second author and Peng [7]. It was used in a cohomological calculation by Okuyama and Sasaki [10], and further generalized in a paper of the second author with Peng and Wheeler [8] to the notion of virtual projectivity relative to a module. That concept and another concerning relative projectivity of maps are the main objects of study for this paper. They have a connection to the question of bounding the generation number for the stable module category stmod(kG).

We introduce the notion of a nearly null map in stmod(kG), and show how it is related to virtual projectivity. A map f: M → N of kG-modules is null if it factors through a projective module and is nearly null if given any kG-module X, the map

f_{∗}: Ex̂t_{kG}^{n}(X, M) → Ex̂t_{kG}^{n}(X, N)

2010 Mathematics Subject Classification. 20C20, 20J06.

Key words and phrases. Finite groups, modular representation theory, virtual projectivity, stable module category, nearly null map.

is zero for all but a finite number of n ∈ ℤ. We show using Tate duality that the vanishing of this map for all X for all n large enough positive is equivalent to the vanishing for all X for all n large enough negative. A map f is nearly null if and only if the H^{∗}(G, k)-submodule of Ext_{kG}^{∗}(M, N) that it generates has finite dimension. Let ℓ denote the radical length of the group algebra kG. A key property is that the composite of any ℓ − 1 nearly null maps is null, i.e., it factors through a projective module.

If M and X are kG-modules, then X is virtually M-projective if the trace map

Tr_{M}: Ext_{kG}^{∗}(M ⊗ X, M ⊗ X) → Ext_{kG}^{∗}(X, X)

is surjective in large degrees. The condition is equivalent to the statement that for n sufficiently large, any map Ω^{n}(X) → X factors through M ⊗ Y for some Y. In the case that k is virtually M-projective, we show that M tensor generates the stable category in fewer than ℓ steps. On the other hand, for most groups, there exist modules M that tensor generate the stable category but with k not virtually M-projective. We show that k is virtually M-projective if and only if the map f in the triangle of the unit map ι_{M},

\xymatrix{N \ar[r]^{f} & k \ar[r]^{ι_{M}} & M^{∗} ⊗ M}

is nearly null.

Some other results of the paper are concerned with degrees of virtual projectivity. In general this is not bounded in any universal sense. Finally, we introduce the notion of the virtual M-variety of X, denoted W_{M}(X). This is a subset of the cohomology variety V_{G}(k) = Spec^{∗}H^{∗}(G, k). It has many interesting properties such as that W_{M⊕N}(X) = W_{M}(X) ∩ W_{N}(X) and that W_{M⊗N}(X) = W_{M}(X) ∪ W_{N}(X)

Although the results of this paper are stated for modules over group algebras, many of the basic constructions make sense in any tensor triangulated category. In particular, many of the main results are true for modules over any finite group scheme or supergroup scheme.

Acknowledgment. Both authors wish to thank the University of Bielefeld for kind hospitality during a visit when some of the ideas for this paper were generated. This material is partly based on work supported by the National Science Foundation under Grant No. DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2018 semester. The second author was also supported by the Simons Foundation under Grant No. 054813-01.

2. Preliminaries

Throughout the paper, G is a finite group and k is a field of characteristic p > 0. Unless otherwise indicated all kG-modules are assumed to be finitely generated. Let mod(kG) denote the category of finitely generated left kG-modules and module homomorphisms. Let stmod(kG) be the corresponding stable category. The objects in stmod(kG) are finitely generated kG-modules. If M and N are kG-modules, the morphisms from M to N in the stable category are given by

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

where PHom denotes the morphisms that factor through a projective module.

Recall that kG is a Hopf algebra with comultiplication g ↦ g ⊗ g for g ∈ G. Thus if M and N are kG-modules, so also is M ⊗_{k} N. The symbol “⊗” with no subscript means ⊗_{k}. We use the notation M ⊕ (proj) to indicate the direct sum of a module M with some projective module, which might be the zero module.

For M a kG-module, let Ω(M) denote the kernel of a projective cover P → M. For n > 1, define Ω^{n} inductively by Ω^{n}(M) = Ω(Ω^{n−1}(M)). Similarly, let Ω^{−1}(M) denote the cokernel of an injective hull M ↪ Q, and for n > 1, define Ω^{−n}(M) inductively by Ω^{−n}(M) = Ω^{−1}(Ω^{−n+1}(M)). For n = 0 we define Ω^{0}(M) to be sum of the nonprojective indecomposable summands of the module M. In this way, using the facts that injective kG-modules coincide with projective kG-modules and that a tensor product of any module with a projective module is projective, it can be seen that

Ω^{n}(Ω^{m}(M)) ≅ Ω^{n+m}(M)

and

Ω^{m}(M) ⊗ Ω^{n}(N) ≅ Ω^{m+n}(M ⊗ N) ⊕ (proj)

for m, n ∈ ℤ.

The k-dual of a kG-module M is denoted M^{∗} = Hom_{k}(M, k). Duals of projective kGmodule are projective (and injective). It can be checked that (Ω^{n}(M))^{∗} ≅ Ω^{−n}(M^{∗}).

The tensor product M ⊗ M^{∗} (which we identify with M^{∗} ⊗ M by swapping the factors) is isomorphic to End_{k}(M) with conjugation action g(f)(m) = g(f(g^{−1}m)). The inclusion of the identity ι_{M}: k → End_{k}(M) and the trace map Tr_{M}: End_{k}(M) → k give kG-module homomorphisms k → M ⊗ M^{∗} → k. The composite is multiplication by Dim(M) on k, and is hence zero if p divides Dim(M). Else it is an isomorphism. Hence, if p does not divide the dimension of M, k is a direct summand of M ⊗ M^{∗}. One of the motivating ideas for this paper is the following strong converse to this observation.

Theorem 2.1. [1] Suppose that M and N are absolutely indecomposable kG-modules such that k is a direct summand of M ⊗ N. Then N ≅ M^{∗}, the dimension of M is not divisible by p, and the multiplicity of k as a direct summand of M ⊗ M^{∗} is one.

The Tate cohomology Ex̂t_{kG}^{∗}(M, N) is the cohomology of the cochain complex Hom_{kG}(P_{∗}, N) where P_{∗} is a complete projective resolution of the kG-module M. That is,

\xymatrix{⋯ \ar[r] & P_{2} \ar[r]^{∂_{2}} & P_{1} \ar[r]^{∂_{1}} & P_{0} \ar[r]^{∂_{0}} & P_{−1} \ar[r]^{∂_{−1}} & P_{−2} \ar[r] & ⋯}

is an exact sequence of projective modules such that the image of ∂_{0} is isomorphic to M. Also, the image of ∂_{n} is Ω^{n}(M) ⊕ (proj). For any integer n, and modules M and N, Ex̂t_{kG}^{n}(M, N) ≅ \underline{Hom}_{kG}(Ω^{n}(M), N), since composing any homomorphism from Ω^{n}(M) to N with the surjection P_{n} → Ω^{n}(M) gives a cocycle P_{n} → N. We write Ĥ^{n}(G, k) for Ex̂t_{kG}^{n}(k, k). This is isomorphic to Hom_{kG}(Ω^{n}(k), k), since in this case there are no homomorphisms that factor through projective modules.

For n > 0, Ex̂t_{kG}^{n}(M, N) agrees with Ext_{kG}^{n}(M, N), and for n ⩾ 0, Ĥ^{n}(G, k) agrees with ordinary group cohomology H^{n}(G, k). The cohomology ring H^{∗}(G, k) is a finitely generated graded commutative k-algebra, and Ext_{kG}^{∗}(M, N) is a finitely generated module over it.

The maps ι_{M}: k → M ⊗M^{∗} and Tr_{M}: M ⊗M^{∗} → k give rise to maps in Tate cohomology, again denoted

ι_{M}: Ex̂t_{kG}^{∗}(k, k) → Ex̂t_{kG}^{∗}(M, M), Tr_{M}: Ex̂t_{kG}^{∗}(M, M) → Ex̂t_{kG}^{∗}(k, k).

More generally, if X and Y are modules, the transfer map is the composite

Tr_{M}: Ex̂t_{kG}^{∗}(M ⊗ X, M ⊗ Y) ⟶^{≅} Ex̂t_{kG}^{∗}(X, M ⊗ M^{∗} ⊗ Y) ⟶^{(Tr_{M} ⊗1)_{∗}} Ex̂t_{kG}^{∗}(X, Y).

If C is a finite complex of kG-modules, there is an analogous trace map of complexes Tr_{C}: C ⊗ C^{∗} → 𝕜, where 𝕜 is the complex with only one nonzero term, isomorphic to k in degree zero. This is really a “supertrace” in the sense that in odd degrees there is a minus sign attached; this is necessary in order to make Tr_{C} commute with the differential.

The following properties of Tr_{M} are easy to verify.

Lemma 2.2. Let f_{1} : W → X, f_{2} : Y → Z, ξ ∈ Ex̂t_{kG}^{∗}(M ⊗ X, M ⊗ Y) and ζ ∈ Ĥ^{∗}(G, k) Then

(i) (f_{2})_{∗} ∘ Tr_{M}(ξ) ∘ (f_{1})_{∗} = Tr_{M}((1 ⊗ f_{2})_{∗} ∘ ξ ∘ (1 ⊗ f_{1})_{∗})
(ii) Tr_{M}(ζ.ξ) = ζ.Tr_{M}(ξ).

In particular, if ζ is in the kernel of Ĥ^{∗}(G, k) → Ex̂t_{kG}^{∗}(M, M) then ζ.Tr_{M}(ξ) = 0. □

Proof. Assume that ξ is represented by its adjoint ξ : Ω^{n}(X) → M^{∗} ⊗ M ⊗ Y. Then the assertion that Tr_{M}(ξ) ∘ (f_{1})_{∗} = Tr_{M}(ξ ∘ (1 ⊗ f_{1})_{∗}) is seen from the commutative diagram

\xymatrix{ & M^{∗} ⊗ M ⊗ Y \ar[rr]^{1⊗f_{1}} \ar[dr]^{Tr_{M^{∗}⊗M} ⊗1} &  & M^{∗} ⊗ M ⊗ Z \ar[dr]^{Tr_{M^{∗}⊗M} ⊗1} \\ Ω^{n}(X) \ar@{-->}[ur]^{ξ} \ar[rr] &  & Y \ar[rr]^{f_{1}} &  & Z}

The proof of the assertion concerning f_{2} is similar and part (ii) follows from part (i). □

Setting W = X = Y = Z, we obtain the following.

Lemma 2.3. The image of

Tr_{M}: Ex̂t_{kG}^{∗}(M ⊗ X, M ⊗ X) → Ex̂t_{kG}^{∗}(X, X)

is a two-sided ideal. □

For ζ ∈ Ĥ^{n}(G, k), let L_{ζ} denote the kernel of the map ζ: Ω^{n}(k) → k. So there is an exact sequence,

\xymatrix{0 \ar[r] & L_{ζ} \ar[r] & Ω^{n}(k) \ar[r]^{ζ} & k \ar[r] & 0,}

and by translating, there is a triangle in stmod(kG) having the form

(2.4)  \xymatrix{⋯ \ar[r] & Ω^{−n}(L_{ζ}) \ar[r] & k \ar[r]^{ζ} & Ω^{−n}(k) \ar[r] & ⋯}

In parts of this paper we require the concept of the support variety of a module, which is a subvariety of the spectrum of the cohomology ring. The definitions are as follows. The projectivized prime ideal spectrum of H^{∗}(G, k) is denoted V_{G}(k). For M a kG-module, let J(M) denote the annihilator in H^{∗}(G, k) of the (finitely generated) module Ext_{kG}^{∗}(M, M). Then V_{G}(M) is the subvariety of this ideal, namely the collection of all homogeneous prime ideals that contain J(M).

The variety has a number of useful properties, among them being the tensor product theorem, which states that V_{G}(M ⊗ N) = V_{G}(M) ∩ V_{G}(N). We recall the theorem of the authors with Rickard [4], which says that a tensor ideal thick subcategory of the stable category is determined entirely by the support varieties of it objects. A subcategory is thick if it is triangulated and closed under taking direct summands. It is a tensor ideal if the tensor product of an object in the subcategory with any module is also in the subcategory. One of the consequences of the theorem is that if M is a kG-module and V_{G}(M) = V_{G}(k) then M generates all of stmod(kG). That is, the smallest tensor ideal thick subcategory of stmod(kG) that contains M is stmod(kG).

We end the section with a reminder of Tate duality. Let M and N be kG-modules. For any n, there is a nondegenerate pairing

\xymatrix{Ex̂t_{kG}^{n}(N, M) ⊗ Ex̂t_{kG}^{−n−1}(M, N) \ar[r] & k}

given by composing and taking trace:

Ex̂t_{kG}^{n}(N, M) ⊗ Ex̂t_{kG}^{−n−1}(M, N) ⟶ Ex̂t_{kG}^{−1}(M, M) ⟶^{Tr_{M}} Ex̂t_{kG}^{−1}(k, k) ≅ k.

In particular, we have

Ex̂t_{kG}^{−n−1}(M, N) ≅ (Ex̂t_{kG}^{n}(N, M))^{∗}.

Notice also that Ex̂t_{kG}^{n}(N, M) ≅ Ex̂t_{kG}^{n}(M^{∗}, N^{∗}).

3. Nearly null maps

In this section we define the notion of a nearly null map and develop some of its main properties.

Definition 3.1. We say that a map f: M → N of kG-modules is nearly null if given any kG-module X, the map

(3.2)  f_{∗}: Ex̂t_{kG}^{n}(X, M) → Ex̂t_{kG}^{n}(X, N)

is zero for all but a finite number of n ∈ ℤ.

In the stable category, an element ζ ∈ \underline{Hom}_{kG}(M, N) is nearly null if it is represented by a nearly null homomorphism ζ′ : M → N. Likewise, an element ζ ∈ Ex̂t_{kG}^{n}(M, N) is nearly null if a representing cocycle Ω^{n}(M) → N is nearly null. There is no ambiguity here, since any two representing cocycles differ by a null map.

Remarks 3.3. (i) If the induced map is zero on the cohomology as in the definition for all X and all n ∈ ℤ, then in particular it is zero for X = M and n = 0, and so f is null.

(ii) We see in Proposition 3.5 that requiring the map in the definition to be zero for all X and all large enough positive n implies that it is also zero for all X and for all large enough negative n, and vice versa.

(iii) The composite of a nearly null map with any map, on either side, is a nearly null map. So the nearly null maps form a two sided ideal in the stable module category.

(iv) If f : M → N is nearly null and g : X → Y is any homomorphism then f ⊗ g : M ⊗ X → N ⊗ Y is a nearly null map.

Example 3.4. Let M be an indecomposable kG-module, and let f: M → Ω(M) be the connecting homomorphism for the almost split sequence terminating in M. Then it follows directly from the definitions that f is nearly null.

Proposition 3.5. Let f: M → N with adjoints f′: k → M^{∗} ⊗ N and f′′: M ⊗ N^{∗} → k. Then the following are equivalent:

(i) For all X there exists m ∈ ℤ such that for all n ⩾ m the map

f_{∗}: Ex̂t_{kG}^{n}(X, M) → Ex̂t_{kG}^{n}(X, N)

is zero. (ii) For all X there exists m ∈ ℤ such that for all n ⩾ m the map

f′_{∗}: Ex̂t_{kG}^{n}(X, k) → Ex̂t_{kG}^{n}(X, M^{∗} ⊗ N)

is zero. (iii) For all X there exists m ∈ ℤ such that for all n ⩾ m the map

f′′_{∗}: Ex̂t_{kG}^{n}(X, M ⊗ N^{∗}) → Ex̂t_{kG}^{n}(X, k)

is zero. (iv) For all X there exists m ∈ ℤ such that for all n ⩾ m the map

f_{∗}^{t}: Ex̂t_{kG}^{n}(X, N^{∗}) → Ex̂t_{kG}^{n}(X, M^{∗})

is zero.

(v) For all X there exists m ∈ ℤ such that for all n ⩾ m the map

(f ⊗ 1)_{∗}: Ex̂t_{kG}^{n}(X, M ⊗ X) → Ex̂t_{kG}^{n}(X, N ⊗ X)

is zero. (vi) For all X there exists m ∈ ℤ such that for all n ⩽ m the map

f_{∗}: Ex̂t_{kG}^{n}(X, M) → Ex̂t_{kG}^{n}(X, N)

is zero.

Proof. We first show that (i) and (ii) are equivalent. The adjoint map f′: k → M^{∗} ⊗ N factors as the natural map ι_{M}: k → M^{∗} ⊗ M followed by 1 ⊗ f: M^{∗} ⊗ M → M^{∗} ⊗ N. Thus we have a commutative diagram:

\xymatrix{Ex̂t_{kG}^{n}(X, k) \ar[r]^{f′_{∗}} \ar@{-->}[dr]_{(ι_{M})_{∗}} & Ex̂t_{kG}^{n}(X, M^{∗} ⊗ N) \ar[r]^{≅} & Ex̂t_{kG}^{n}(M ⊗ X, N) \\  & Ex̂t_{kG}^{n}(X, M^{∗} ⊗ M) \ar[r]^{≅} \ar[u]_{(1⊗f)_{∗}} & Ex̂t_{kG}^{n}(M ⊗ X, M). \ar[u]_{f_{∗}}}

It follows that if f_{∗} is zero then so is f′_{∗}. Conversely, f: M → N factors as

M ≅ M ⊗ k ⟶^{1⊗f′} M ⊗ M^{∗} ⊗ N ⟶^{Tr_{M} ⊗1} k ⊗ N ≅ N.

We have

\xymatrix{Ex̂t_{kG}^{n}(M^{∗} ⊗ X, k) \ar[r]^{≅} \ar[d]_{f′_{∗}} & Ex̂t_{kG}^{n}(X, M) \ar[d]^{(1⊗f′)_{∗}} \ar[r]^{f_{∗}} & Ex̂t_{kG}^{n}(X, N) \\ Ex̂t_{kG}^{n}(M^{∗} ⊗ X, M^{∗} ⊗ N) \ar[r]^{≅} & Ex̂t_{kG}^{n}(X, M ⊗ M^{∗} ⊗ N) \ar@{-->}[ur]_{(Tr_{M} ⊗1)_{∗}}}

and it follows that if f′_{∗} = 0 then so is f_{∗}.

The equivalence of (i) and (iii) is similar. The equivalence of (iv) and (ii) is the same as the equivalence of (i) and (ii), with M and N replaced by N^{∗} and M^{∗}.

Next, Ex̂t_{kG}^{n}(X, M ⊗ X) is isomorphic to Ex̂t_{kG}^{n}(X ⊗ X^{∗}, M). In condition (i), it suffices to test against modules of the form X ⊗ X^{∗}, since X is isomorphic to a direct summand of (k ⊕ X) ⊗ (k ⊕ X)^{∗}. This proves that (i) and (v) are equivalent.

Finally, Tate duality shows that f_{∗}^{t}: Ex̂t_{kG}^{n}(X, N^{∗}) → Ex̂t_{kG}^{n}(X, M^{∗}) is zero if and only if the dual map f_{∗}: Ex̂t_{kG}^{−n−1}(X^{∗}, M) → Ex̂t_{kG}^{−n−1}(X^{∗}, N) is zero. This shows that (iv) and (vi) are equivalent. □

Proposition 3.6. Let f: M → N. Then the following are equivalent:

(i) For all X there exists m such that for all n ⩾ m, f_{∗}: Ex̂t_{kG}^{n}(X, M) → Ex̂t_{kG}^{n}(X, N) is zero.

(ii) There exists m such that for all n ⩾ m, f′_{∗}: Ĥ^{n}(G, k) → Ĥ^{n}(G, M^{∗} ⊗ N) is zero.

Proof. Applying Proposition 3.5 and then setting X = k, it is clear that (i) implies (ii). Conversely, if (ii) holds, we argue as follows. We may assume that m > 0 and replace Ĥ^{n} with H^{n}. Now for a given X, Ext_{kG}^{∗}(X, k) is a finitely generated module over H^{∗}(G, k). Suppose that it is generated by elements of degree at most t ⩾ 0. If n ⩾ t + m then any element of Ext_{kG}^{n}(X, k) is a sum of elements of the form xξ with ξ a generator of Ext_{kG}^{∗}(X, k) of degree at most t and x ∈ H^{∗}(G, k) of degree at least m. So we have f′_{∗}(xξ) = f′_{∗}(x)ξ = 0 in Ex̂t_{kG}^{∗}(X, M^{∗} ⊗ N). It follows that for n ⩾ t + m the map

f′_{∗}: Ex̂t_{kG}^{n}(X, k) → Ex̂t_{kG}^{n}(X, M^{∗} ⊗ N)

is zero. Applying Proposition 3.5 again, we see that (i) holds. □

Example 3.7. Suppose that H^{∗}(G, k) has depth at least two (for example, this happens if the center of a Sylow p-subgroup of G is not cyclic). If f: k → Ω^{m}(k) represents an element of Ĥ^{−m}(G, k) with m > 0 then by the main theorem of [1], the product of f with any element of Ĥ^{n}(G, k) with n ⩾ m is zero. Thus, by definition, f is nearly null.

Theorem 3.8. Let f: M → N with adjoint f′: k → M^{∗} ⊗ N. The following are equivalent: (i) f is nearly null. (ii) f′ is nearly null. (iii) For all X there exists m such that for all n ⩾ m, f_{∗}: Ex̂t_{kG}^{n}(X, M) → Ex̂t_{kG}^{n}(X, N)

is zero. (iv) For all X there exists m such that for all n ⩽ m, f_{∗}: Ex̂t_{kG}^{n}(X, M) → Ex̂t_{kG}^{n}(X, N)

is zero.

(v) There exists m such that for all n ⩾ m the map f′_{∗}: H^{n}(G, k) → H^{n}(G, M^{∗} ⊗ N) is zero.

Proof. This follows from Propositions 3.5 and 3.6. □

Remark 3.9. The conditions of the theorem imply, but are not equivalent to the condition that there exists m such that for all n ⩽ m, the map f′_{∗}: Ĥ^{n}(G, k) → Ĥ^{n}(G, M^{∗} ⊗ N) is zero. For example, let G have p-rank at least two, and let ζ ∈ H^{s}(G, k) be a non zero-divisor of positive degree s (this exists by a theorem of Duflot [9]), and complete the corresponding map Ω^{s}(k) → k to a triangle Ω^{s}(k) → k ⟶^{f} N in the stable module category. Then the induced map f_{∗}: Ĥ^{n}(G, k) → Ĥ^{n}(G, N) is zero in negative degrees but nonzero in infinitely many positive degrees.

Question 3.10. Is there some natural condition on a map f: M → N which would make the reverse implication true? For example, is it true when k is algebraically closed, and every direct summand of the cokernel of f′: k → M^{∗} ⊗ N has dimension divisible by p?

Proposition 3.11. Let Rad(kG) be the radical of the group algebra. Suppose that we have Rad(kG)^{ℓ} = 0. Then the composite of any ℓ − 1 nearly null maps is null.

Proof. Let f_{1} : M_{1} → M_{2}, …, f_{ℓ−1} : M_{ℓ−1} → M_{ℓ} be nearly null maps. Choose n large enough so that each f_{i} induces the zero map from Ex̂t_{kG}^{n}(S, M_{i}) → Ex̂t_{kG}^{n}(S, M_{i+1}) for each simple module S. Without affecting the hypothesis or the conclusion, we may replace each M_{i} by Ω^{−n}(M_{i}) and f_{i} by Ω^{−n}(f_{i}). Then each f_{i} induces the zero map

Ex̂t_{kG}^{0}(S, M_{i}) → Ex̂t_{kG}^{0}(S, M_{i+1}).

We may assume that none of the M_{i} has any projective summands, so

Hom_{kG}(S, M_{i}) → Hom_{kG}(S, M_{i+1})

is the zero map. It follows that the kernel of f_{i} contains the socle of M_{i}. By induction on i, the kernel of f_{i} ∘ ⋯ ∘ f_{2} ∘ f_{1} contains the ith socle of M_{1}. Since Rad^{ℓ}(kG) = 0, and M_{1} contains no projective direct summands, the (ℓ − 1)^{st} socle of M_{1} is the whole of M_{1}, and so the composite f_{ℓ−1} ∘ ⋯ ∘ f_{1} is zero. □

Proposition 3.12. Given any modules M and N, there exists m ∈ ℤ such that for all n ⩾ m, if f: Ω^{n}(M) → N is nearly null then it is null.

Proof. By Theorem 3.8, f: Ω^{n}(M) → N is nearly null if and only if its adjoint

k → Ω^{−n}(M^{∗} ⊗ N)

is nearly null. So it suffices to show that given a kG-module N there exists m > 0 such that for all n ⩾ m, if f: k → Ω^{−n}(N) is nearly null then it is null. Recall that for n > 0 we have

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

Let U ⊆ Ext_{kG}^{∗}(k, N) be the subset consisting of those f: k → Ω^{−n}(N) such that f is nearly null. Then U is an Ext_{kG}^{∗}(k, k)-submodule of Ext_{kG}^{∗}(k, N). Since Ext_{kG}^{∗}(k, N) is a Noetherian module over the Noetherian ring Ext_{kG}^{∗}(k, k), it follows that U is finitely generated as an Ext_{kG}^{∗}(k, k)-module, say by elements ξ_{1}, …, ξ_{s}. Each ξ_{i} is nearly null, so it annihilates elements of Ext_{kG}^{∗}(k, k) of degree at least n_{i} for some n_{i} ⩾ 0. Let t be the largest of the numbers deg(ξ_{i}) + n_{i}. If f is nearly null and has degree larger than t then f = ∑ ξ_{i}a_{i} with a_{i} ∈ Ext_{kG}^{∗}(k, k), and a_{i} has degree larger than n_{i} so ξ_{i}a_{i} = 0. Hence f = 0 in Ext_{kG}^{∗}(k, N), so f is null. □

4. The space of nearly null maps

If X and Y are kG-modules, let N (X, Y) denote the collection of all nearly null maps γ: X → Y, as a subspace of Hom_{kG}(X, Y). Let N^{n}(X, Y) denote the space of nearly null maps in Ext_{kG}^{n}(X, Y). Let N̂^{n}(X, Y) be the similar thing for Ex̂t_{kG}^{n}(X, Y). So, for example, N̂^{n}(X, Y) is the quotient of N (Ω^{n}(X), Y) by the subspace of all null maps from Ω^{n}(X) to Y.

Then we easily have the following.

Lemma 4.1. For any n ∈ ℤ, N̂^{n}(X, Y) is a subfunctor of Ex̂t_{kG}^{n}(X, Y). Furthermore, N^{∗}(X, Y) and N̂^{∗}(X, Y) are H^{∗}(G, k)-submodules of Ext_{kG}^{∗}(X, Y) and Ex̂t_{kG}^{∗}(X, Y), respectively. In addition, they are both Ext_{kG}^{∗}(Y, Y)-Ext_{kG}^{∗}(X, X)-bimodules. The space N^{∗}(X, Y) is finitely generated when regarded as a module over any of the rings H^{∗}(G, k), Ext_{kG}^{∗}(Y, Y) or Ext_{kG}^{∗}(X, X).

An easy example is contained in Proposition 3.7. Here it is, stated in functorial terms.

Proposition 4.2. Suppose that the depth of the cohomology ring H^{∗}(G, k) is at least two. Then N̂^{∗}(k, k) = ∑_{n<0} Ex̂t_{kG}^{n}(k, k).

A principal result is the following.

Theorem 4.3. A homomorphism f : M → N is nearly null if and only if the H^{∗}(G, k)- submodule that it generates in Ext_{kG}^{∗}(M, N) has finite k-dimension. In particular, any finitely generated H^{∗}(G, k)-submodule of N̂^{∗}(M, N) has finite dimension.

Proof. The “only if” part of the first statement is a direct consequence of Proposition 3.12. The proof of the remainder is very similar to that of Theorem 3.8. That is, for any kGmodule X, the H^{∗}(G, k)-module Ext_{kG}^{∗}(X, M) is finitely generated. By hypothesis, there is a number t such that the product f with any element of H^{n}(G, k), for n > t, is zero in Ext_{kG}^{∗}(M, N). So if m is the maximum of the degrees of any of the generators of Ext_{kG}^{∗}(X, M), then any element of Ext_{kG}^{t+m}(X, M) will be a sum of products of generators of Ext_{kG}^{∗}(X, M) with elements of H^{∗}(G, k) of degree greater than t. Any such element is annihilated by f. Hence by Theorem 3.8 (iv), f is nearly null. □

Theorem 4.4. Suppose that M and N are kG-modules such that Ext_{kG}^{∗}(N, M) is neither periodic nor finite dimensional. Then, for infinitely many values of n, there exist nonzero maps Ω^{n}(M) → N that are nearly null but not null.

Remark 4.5. By periodic, we mean that for some number s > 0 and for all n > 0, we have that Dim(Ext_{kG}^{n}(N, M)) = Dim(Ext_{kG}^{n+s}(N, M)). Any such periodicity, must (at least for some multiple of s) be induced by multiplication by an element of H^{s}(G, k). In the case that G is a p-group, the hypothesis that Ext_{kG}^{∗}(N, M) is neither periodic nor finite dimensional is equivalent to the assumption that the support varieties of M and N intersect in a variety of dimension at least two.

Proof. We recall that in Tate duality, Ex̂t_{kG}^{n}(M, N) is dual to Ex̂t_{kG}^{−n−1}(N, M) and hence they have the same dimension. Suppose that r is the p-rank of G. Then in some degree d, we can choose a homogeneous system of parameters ζ_{1}, …, ζ_{r} ∈ H^{d}(G, k). The assumption that Ext_{kG}^{∗}(N, M) is neither periodic nor finite dimensional implies that in some degree s, there is an element γ ∈ Ext_{kG}^{s}(N, M) such that the annihilator of γ is an ideal J in H^{∗}(G, k) such that H^{∗}(G, k)/J has Krull dimension at least two. Thus, there is some j such that

Dim(Ext_{kG}^{s+jd}(N, M)) > rDim(Ext_{kG}^{s}(N, M)),

or, using the duality,

Dim(Ex̂t_{kG}^{−s−jd−1}(N, M)) > rDim(Ex̂t_{kG}^{−s−1}(N, M)).

For each i = 1, …, r, there is the multiplication

\xymatrix{ζ_{i}^{j} : Ex̂t_{kG}^{−s−jd−1}(N, M)) \ar[r] & Ex̂t_{kG}^{−s−1}(N, M)),}

whose kernel V_{i} has codimension at most Dim(Ex̂t_{kG}^{−s−1}(N, M)) in Ex̂t_{kG}^{−s−jd−1}(N, M). Thus, by our assumption, the intersection V_{1} ∩ ⋯ ∩ V_{r} is not zero. That is, there is a nonzero element θ in Ex̂t_{kG}^{−s−jd−1}(N, M)) such that ζ_{i}θ = 0 for all i, and the submodule generated by θ has finite k-dimension. Hence, θ is nearly null.

The statement about infinitely many degrees is a consequence of the observation that we can replace s by s + id for any positive value of i. □

5. Relative projectivity and virtual projectivity

In this section we recall the definition of projectivity relative to a module, and we review the notion of virtual relative projectivity. A large part of the material is founded on one very basic fact: for any kG-module M, the map

\xymatrix{Id_{M} ⊗ Tr_{M}: M ⊗ M^{∗} ⊗ M \ar[r]^{≃} & M ⊗ Hom_{k}(M, M) \ar[r] & M ⊗ k ≅ M}

is a split epimorpism. It is split by the unit map

ι_{M} ⊗ Id_{M}: M ≅ k ⊗ M → M ⊗ M^{∗} ⊗ M.

Definition 5.1. Suppose that M is a kG-module. We say that a kG-module X is projective relative to M, or just M-projective, if X is isomorphic to a direct summand of M ⊗ U for some kG-module U.

For M a kG-module, an exact sequence in mod(kG) is said to be M-split if after tensoring with M it splits. The category mod(kG) together with the collection of M-split exact sequences is an exact category in the sense of Quillen. From the definition, we see that the collection of modules that are projective relative to the set of M-split sequences is precisely the collection of M-projective modules. In the accompanying stable category, the set of morphisms between two objects (kG-modules) is the quotient of the space of all homomorphisms from X to Y by the subspace of those homomorphisms that factor through an M-projective module.

We say that a map that factors through an M-projective module is an M-projective map. The M-projective maps clearly form an ideal in the mod(kG). More importantly, we need the following.

Lemma 5.2. Let M, X and Y be kG-modules and let θ: X → Y be a homomorphism. Then the following are equivalent:

(i) θ factors through an M-projective module.

(ii) θ factors through the trace map

Tr_{M} ⊗ Id_{Y} : M ⊗ M^{∗} ⊗ Y → k ⊗ Y ≅ Y.

(iii) θ factors through the unit map

ι_{M} ⊗ Id_{X}: X ≅ k ⊗ X → M ⊗ M^{∗} ⊗ X.

(iv) θ is in the image of the transfer map

Tr_{M}: Hom_{kG}(M ⊗ X, M ⊗ Y) → Hom_{kG}(X, Y).

Proof. See [7, 8] for more details. The main argument is based on the fact, stated at the beginning of the section, which implies that Tr_{M} ⊗ Id_{Y} is an M-split epimorphism and ι_{M} ⊗ Id_{X} is an M-split monomorphism. □

From this we deduce basic properties of relative projectivity. Again, see [7, 8] for details.

Lemma 5.3. Let M and X be kG-modules. The following are equivalent:

(i) X is M-projective.

(ii) X is a direct summand of M^{∗} ⊗ M ⊗ X.

(iii) The identity map Id_{X}: X → X factors through an M-projective module.

(iv) For n ⩾ 0, the transfer map

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

is surjective.

There is a weaker condition that is more closely related to the theory of support varieties for modules. It is that of virtual projectivity, first introduced by the second author, Peng and Wheeler [8]. We see in Proposition 5.10 that this notion bears the same relation to nearly null maps that relative projectivity does to null maps.

Definition 5.4. We say that a module X is virtually M-projective if for n sufficiently large, any homomorphism Ω^{n}(X) → X factors through an M-projective module. The degree of virtual M-projectivity of X is the least m ⩾ 0 such that this holds for all n ⩾ m.

Lemma 5.5. Let M and X be kG-modules. The following are equivalent:

(i) X is virtually M-projective of degree m.

(ii) If n ⩾ m every map Ω^{n}(X) → X factors through the transfer map

M^{∗} ⊗ M ⊗ X ⟶^{Tr_{M} ⊗1} k ⊗ X ≅ X.

(iii) If n ⩾ m every map X → Ω^{−n}(X) factors through the map

X ≅ k ⊗ X ⟶^{ι_{M} ⊗1} M^{∗} ⊗ M ⊗ X.

(iv) For n ⩾ m the transfer map

Tr : Ext_{kG}^{n}(M ⊗ X, M ⊗ X) ⟶^{Tr_{M}} Ext_{kG}^{n}(X, X)

is surjective.

Concerning varieties, we emphasize the following consequence of Lemma 2.2.

Lemma 5.6. Let J(M) be the annihilator in H^{∗}(G, k) of Ext_{kG}^{∗}(M, M). Then,

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

Proposition 5.7. If X is virtually M-projective then V_{G}(X) ⊆ V_{G}(M).

Proof. Choose ζ ∈ J(M) homogeneous of degree d > 0. Then for n suffiently large, Ext_{kG}^{n}(X, X) is in the image of the transfer Tr_{M} and hence is annihilated by ζ. Therefore, for m sufficiently large ζ^{m} annihilates all of Ext_{kG}^{n}(X, X). Thus J(M) ⊆ \sqrt{J(X)}, and V_{G}(M) ⊇ V_{G}(X). □

Remark 5.8. A theorem of both authors and Rickard [4] implies that if V_{G}(X) ⊆ V_{G}(M) then M tensor generates X in the stable module category. Combining this with Proposition 5.7, we deduce that if X is virtually M-projective then M tensor generates X in the stable module category. We quantify this in the case X = k in Theorem 5.12.

Proposition 5.9. Suppose that k is virtually M-projective of degree d. Let H be a subgroup of G, and assume that H^{∗}(H, k) is generated as a module over R = res_{G,H}(H^{∗}(G, k)) by elements in degree at most n. Then k_{H} is virtually M_{↓H}-projective of degree at most d + n.

Proof. Assume that γ_{1}, …, γ_{m} are homogeneous elements having degree at most n in H^{∗}(H, k) such that H^{∗}(H, k) = ∑_{i=1}^{m} Rγ_{i}. Then any element of degree at least d + n in H^{∗}(H, k) is a sum ∑_{i=1}^{m} ζ_{i}γ_{i}, where each ζ_{i} is the restriction to H of an element ζ̂_{i}: Ω^{d_{i}}(k) → k of degree d_{i} ⩾ d. Because ζ̂_{i} factors through M^{∗} ⊗ M, its restriction ζ_{i} to H factors through (M_{↓H})^{∗} ⊗ M_{↓H}. The rest follows from Lemma 2.3. □

Now let M be a kG-module, and form the triangles containing the unit map and the trace map:

N ⟶^{f} k ⟶^{ι_{M}} M ⊗ M^{∗}, M ⊗ M^{∗} ⟶^{Tr_{M}} k ⟶^{f^{t}} N^{∗}

in stmod(kG). Here f^{t} denotes the dual of f.

Proposition 5.10. For a kG-module X and an integer m, the following are equivalent: (i) X is virtually M-projective of degree at most m.

(ii) Ex̂t_{kG}^{n}(X, X) ⟶^{(f⊗1)^{∗}} Ex̂t_{kG}^{n}(N ⊗ X, X) is the zero map for n ⩾ m. In particular, k is virtually M-projective if and only if f: N → k is nearly null.

Proof. By Lemma 5.5, the module X is virtually M-projective of degree at most m if and only if for all n ⩾ m every map X → Ω^{−n}(X) factors through X → M ⊗ M^{∗} ⊗ X. The diagram

\xymatrix{N ⊗ X \ar[r] \ar[dr]_{0} & X \ar[r] \ar[d]^{ζ} & M ⊗ M^{∗} ⊗ X \ar@{.>}[dl] \\  & Ω^{−n}(X)}

now shows that this happens if and only if the map

Ex̂t_{kG}^{n}(X, X) → Ex̂t_{kG}^{n}(N ⊗ X, X)

is zero for n ⩾ m.

For the last statement, if k is virtually M-projective then every module X is virtually M-projective and so condition (ii) holds for all X. Taking adjoints, this shows that for all X there exists m such that

Ex̂t_{kG}^{n}(X, X) → Ex̂t_{kG}^{n}(X, N^{∗} ⊗ X)

is zero for n ⩾ m. So Theorem 3.8 implies that f^{∗}: k → N^{∗} and hence also f: N → k are nearly null. □

Remark 5.11. The same argument as the last part of the proof shows that X is virtually M-projective if and only if the map

1 ⊗ 1 ⊗ f: X ⊗ X^{∗} ⊗ N → X ⊗ X^{∗} ⊗ k ≅ X ⊗ X^{∗}

is nearly null.

Theorem 5.12. Suppose that Rad(kG)^{ℓ} = 0 and that k is virtually M-projective. Then M tensor generates k in the stable module category, and the tensor M-generation number is as most ℓ − 1.

Proof. We examine the triangle N ⟶^{f} k → M ⊗ M^{∗}. By Proposition 5.10, f is nearly null. We can view f^{⊗(ℓ−1)} as a composite

N^{⊗ℓ−1} ⟶^{f⊗1^{ℓ−2}} N^{⊗(ℓ−2)} → ⋯ → N^{⊗2} ⟶^{f⊗1} N ⟶^{f} k

where each map is nearly null. By Proposition 3.11, it follows that f^{⊗(ℓ−1)} is null. By Theorem 8.10 of [3], this is equivalent to the statement that the tensor M-generation number is at most ℓ − 1. □

6. Degrees of virtual projectivity

The aim of this section is to show that among the modules M such that k is virtually M-projective, there is no universal bound (independent of M) on the degree of virtual M-projectivity. Specifically, we show the following.

Theorem 6.1. Assume that p ⩾ 3, and suppose that G is a finite group of p-rank at least 2. Let B be any natural number. Then there exists a kG-module M having the properties that

(i) the trivial module k is virtually M-projective, and

(ii) for some n ⩾ B, there is an element ζ ∈ H^{n}(G, k) that is not in the image of Tr_{M}: Ext_{kG}^{n}(M, M) → H^{n}(G, k). That is, the degree of virtual M-projectivity of k is greater than B.

The proof of the theorem comes in two basic steps. The first thing that we need to do is to establish a class of modules relative to which k is virtually projective. The class that we produce here is a generalization of Example 7.1 of [8]. We recall that the ordinary cohomology ring H^{∗}(G, k) has Krull dimension equal to the p-rank of G. The depth of H^{∗}(G, k) is defined to be the length of the longest regular sequence of homogeneous elements of positive degree. By the theorem of Duflot [9], the depth is always at least equal to the p-rank of the center of a Sylow p-subgroup of G, and hence, it is always positive.

Theorem 6.2. Suppose that the group G has p-rank at least 2. Let ζ be a nonzero element of even degree in negative Tate cohomology, i.e., ζ ∈ Ĥ^{−2n}(G, k) for n > 0. Assume, additionally, that one of the following holds:

(i) The depth d of the ordinary cohomology ring H^{∗}(G, k) is at least 2.

(ii) The depth d = 1, and the cohomology ring H^{∗}(G, k) has a regular element γ in degree

m > 0 such that γζ = 0. Then the trivial module k is relatively L_{ζ}-projective.

The hypothesis of Theorem 6.2 is needed to establish the following result that is crucial to our argument.

Proposition 6.3. Suppose that G and ζ satisfy the hypotheses of Theorem 6.2. If μ ∈ Ĥ^{m}(G, k) for m ⩾ 2n, then ζμ = 0.

Proof. In the case that G satisfies conditions (i) of the hypothesis, the proof follows directly from results in [2]. That is, Lemma 2.1 of that paper implies that the conclusion fails if and only if there exist two elements in negative Tate cohomology whose product is not zero. By Theorem 3.3 of [2], the latter happens only if the depth d is one. In case (ii), the proof is established by noting that if there exists a regular element γ ∈ H^{∗}(G, k) such that γζ = 0, then γζμ = 0, for any μ. If we also have that ζμ has positive degree, then the regularity of γ insures that ζμ = 0. □

Proof of Theorem 6.2. Suppose that γ ∈ H^{m}(G, k) for m > −2n. Then we have a diagram

[figure: Ω^{m}(k); γ; Ω^{−2n}(L_{ζ}); τ; k; ζ; Ω^{−2n}(k)]

where the bottom row is a shift of the triangle as in (2.4). The point is that by Propositon 6.3, ζγ factors through a projective module. Hence, there is a map μ: Ω^{m}(k) → Ω^{−2n}(L_{ζ}) such that τμ = γ in the stable category. Thus γ is L_{ζ}-projective. □

Remark 6.4. It should be noted that if the depth of G is one, then there is an abundance of elements ζ that satisfy the condition (ii) of Proposition 6.2. For suppose that γ is a regular element in H^{n}(G, k). If d < −n, then an easy argument using Tate duality shows that the map Ĥ^{d}(G, k) → H^{d+n}(G, k) given by multiplication by γ is surjective (see Lemma 3.5 of [2]). However, because the Krull dimension of H^{∗}(G, k) is at least 2, the dimension of Ĥ^{d}(G, k) is usually larger than that of Ĥ^{d+n}(G, k) and hence, multiplication by γ on Ĥ^{d}(G, k) is seldom injective. For d large enough there must exist elements ζ in the kernel satisfying condition (ii). Indeed, the method described above can be used, as in Proposition 3.4 of [2], to show that the subspace of Ĥ^{d}(G, k), d < 0, of elements satisfying the condition (ii) has bounded codimension in Ĥ^{d}(G, k).

Proof of Theorem 6.1. Assume that ζ satisfies the hypothesis of the theorem. Then we have an exact sequence

\xymatrix{0 \ar[r] & L_{ζ} \ar[r] & Ω^{−2n}(k) \ar[r]^{ζ} & k \ar[r] & 0.}

Let C denote the complex

\xymatrix{⋯ \ar[r] & 0 \ar[r] & Ω^{−2n}(k) \ar[r]^{ζ̂} & k ⊕ P \ar[r] & 0 \ar[r] & ⋯}

where P is a projective module large enough that the map ζ̂, which is in the class of ζ in the stable category, is injective. The nonzero terms are in degrees one and zero (reading left to right). The homology of C is isomorphic to Ω^{−1}(L_{ζ}), all in degree zero. Recall that we have a (super) trace map Tr_{C}: C ⊗ C^{∗} → 𝕜 where 𝕜 is the complex with only one nonzero term isomorphic to k in degree zero. It restricts to the trace map on homology

Tr_{Ω^{−1}(L_{ζ})} : Ω^{−1}(L_{ζ}) ⊗ (Ω^{−1}(L_{ζ}))^{∗} → k.

Note that Ω^{−1}(L_{ζ}) ⊗ (Ω^{−1}(L_{ζ}))^{∗} ≅ L_{ζ} ⊗ L_{ζ}^{∗} in stmod(kG). Also the complex

\xymatrix{C^{∗}: 0 \ar[r] & k \ar[r]^{ζ} & Ω^{2n}(k) \ar[r] & 0}

14

is the shift Ω^{2n}(C), except that the nonzero terms are now in degrees 0 and −1. Thus C ⊗ C^{∗} ≅ Ω^{2n}(C ⊗ C). The complex C ⊗ C has an action of the cyclic group C_{2} that switches the factors. This action was analyzed in [5]. The results show that this tensor product splits into the direct sum of two complexes, according to the eigenvalue of the action of C_{2}, that is, either 1 or −1 (here is where we use the assumption that p > 2). Translating appropriately, we can show that up to homotopy C ⊗ C^{∗} ≅ C ⊕ Ω^{2n}(C). That is we have a diagram

\xymatrix{C ⊗ C^{∗}: & 0 \ar[r] & Ω^{−2n}(k) \ar[r]^{(ζ,0)} & k ⊕ k ⊕ (proj) \ar[d]^{_{Tr_{C}}} \ar[r]^{(0,ζ)} & Ω^{2n}(k) \ar[r] & 0 \\  &  &  & 𝕜}

Next, notice that the composition Tr_{C}(ζ, 0) is the zero map. The splitting implies that homology of C ⊗C^{∗} is Ω^{−1}(L_{ζ})⊕Ω^{2n}(L_{ζ}). Thus, the trace map vanishes on the term Ω^{−1}(L_{ζ}) and has the form

\xymatrix{Tr: Ω^{−1}(L_{ζ}) ⊕ Ω^{2n}(L_{ζ}) \ar[r]^{(0,τ)} & k}

where τ is the map in the triangle

[figure: Ω^{2n}(L_{ζ}); τ; k; ζ; Ω^{2n}(k)]

Consequently, a map M → k is L_{ζ}-projective if and only if it factors through τ. Significantly, this happens if and only if its composition with ζ is zero in the stable category. For any bound B, by Remark 6.4, for some 2n ⩾ B, there exists an element ζ in negative Tate cohomology Ĥ^{−2n}(G, k), satisfying the hypothesis of Proposition 6.2. Thus, the trivial module k is virtually L_{ζ}-projective of degree d ⩽ 2n. We need only be certain that d ⩾ B. For this let η ∈ H^{2n−1}(G, k) be an element that is dual to ζ under Tate duality, so that ζη ≠ 0 in Ĥ^{−1}(G, k). Then from the above, we see that η: Ω^{2n−1}(k) → k does not factor through τ, and hence it is not L_{ζ}-projective. Thus d = 2n and the proof is complete. □

In the proof we have demonstrated the following.

Corollary 6.5. Suppose that ζ is an element of Ĥ^{−2n}(G, k) for n > 0, satisfying the hypothesis of Theorem 6.2. Then the ideal I of all elements in H^{∗}(G, k) that are L_{ζ}-projective consists of exactly those elements γ such that ζγ = 0 in Ĥ^{∗}(G, k).

7. Modules that generate the stable category

The purpose of this section is to show that there exist modules that generate the stable category, but relative to which, the trivial module k is not virtually projective. We prove this for any elementary abelian p-group of rank at least 2 for p odd. The same holds in the case that p = 2 and the rank is at least 4. In the proof we provide explicit examples. Recall that M tensor generates the stable category stmod(kG) if and only if V_{G}(M) = V_{G}(k). If p fails to divide the dimension of a module M, then k is M-projective and hence also virtually M-projective. So the object is to construct modules M that are absolutely indecomposable and have the property that the ideal of elements of H^{∗}(G, k) that are not M-projective is not big. We begin with a general method of construction.

Theorem 7.1. Suppose that G ≅ H × J where H and J are subgroups both of which have order divisible by p. Let N be an absolutely indecomposable kH-module with the property that p divides the dimension of N. Let M be the inflation of N to a kG-module. If a cohomology element ζ ∈ H^{n}(G, k) factors through an M-projective module, then the restriction of ζ to H^{∗}(J, k) is zero. Moreover, k is not virtually M-projective.

Proof. Suppose that P_{∗} is a projective kG-resolution of k and that Q_{∗} is the inflation to kG of a projective kJ-resolution of k. That is, the restriction of Q_{∗} to J is a projective resolution of k and the subgroup H acts trivially on Q_{∗}. Then there is a kG-chain map from P_{∗} to Q_{∗} that lifts the identity on k. This chain map induces a surjective homomorphism

\xymatrix{ϑ: Ω^{n}(k) \ar[r] & Ω_{J}^{n}(k_{J})}

where by Ω_{J}^{n}(k_{J}) we mean the n^{th} syzygy kJ-module inflated to a kG-module, so that H acts trivially on it. Note that when restricted to kJ, ϑ is a split surjection. That is,

Ω^{n}(k)_{↓J} ≅ P ⊕ Ω_{J}^{n}(k_{J})

where P is the kernel of ϑ and is a projective kJ-module.

Let M be as in the hypothesis. Let ζ: Ω^{n}(k) → k be a cocycle that factors through an M-projective module. Thus there are maps σ: Ω^{n}(k) → M^{∗} ⊗ M and τ: M^{∗} ⊗ M → k such that τσ = ζ. Assume that the restriction of ζ to J is not null. Our task is to show that this last assumption leads to a contradiction.

The assumption on the restriction of ζ to a kJ-cocycle means that ζ does not vanish on the kJ-submodule Ω_{J}^{n}(k_{J}). That is, there is an element u ∈ Ω_{J}^{n}(k_{J}) such that ζ(u) ≠ 0 and in particular σ(u) ≠ 0. By hypothesis, J acts trivially on M^{∗} ⊗ M and hence has trivial action on the element σ(u). For any h ∈ H, multiplication by h − 1 is a kJ-endomorphism of Ω^{n}(k) whose image is in the kernel of ϑ, the projective kJ-module P. In particular, (h − 1)(Ω^{n}(k)_{↓J}) ⊆ Rad_{kJ}(P). Because M^{∗} ⊗ M is a trivial kJ-module, it must be that σ((h − 1)u) = (h − 1)σ(u) = 0. Thus both H and J acts trivially on σ(u) and because τ(σ(u)) ≠ 0, we have that k is a direct summand of M^{∗}⊗M. This contradicts the hypothesis that p divides the dimension of M (see Theorem 2.1). □

For G elementary abelian of rank 2 and p > 2, there are other constructions of modules M having full variety and also the property that k is not virtually M-projective. The idea of the proof is basically the same as for that of Theorem 7.1, but requires more care.

Proposition 7.2. Assume that p > 2. Suppose that G ≅ C_{p}^{2} = ⟨x, y⟩ is an elementary abelian group of order p^{2}. Write X = x − 1, Y = y − 1, so that kG ≅ k[X, Y]/(X^{p}, Y^{p}) is a truncated polynomial ring. Assume that M is a kG-module with the property that there is no element m ∈ M ⊗ M^{∗} satisfying all three of the conditions:

(i) Tr_{M}(m) = 1, (ii) Y m = 0, and (iii) Xm ∈ Y^{p−1}(M ⊗ M^{∗}).

Then k is not virtually M-projective.

Proof. The group algebra kG can be written as a tensor product of rings

kG ≅ k[X]/(X^{p}) ⊗ k[Y]/(Y^{p})

and the minimal projective kG-resolution P_{∗} of the trivial module k can be written as a tensor product P_{∗} ≅ P_{∗}^{X} ⊗ P_{∗}^{Y} where P_{∗}^{X} and P_{∗}^{Y} are minimal projective k[X]/(X^{p})- and k[Y]/(Y^{p})-resolutions of k. Note that P_{∗}^{X} has the form

\xymatrix{⋯ \ar[r]^{X} & P_{2}^{X} \ar[r]^{X^{p−1}} & P_{1}^{X} \ar[r]^{X} & P_{0}^{X} \ar[r] & k \ar[r] & 0}

where every P_{i}^{X} is isomorphic to k[X]/(X^{p}) and the boundary maps alternate between multiplication by X and by X^{p−1}. The resolution over k[Y]/(Y^{p}) is the same except with X replaced by Y.

With the above construction, we can write a presentation for Ω^{n}(k) for n > 0. Specifically we have, for n = 2m, even, that Ω^{n}(k) is generated by elements a_{0}, …, a_{m} and b_{1}, …, b_{m}, with relations given as

Y a_{0} = 0, Y a_{i} = X^{p−1}b_{i}, Xa_{i−1} = Y^{p−1}b_{i}, Xa_{m} = 0

for 1 ⩽ i ⩽ m. We leave it to the reader to verify the details, adding only the hint that, up to a sign, a_{i} can be taken to be the image in P_{n−1} of 1 ⊗ 1 in P_{2i}^{X} ⊗ P_{n−2i}^{Y} while b_{i} is the is the image of 1 ⊗ 1 in P_{2i−1}^{X} ⊗ P_{n−2i+1}^{Y}.

Now for the proof of the proposition, choose any m > 0 and let ζ: Ω^{2m}(k) → k be the cocycle given on generators by ζ(a_{0}) = 1, ζ(a_{i}) = 0 = ζ(b_{i}) for i = 1, …, m. Then ζ represents a nonzero element of cohomology in H^{2m}(G, k). We claim that ζ is not Mprojective. The reason is that if ζ were a composition

\xymatrix{Ω^{2m}(k) \ar[r]^{σ} & M^{∗} ⊗ M \ar[r]^{^{Tr^{M}}} & k}

then from the computed relation, it would be necessary that σ(a_{0}) satisfy all three of the listed conditions of the proposition. □

Proposition 7.3. Suppose that G is an elementary abelian p-group of order p^{2} for p > 2. There exists a kG-module M that generates stmod(kG) having the property that k is not virtually M-projective.

Proof. We use that notation of the previous result, so that kG = k[X, Y]/(X^{p}, Y^{p}). Define the module M as M = kG/(X^{p−1}, Y^{2}, XY). Then M has a k-basis B = {1, Y, X, …, X^{p−2}} and has dimension p. Let ℓ ∈ M be the class of 1 ∈ kG so that M = kGℓ. It can be checked that M^{∗} is generated by λ_{1}, λ_{2}, where we let λ_{1}(u) = 0 for all u in B except λ_{1}(X^{p−2}) = 1, and λ_{2}(u) = 0 for all u in B except λ_{2}(Y) = 1. It is easy to see that Y λ_{1} = 0, Xλ_{2} = 0, and X^{p−2}λ_{1} = Y λ_{2}.

We suppose that m is an element of M^{∗} ⊗ M satisfying the three conditions of Proposition 7.2. Notice that if p > 3, then Y^{p−1}(M^{∗} ⊗ M) = {0}, and hence, Xm = 0. The same holds if p = 3. For in this case, Y^{2}(M^{∗} ⊗ M) has dimension one and is spanned by Y λ_{2} ⊗ Y ℓ = X^{p−2}λ_{1} ⊗ Y ℓ. Thus we may assume that m = λ_{1} ⊗ Y ℓ + m̂ where Xm̂ = 0. But Tr_{M}(λ_{1} ⊗ Y ℓ) = λ_{1}(Y ℓ) = 0. Therefore m̂ also satisfies the three conditions, and we may assume that m = m̂.

We have that m has the property that Xm = 0 = Y m and Tr_{M}(m) = 1. Hence, viewed as an element of Hom_{k}(M, M), m is a kG-homomorphism which is not nilpotent since its trace is not zero. However, because the identity homomorphism has zero trace, this is not possible. □

Remark 7.4. We should note from the above proof that if p > 3 and M is any indecomposable module having Loewy length at most (p − 1)/2, then k is not virtually M-projective. That is, in such a case Y^{p−1}(M^{∗} ⊗ M) = {0} and we are forced to the same conclusion as above.

The above results naturally extend to elementary abelian groups of larger rank.

Corollary 7.5. Suppose that p ⩾ 3 and that G is an elementary abelian group having rank at least 2. Then there is a kG-module M such that M generates stmod(kG), and k is not virtually M-projective.

Proof. In the case that G has rank 2 and p > 2, the previous proposition proves the result. Otherwise use Theorem 7.1. □

8. Additive tensor ideals

In this section we show that virtual projectivity defines additive tensor ideals in the stable category stmod(kG). An additive tensor ideal C is a full subcategory that is closed under arbitrary tensor (meaning that if X is in C, then X ⊗ Y is in C for any object Y) and finite direct sums and direct summands. Note that an additive tensor ideal is not assumed to be triangulated. Indeed, if an additive tensor ideal C were also triangulated (meaning that if two objects in a triangle were in C, then the third would also be in C), then it would be a thick tensor ideal. One result of this section is that, in general, there is an infinite decreasing nested sequence of additive tensor ideals related to relative projectivity in stmod(kG).

A motivation for the following notion can be found in [6]. Suppose that H is a subgroup of G. Let M be the induced module M = k_{H}^{↑G} ≅ kG ⊗_{kH} k_{H}. The relative stable category stmod_{H}(kG) is the category of all kG-modules, with the morphisms between the modules being the quotient of the ordinary kG-homomorphisms by the subspace of those that factor through an M-projective module. In [6] it is shown that if J is a subgroup of H and if C is any additive tensor ideal in stmod(kJ), then the full subcategory of all kG-modules whose restriction to J is in C is a thick tensor ideal in the relative category stmod_{H}(kG).

We begin with an easy observation. The proof is a consequence of the fact that if a map factors through an L ⊗ M-projective module, then it factors though an L-projective module and an M-projective module.

Lemma 8.1. Let X, L and M be kG-modules. If X is virtually L ⊗ M projective, then it is both virtually L-projective and virtually M-projective.

Concerning degrees we have the following.

Lemma 8.2. Suppose that X is virtually L-projective of degree d and virtually M-projective of degree e. Let n be a number such that the ring Ext_{kG}^{∗}(X, X) is generated as a k-algebra in degrees at most n. Then X is vitually L ⊗ M-projective of degree at most d + e + n and at least the maximum of d and e.

Proof. Suppose that X is virtually L ⊗ M-projective of degree m. Then given any ζ ∈ Ext_{kG}^{t}(X, X) with t ⩾ m, we have that ζ is both L and M-projective. By the definition of the degree, for any t < d there exists such a ζ that is not L-projective. Therefore, m ⩾ d. Similarly, m ⩾ e. Thus we have the lower bound on the degree.

Any monomial ζ in the generators of Ext_{kG}^{∗}(X, X), having degree at least d + e + n can be written as a product ζ = ζ_{1} ⋯ ζ_{m} where each ζ_{i} has degree at most n. Any such product can be rewritten as ζ = η_{1}η_{2} such that for some i, η_{1} = ζ_{1} ⋯ ζ_{i} has degree u at least d and η_{2} = ζ_{i+1} ⋯ ζ_{m} has degree v at least e. Thus we have maps α and β and a commutative diagram

\xymatrix{ &  & Ω^{u+v}(X) \ar[d]^{η_{2}} \ar[dl]_{α} \\  & M^{∗} ⊗ M ⊗ Ω^{v}(X) \ar[r]^{Tr_{M} ⊗Id_{Ω^{v}(X)}} \ar[d]^{Id_{M} ⊗η_{1}} \ar[dl]_{β} & Ω^{v}(X) \ar[d]^{η_{1}} \\ M^{∗} ⊗ M ⊗ L^{∗} ⊗ L ⊗ X \ar[r]^{Id_{M} ⊗Tr_{L}⊗Id_{X}} & M^{∗} ⊗ M ⊗ X \ar[r]^{Tr_{M} ⊗Id_{X}} & X}

The Lemma follows. □

One problem with the notion of degree is that it does not behave well with respect to direct sums. A good example is the following.

Example 8.3. Let G = ⟨g, h⟩ be an elementary abelian group of order p^{2} for p > 2. The cohomology ring of G is a product H^{∗}(G, k) = k[ζ_{1}, ζ_{2}] ⊗ Λ(η_{1}, η_{2}) of a polynomial ring generated by ζ_{1}, ζ_{2} in degree two and an exterior algebra generated by η_{1}, η_{2} in degree one. Hence, every element is a linear combination of the monomials η_{1}^{j_{1}}η_{2}^{j_{2}}ζ_{1}^{ℓ_{1}}η_{2}^{ℓ_{2}} where 0 ⩽ j_{1}, j_{2} ⩽ 1. The negative cohomology is generated by the Tate dual elements θ̂ ∈ Ĥ^{−n−1}(G, k) where θ ∈ H^{n}(G, k). Here the assumption is that θθ̂ = ω where ω is the generator of Ĥ^{−1}(G, k) dual to 1 ∈ H^{0}(G, k). Moreover, the Tate duality (see the end of Section 2) has ⟨γ, θ̂⟩ = 0 if γ is a monomial in the generators of degree n other than θ. The Tate duality gives us that

⟨θγ, θ̂γ⟩ = ⟨γ, θθ̂γ⟩ = ω

so that θθ̂γ = γ̂. Note that all products in negative cohomology are zero by [2]. Our main interest is in two elements γ_{1} = η̂_{1}ζ_{1}^{n} and γ_{2} = η̂_{2}ζ_{2}^{n} in degree −2n−2 for n ⩾ 1. Both satisfy the conditions of Theorem 6.2. Let L_{i} = L_{γ_{i}} and let I_{i} be the ideal of elements in H^{∗}(G, k) that are L_{i}-projective. By Corollary 6.5, k is virtually L_{i}-projective of degree 2n + 1. On the other hand, both η_{1} and ζ_{1} annihilate γ_{2}, while both η_{2} and ζ_{2} annihilate γ_{1}. It follows that the ideal of H^{∗}(G, k) of all elements that are (L_{1} ⊕ L_{2})-projective consists of all of the elements of positive degree. Hence k is virtually (L_{1} ⊕ L_{2})-projective of degree 1.

On the other hand it is not difficult to find additive tensor ideals in the stable category using the information in Proposition 2.3. For example, we have the following.

Proposition 8.4. Let X be a kG-module, and let V be a linear subspace of Ext_{kG}^{n}(X, X). For another kG-module M, let I_{M} be the ideal of Ext_{kG}^{∗}(X, X) of all elements that are Mprojective. Let V be the full subcategory of all kG-modules M such that I_{M} ∩ Ext_{kG}^{n}(X, X) ⊆ V. Then V is an additive tensor ideal.

Proof. The fact that V is close under direct sums uses the fact that I_{M} is a graded ideal. Its closure under direct summands is obvious. Finally, suppose that M ∈ V and L is any module. If ζ ∈ I_{L⊗M} ∩ Ext_{kG}^{n}(X, X), then by Proposition 2.3, ζ is M-projective and hence, ζ ∈ V. Thus, L ⊗ M is in V. This completes the proof. □

It remains to show that X and V can be found, as in the Proposition, such that the additive tensor ideal V is nontrivial, perhaps even interesting. The next example provides some hint. In the next section, we see that this all leads to a variety that could be of some interest.

Example 8.5. For the example, consider the situation in Example 8.3. For n large, let γ = θ̂ where θ = ζ_{1}^{n}ζ_{2}^{n}η_{1} and let M = L_{γ}. Then the annihilator of γ in H^{∗}(G, k) is generated by η_{2}, ζ_{1}^{n+1} and ζ_{2}^{n+1}. Let V_{s} = η_{1}η_{2}H^{2s}(G, k) for some s. Let V_{s} be the full subcategory of all L such that I_{L} ∩ H^{2s}(G, k) ⊂ V_{s}. Then M ∈ V_{s} if and only if s ⩽ n.

Next notice that for any s, V_{s+1} ⊆ V_{s}. For if M is not in V_{s}, then there is some element ϑ in I_{M} ∩ H^{2s}(G, k) such that ϑ ∉ V_{s}. But then, ζ_{1}ϑ is not in V_{s+1} and M is not in V_{s+1}.

The above argument is easily generalized to elementary abelian p-groups of larger rank. Hence, we have the following.

Theorem 8.6. Suppose that p > 3 and that G is an elementary abelian p-group of rank at least two. There exist a collection of k-subspaces {V_{s}}, with V_{s} ⊆ H^{2s}(G, k) such that if V_{s} is the full subcategory of stmod(kG) of all modules M such that every element of H^{2s}(G, k) that is M-projective is in V_{s}, then every V_{s} is an additive tensor ideal and

V_{s} ⊃ V_{s+1} ⊃ V_{s+2} ⊃ …

9. Virtual complexity and virtual varieties

By considering the virtual projectivity of a fixed module, we can define a variety in the spectrum of the cohomology. The definition is the following. Let X and M be kG-modules. Recall that Ext_{kG}^{∗}(X, X) is a finitely generated module over H^{∗}(G, k), and the image of

Tr_{M}: Ext_{kG}^{∗}(M ⊗ X, M ⊗ X) → Ext_{kG}^{∗}(X, X)

is an ideal. It follows that Ext_{kG}^{∗}(X, X)/Im(Tr_{M}) is also finitely generated as a module over H^{∗}(G, k).

Definition 9.1. The virtual M-complexity of X is the polynomial rate of growth of the quotient ring Ext_{kG}^{∗}(X, X)/Im(Tr_{M}), namely the smallest integer c such that there is a constant B with

Dim_{k}Ext_{kG}^{n}(X, X)/Im(Tr_{M}) ⩽ B.n^{c−1}

for all n.

Lemma 9.2. The module X is virtually M-projective if and only if its virtual M-complexity is equal to zero.

Proof. From the definition, we see that X has virtual M-complexity zero if and only if Ext_{kG}^{n}(X, X)/Im(Tr_{M}) is eventually zero, namely if and only if every element of large enough degree in Ext_{kG}^{n}(X, X) is in the image of Tr_{M}. This is the definition of virtual projectivity. □

Definition 9.3. We define J_{M}(X) to be the kernel of

H^{∗}(G, k) → Ext_{kG}^{∗}(X, X)/Im(Tr_{M}).

Thus, for example, J(X) = J_{kG}(X) where J(X) is the annihilator in H^{∗}(G, k) of the cohomology ring of X..

We define the virtual M-variety of X, denoted W_{M}(X), to be the subvariety of V_{G}(k) defined by the ideal J_{M}(X).

Thus we have the following.

Lemma 9.4. The virtual M-complexity of X is equal to the dimension of W_{M}(X). □

Lemma 9.5. For kG-modules, M, X and Y, W_{M}(X ⊕ Y) = W_{M}(X) ∪ W_{M}(Y).

Proof. Examining the map

H^{∗}(G, k) → Ext_{kG}^{∗}(X, X) ⊕ Ext_{kG}^{∗}(Y, Y) ⊆ Ext_{kG}^{∗}(X ⊕ Y, X ⊕ Y),

we see that J_{M}(X ⊕ Y) = J_{M}(X) ∩ J_{M}(Y). □

Lemma 9.6. Given M and X, W_{M}(X) ⊆ V_{G}(X) ∩ W_{M}(k).

Proof. Since J_{M}(X) ⊇ J(X) we have W_{M}(X) ⊆ V_{G}(X). The commutativity of the diagram

\xymatrix{Ext_{kG}^{∗}(M, M) \ar[r]^{^{Tr^{M}}} \ar[d] & Ext_{kG}^{∗}(k, k) \ar[d]^{ι_{X}} \\ Ext_{kG}^{∗}(M ⊗ X, M ⊗ X) \ar[r]^{_{Tr_{M}}} & Ext_{kG}^{∗}(X, X)}

proves that J_{M}(X) ⊇ J_{M}(k), and so W_{M}(X) ⊆ W_{M}(k). □

Remark 9.7. Equality does not always hold in the above lemma. For example, taking X = M, we have W_{X}(X) = 0. But there are modules X such that V_{G}(X) = V_{G}(k) but k is not virtually X-projective so W_{X}(k) ≠ 0. For an example, take G = H × J where H and J are p-groups. Let X = M = Ω_{J}^{n}(k_{J}) as in the proof of Theorem 7.1. Then J_{M}(k) is the ideal generated by the images of the inflation maps H^{d}(G/H, k) → H^{d}(G, k) for all d ≥ n. It follows that W_{M}(k) ≅ V_{H}(k).

Lemma 9.8. For kG-modules M and X, V_{G}(X) ⊆ V_{G}(M) ∪ W_{M}(X).

Proof. It follows from Lemma 2.2 that J(M) · Im(Tr_{M}) = 0 in Ext_{kG}^{∗}(X, X). Thus we have J(M) · J_{M}(X) ⊆ J(X), and so V_{G}(M) ∪ W_{M}(X) ⊇ V_{G}(X). □

Lemma 9.9. For kG-modules, M, X and Y, we have W_{M⊕N}(X) = W_{M}(X) ∩ W_{N}(X) and W_{M⊗N}(X) = W_{M}(X) ∪ W_{N}(X).

Proof. The first statement follows from the fact that J_{M⊕N}(X) = J_{M}(X) + J_{N}(X). The second one follows from the inclusions

J_{M}(X) · J_{N}(X) ⊆ J_{M⊗N}(X) ⊆ J_{M}(X) ∩ J_{N}(X).  □

Question 9.10. Given a kG-module X, what subvarieties of V_{G}(X) can be realized as W_{M}(X)? In particular, given a closed homogeneous subvariety V of V_{G}(k), does there exist M with W_{M}(k) = V? It follows from Lemma 9.9 that for an affirmative answer to this question, it is only necessary to realize irreducible hypersurfaces.

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Institute of Mathematics, Fraser Noble Building, Univeristy of Aberdeen, Aberdeen AB24 3UE, United Kingdom

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