NILPOTENCE AND GENERATION IN THE STABLE MODULE CATEGORY

DAVID J. BENSON AND JON F. CARLSON

Abstract. Nilpotence has been studied in stable homotopy theory and algebraic geometry. We study the corresponding notion in modular representation theory of finite groups, and apply the discussion to the study of ghosts, and generation of the stable module category. In particular, we show that for a finitely generated kG-module M, the tensor M-generation number and the tensor M-ghost number are both equal to the degree of tensor nilpotence of a certain map associated with M.

1. Introduction

The study of nilpotence started in stable homotopy theory [11, 13, 20]. Analogues have also been studied in algebraic geometry, see for example Theorems 3.6 and 3.8 of Thomason [22]. In this paper, we study the analogous notion in modular representation theory, and relate it to tensor notions of ghost maps and categorical generations. The emphasis on concepts involving tensor closure seems practical in light of the fact that the thick tensor ideal generated by the trivial module k is the entire stable category, even though the thick subcategory generated by k (without closure under arbitrary tensors) is not.

The context here is the stable category stmod(kG) of finitely generated kG-modules. This is a tensor triangulated category with the tensor being ⊗ = ⊗_{k}, the action of the group on a tensor product of modules being given by the usual diagonal Hopf algebra structure g ↦ g ⊗ g for g ∈ G. It is the quotient category of the ordinary category of kG-modules and kG-homomorphisms by the projective modules. That is, the maps of kG-modules in which are zero in stmod(kG) are those that factor through some projective module. We call these null maps. A map f: N → M is said to be tensor nilpotent if for some value n, f^{⊗n}: N^{⊗n} → M^{⊗n} is null.

As with many notions in modular representation theory, tensor nilpotence is controlled by elementary abelian p-subgroups. Using some rather well known methods, we prove in Section 2 that there is a constant C, depending only on G, such that if f: N → M is a map of kG-modules which is null on restriction to every elementary abelian p-subgroup E of G then f^{⊗C}: N^{⊗C} → M^{⊗C} is null. On the other hand, even for an elementary abelian p-group E there is no bound on the possible tensor nilpotence of maps in the stable module category. We give examples for the Klein four group ℤ/2 × ℤ/2 of maps with arbitrarily large degree of tensor nilpotence.

The same group provides examples of maps f: k → M where the question of whether f is tensor nilpotent depends on the Hopf algebra structure we put on kE. In our example, M is two dimensional. With the group theoretic Hopf algebra structure f^{⊗3} is null, whereas with the Hopf algebra structure coming from regarding kE as a restricted enveloping of a restricted p-Lie algebra, no power of f is null.

The second author was partially supported by NSA grant H98230-15-1-0007 and Simons Foundation grant 315728.

We develop a related notion of strong nilpotence of maps from the trivial module over elementary abelian groups. This concept concerns factorization through the second socle of the projective module and is better behaved than ordinary nilpotence. In Section 6 we prove the following.

Theorem 1.1. (i) If a map of kG-modules is strongly nilpotent then it is tensor nilpotent.

(ii) The strong nilpotence of a particular map f: k → M (or f: N → k) does not depend on the Hopf algebra structure on kG.

(iii) There is a bound C depending only on G and not on f such that if f is strongly nilpotent then f^{⊗C} is null.

In the latter part of the paper, we apply these notions to questions about ghosts and generation. A tensor M-ghost is a map f: X → Y such that for all modules U,

f_{∗}: Ex̂t_{kG}^{∗}(M ⊗ U, X) → Ex̂t_{kG}^{∗}(M ⊗ U, Y)

is the zero map. It turns out that this is equivalent to the statement that

Id_{M} ⊗ f: M ⊗ X → M ⊗ Y

is null. For example, a tensor k-ghost is the same as a null map. Given a kG-module M, take the adjoint k → M ⊗ M^{∗} of the identity map on M, and complete to a triangle

N ⟶^{f} k → M ⊗ M^{∗}

in stmod(kG). Then f is a tensor M-ghost. The tensor M-ghost number is the smallest n such that the composite of n tensor M-ghosts is always null, or infinity if there is no such n.

For a kG-module M, the tensor M-generation number is the smallest n such that every module can be built in at most n steps from modules of the form M ⊗ X, or infinity if there is no such n. These notions are related by the following theorem, proved in Section 8.

Theorem 1.2. Given a kG-module M, form the triangle N ⟶^{f} k → M ⊗ M^{∗} in stmod(kG) as above. Then the following statements are equivalent.

(i) The tensor M-generation number is at most n.

(ii) The map f^{⊗n} is null.

(iii) The tensor M-ghost number is at most n.

These statements hold for some n ≥ 1 if and only if the support variety V_{G}(M) is equal to V_{G}(k).

This leads us to the following question, which we study in Section 9.

Question 1.3. For kG-modules with V_{G}(M) = V_{G}(k), is there a bound on the tensor Mgeneration number that depends only on G and not on M?

We show that an affirmative answer to this question would follow from the conjecture below.

Conjecture 1.4. If M is a kG-module with V_{G}(M) = V_{G}(k) then the map f in the triangle N ⟶^{f} k → M ⊗ M^{∗} is strongly nilpotent.

This conjecture reduces to the case that G = E is elementary abelian, and in that case it is equivalent to the surjectivity of the trace map

Tr_{M}: Ext_{kE}^{1}(M, M) → Ext_{kE}^{1}(k, k)

for modules M with V_{E}(M) = V_{E}(k). We provide some evidence for this conjecture, based on extensive computations.

The authors wish to thank Paul Balmer for several helpful suggestions.

2. Tensor nilpotence

Let G be a finite group and k a field of characteristic p, which we assume divides the order of G. The major portion of the results in this paper concerns finitely generated kG-modules. However, we employ some constructions, such as the colimit in Lemma 2.3, below, that require a larger context. The larger context is the big stable module category StMod(kG) whose objects are all kG-modules and whose morphisms are module homomorphisms modulo those that factor through some projective module. We say that a map of kG-modules is null if it is zero in StMod(kG), namely when it factors through a projective module. We write \underline{Hom}_{kG}(N, M) for the homomorphisms modulo null homomorphisms, namely the Hom sets in StMod(kG).

We write stmod(kG) for the full subcategory of StMod(kG) whose objects are the finitely generated modules. Both StMod(kG) and stmod(kG) are triangulated categories in which the triangles come from the short exact sequences in the module category. The shift in these categories is the inverse Ω^{−1} of the Heller operator Ω. See for example, Chapter 1 of Happel [12] for more details.

As in any triangulated category, the homomorphisms are graded using the shift. In the stable module category, this gives us Tate cohomology:

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

In particular, Ex̂t_{kG}^{0}(N, M) is just another name for \underline{Hom}_{kG}(N, M).

Definition 2.1. Let M and N be kG-modules. We say that a map f: N → M is tensor nilpotent if some tensor power f^{⊗n}: N^{⊗n} → M^{⊗n} is null. The degree of tensor nilpotence is the smallest such n.

Remark 2.2. Let f: N → M be a map of finitely generated kG-modules. Then the following are equivalent.

(1) The map f is tensor nilpotent,

(2) The adjoint map k → N^{∗} ⊗ M is tensor nilpotent,

(3) The other adjoint N ⊗ M^{∗} → k is tensor nilpotent.

The following is basically Lemma 5.1.5 of [16].

Lemma 2.3. A map f: k → M is tensor nilpotent if and only if the colimit of

k ⟶^{f} M ⟶^{f⊗1} M ⊗ M ⟶^{f⊗1⊗1} M ⊗ M ⊗ M → ⋯

is projective.

Proof. If the colimit is projective then the map from k to the colimit factors through the injective hull I_{k} of k, which is a finitely generated projective. The map from I_{k} then factors through some finite stage of the filtered system, and so for some n > 0 the composite k → M^{⊗n} is null. This composite is equal to the nth tensor power of f.

Conversely, if f^{⊗n}: k → M^{⊗n} is null then every composite of n consecutive maps in the filtered system is null, and so the colimit is projective. □

The first immediate consequence of this lemma is that we only need consider elementary abelian p-groups, as we see in the following theorem.

Theorem 2.4. Let G be a finite group. Then there is a constant C depending only on G such that given any string of C composable maps of kG-modules that are all null on restriction to every elementary abelian p-subgroup of G, their composite is null as a map of kG-modules.

In particular, if f: N → M is null on restriction to every elementary abelian p-subgroup of G then f^{⊗C}: N^{⊗C} → M^{⊗C} is null.

Proof. The first statement is the essence of Theorem 2.5 of [7], which works equally well for Tate cohomology. For the last statement, note that f^{⊗C} is a composite

N^{⊗C} ⟶^{f⊗Id_{N}^{⊗(C−1)}} M ⊗ N^{⊗(C−1)} ⟶^{Id_{M} ⊗f⊗Id_{N}^{⊗(C−2)}} M^{⊗2} ⊗ N^{⊗(C−2)} → ⋯ → M^{⊗C}

of C maps that are null on restriction to every elementary abelian p-subgroup of G. □

Remark 2.5. If k has characteristic p and E = ⟨g_{1}, …, g_{r}⟩ ≅ (ℤ/p)^{r}, then we have the following test for whether a map f: N → M is null. Let X_{i} = g_{i} − 1 ∈ kE. Then the norm element of kE is equal to

∑_{g∈E} g = X_{1}^{p−1} … X_{r}^{p−1}.

A map f: N → M of kE-modules is null if and only if f is in the image of multiplication by the norm element on Hom_{k}(N, M). If N = k, this is equivalent to saying that f(1) is in the image of the norm element on M. This implies that M has a copy of kE as a direct summand.

3. Hopf structures

For an elementary abelian p-group E, there are many different Hopf algebra structures that one can put on the group algebra kE, and hence many different ways of taking tensor powers and duals of modules.

The usual one has as its diagonal map Δ(g) = g⊗g for group elements g. Thus if X = g−1 then

Δ(X) = X ⊗ 1 + 1 ⊗ X + X ⊗ X.

Often it is easier to work with the Hopf algebra structure which results from regarding kE as the restricted enveloping algebra U^{[p]}(L) of a commutative restricted Lie algebra L with zero p-power operation. The diagonal map is then given by Δ(X) = X ⊗ 1 + 1 ⊗ X, where X is a Lie algebra element. The reason why this is more convenient is that cyclic shifted subgroups then define Hopf subalgebras of kE, and so restriction to any of these objects commutes with taking tensor products and homs.

We first set up some notation, and then address the question of the effect of changing the Hopf algebra structure.

Let E = ⟨g_{1}, …, g_{r}⟩ ≅ (ℤ/p)^{r} be an elementary abelian p-group of rank r. We set

X_{i} = g_{i} − 1 ∈ kE, 1 ≤ i ≤ r.

This allows us to regard kE as a truncated polynomial ring,

kE = k[X_{1}, …, X_{r}]/(X_{1}^{p}, …, X_{r}^{p}).

If 0 ≠ α = (λ_{1}, …, λ_{r}) ∈ 𝔸^{r}(k), we set

X_{α} = λ_{1}X_{1} + ⋯ + λ_{r}X_{r} ∈ kE.

We have X_{α}^{p} = 0, and so setting g_{α} = 1 + X_{α}, the element g_{α} ∈ kE is an element of order p. The group it generates is called a cyclic shifted subgroup of kE with respect to the basis g_{1}, …, g_{r}.

The identification of kE with U^{[p]}(L) depends on the choice of a set of generators for E. This identification is given by letting L be the linear span of the elements X_{1}, …, X_{r} defined above. The primitive elements of U^{[p]}(L) are then just the elements of L. Thus we have

Δ(X_{α}) = X_{α} ⊗ 1 + 1 ⊗ X_{α}.

Theorem 3.1. Endow kE with the Lie theoretic Hopf algebra structure. Then a map f: N → M of finitely generated kE-modules is tensor nilpotent if and only if its restriction to every cyclic shifted subgroup is tensor nilpotent.

Proof. If f is tensor nilpotent, then f restricted to every cyclic shifted subgroup is tensor nilpotent, because restriction to cyclic shifted subgroups commutes with tensor products.

For the converse, by Remark 2.2 we may assume that N = k. So suppose that the restriction of f: k → M to every cyclic shifted subgroup is tensor nilpotent. This is equivalent to the statement that the restriction itself factors through a projective module, without taking tensor powers. Form the colimit C of k → M → M ⊗ M → ⋯ as in Lemma 2.3. Then the restriction of C to every cyclic shifted subgroup is projective.

We wish to apply Theorem 5.2 of [3], which is a version of Dade’s lemma (see Lemma 11.8 of [10]) for infinitely generated modules, to deduce that C is projective. Then we apply Lemma 2.3 again to deduce that f is tensor nilpotent. This version of Dade’s lemma requires the restriction of C to a cyclic shifted subgroup to remain projective after extending to every larger algebraically closed field K ⊇ k.

For this purpose, it remains to prove the following statement. If M is a finitely generated kG-module and f: k → M has the property that for every cyclic shifted subgroup, the restriction of f factors through a projective module, then the same is true after extending to any algebraically closed extension field K ⊇ k.

To see this, consider the statement that for all 0 ≠ α ∈ 𝔸^{r}(k) there exists m ∈ M such that f(1) = X_{α}^{p−1}m. This is equivalent to the restriction to X_{α} factoring through a projective module. Since M is finite dimensional, this is a first order formula with parameters in k. By model completeness of algebraically closed fields (this is essentially the theory of quantifier elimination for algebraically closed fields, see for example Corollary 3.2.3 of Marker [18]), it is true over k if and only if it is true over K. This completes the proof.

Note that one can also deal with this last step directly by writing out matrix equations and looking at ranks. This leads to writing 𝔸^{r}(k) \ {0} as a finite union of subsets, each of which is the intersection of an open and a closed set in the Zariski topology. The fact that this remains true after extending these subsets to K can be checked directly using generic points and Hilbert’s Nullstellensatz, or indirectly using model completeness as before. □

Corollary 3.2. Again, endow kE with the Lie theoretic Hopf structure. Then a map f: k → M is tensor nilpotent if and only if f(1) is in the image of X_{α} for all non-zero α = (λ_{1}, …, λ_{r}).

Unfortunately, the theorem and corollary are false with the group theoretic Hopf structure, as we see in the next section.

We end the section by emphasizing one aspect of the proof of Theorem 3.1 that could be of independent interest.

Proposition 3.3. Let E be an elementary abelian p-group and k an algebraically closed field of characteristic p. Suppose that M and N are finitely generated kE-modules and that f : M → N is a homomorphism such that restriction to every cyclic shifted subgroup of kE is null (factors through a projective for the cyclic shifted subgroup). Then for any extension K of k, the restriction of K ⊗ f : K ⊗ M → K ⊗ N is null on restriction to every cyclic shifted subgroup of KE.

4. Tensor nilpotence depends on Hopf structure

In this section, we give an example to show that for a map f: k → M of kE-modules, different Hopf algebra structures can give different answers to the question of whether f is tensor nilpotent.

Let E = ⟨g_{1}, g_{2}⟩ ≅ (ℤ/2)^{2}, the Klein four group, and let k be a field of characteristic two. Set X_{1} = g_{1} − 1, X_{2} = g_{2} − 1 in kE. Note that a map f: k → M of kE-module is null if and only if f(1) is in the image of the norm element X_{1}X_{2} = 1 + g_{1} + g_{2} + g_{1}g_{2} ∈ kE (see 2.5).

Let λ ∈ k with λ ≠ 0 and λ ≠ 1. Let M_{λ} be the two dimensional kE-module with k-basis elements u and v, such that

X_{1}u = λv, X_{2}u = v, X_{1}v = 0, X_{2}v = 0.

Let f : k → M_{λ} be the kE-module homomorphism sending 1 to v.

If we use the tensor product coming from the group structure then the action on tensor products is given by

X_{i}(a ⊗ b) = X_{i}a ⊗ b + a ⊗ X_{i}b + X_{i}a ⊗ X_{i}b.

With this tensor product we have

X_{1}X_{2}(u ⊗ u ⊗ u) = X_{1}(v ⊗ u ⊗ u + u ⊗ v ⊗ u + u ⊗ u ⊗ v
+ u ⊗ v ⊗ v + v ⊗ u ⊗ v + v ⊗ v ⊗ u + v ⊗ v ⊗ v)
= λ v ⊗ v ⊗ u + λ v ⊗ u ⊗ v + λ^{2} v ⊗ v ⊗ v
+ λ v ⊗ v ⊗ u + λ u ⊗ v ⊗ v + λ^{2} v ⊗ v ⊗ v
+ λ v ⊗ u ⊗ v + λ u ⊗ v ⊗ v + λ^{2} v ⊗ v ⊗ v
+ λ v ⊗ v ⊗ v + λ v ⊗ v ⊗ v + λ v ⊗ v ⊗ v
= λ(λ + 1) v ⊗ v ⊗ v.

Since λ ∉ {0, 1}, we have λ(λ + 1) ≠ 0, and the element v ⊗ v ⊗ v is in the image of X_{1}X_{2}. It follows that f^{⊗3} : k → M_{λ}^{⊗3} is null, and f is tensor nilpotent.

On the other hand, if we use the Lie theoretic tensor product, we have

X_{i}(a ⊗ b) = X_{i}a ⊗ b + a ⊗ X_{i}b.

With this tensor product, X_{1} + λX_{2} acts as zero on M_{λ} ⊗ M_{λ}, from which it follows that

M_{λ} ⊗ M_{λ} ≅ M_{λ} ⊕ M_{λ}.

So no tensor power of M_{λ} has a non-zero projective summand, and f: k → M_{λ} is not tensor nilpotent.

5. Tensor nilpotence is unbounded

In an attempt to answer Question 1.3, one might try to prove that there is a bound on the tensor powers that appear for tensor nilpotent maps, and then apply Theorem 1.2. In this section, we give a family of examples to show that this is not the case. Specifically we show the following. Note that the class of tensor nilpotence of a map f is the smallest d such that f^{⊗d} = 0.

Proposition 5.1. Suppose that p = 2 and that E ≅ (ℤ/2)^{2} is an elementary abelian group of order 4. There exist a sequence {M_{d}} of kE-modules such that for all d > 2 there is a map f_{d} : k → M_{d} that is tensor nilpotent of class d but not of class d − 1.

Proof. Let E = ⟨g_{1}, g_{2}⟩ ≅ ℤ/2 × ℤ/2, and let k be a field of characteristic two. Set X_{1} = g_{1} − 1, X_{2} = g_{2} − 1 in kE. Let M_{d} be the 2d dimensional kE-module with k-basis elements a_{1}, …, a_{d}, b_{1}, …, b_{d} on which E acts

X_{1}(a_{i}) = b_{i} (1 ≤ i ≤ d), X_{2}(a_{i}) = b_{i+1} (1 ≤ i ≤ d − 1), X_{2}(a_{d}) = 0,
X_{1}(b_{i}) = X_{2}(b_{i}) = 0 (1 ≤ i ≤ d).

To draw a diagram of this module, we indicate the action of X_{1} by single lines and of X_{2} by double lines.

[figure: •^{a_{1}}; •^{a_{2}}; •^{a_{d−1}}; •^{a_{d}}; •_{b_{1}}; •_{b_{2}}; ⋯; •_{b_{d}}]

We claim that the map k → M_{d} taking 1 to b_{2} is tensor nilpotent, with degree of nilpotence equal to d. Suppose we are looking at the tensor ith power. There is only one monomial (up to permuting the factors) to which we can apply X_{1}X_{2} and have b_{2}^{⊗i} appear in the answer, namely

X_{1}X_{2}(b_{2}^{⊗(i−2)} ⊗ a_{1} ⊗ a_{2}) = b_{2}^{⊗i} + b_{2}^{⊗(i−2)} ⊗ b_{1} ⊗ b_{3}.

To get rid of the second term on the right, again up to permuting the factors there is only one monomial we can use at each stage. We have

X_{1}X_{2}(b_{2}^{⊗(i−3)} ⊗ a_{1} ⊗ b_{1} ⊗ a_{3}) = b_{2}^{⊗(i−2)} ⊗ b_{1} ⊗ b_{3} + b_{2}^{⊗(i−3)} ⊗ b_{1}^{⊗2} ⊗ b_{4},
X_{1}X_{2}(b_{2}^{⊗(i−4)} ⊗ a_{1} ⊗ b_{1}^{⊗2} ⊗ a_{4}) = b_{2}^{⊗(i−3)} ⊗ b_{1}^{⊗2} ⊗ b_{4} + b_{2}^{⊗(i−4)} ⊗ b_{1}^{⊗3} ⊗ b_{5}

and so on. To get to the other end of the module takes d steps, with

X_{1}X_{2} (∑_{j=2}^{d} b_{2}^{⊗(i−j)} ⊗ a_{1} ⊗ b_{1}^{⊗(j−2)} ⊗ a_{j}) = b_{2}^{⊗d}.  □

6. Strong nilpotence

Since tensor nilpotence is detected in a well controlled manner on elementary abelian subgroups, in this section we formulate a version of tensor nilpotence that is easier to manage, in terms of restriction to elementary abelian subgroups. All modules in this section are finitely generated.

For notation, Rad(M) is the Jacobson radical of the module M, the intersection of all maximal submodules and Rad^{2}(M) = Rad(Rad(M)). The socle Soc(M) is the sum of the minimal submodules of M, and Soc^{2}(M) is the inverse image in M of Soc(M/Soc(M)).

Definition 6.1. Let E = ⟨g_{1}, …, g_{r}⟩ ≅ (ℤ/p)^{r} be an elementary abelian p-group of rank r, and let f: k → M be a kE-module homomorphism. We say that f is strongly nilpotent if, when k is regarded as Soc(kE), the map f extends to a map from Soc^{2}(kE) to M, as in the following diagram:

\xymatrix{k \ar[r]^{f} \ar[d] & M \\ Soc^{2}(kE) \ar@{.>}[ur]}

Lemma 6.2. If a map of kE-modules f: k → M is strongly nilpotent and E′ ⩽ E is a subgroup then f is also strongly nilpotent as a map of kE′-modules.

Proof. Soc^{2}(kE) contains a kE′-submodule isomorphic to Soc^{2}(kE′), and containing k. □

Definition 6.3. For any finite group G, we say that a map f: N → M of kG-modules is strongly nilpotent if and only if its adjoint k → N^{∗} ⊗ M is strongly nilpotent on restriction to every elementary abelian p-subgroup of G.

Lemma 6.4. Let E be an elementary abelian p-group and f: N → k a map of kE-modules. Then the following are equivalent.

(i) f is strongly nilpotent.

(ii) Regarding k as kE/Rad(kE), f lifts to a map N → kE/Rad^{2}(kE), as in the following

diagram:

\xymatrix{ & kE/Rad^{2}(kE) \ar[d] \\ N \ar[r]^{f} \ar@{.>}[ur] & k}

(iii) For all extensions 0 → k → U → k → 0 in Ext_{kE}^{1}(k, k), f factors through U as in

the following diagram:

\xymatrix{ & U \ar[d] \\ N \ar[r]^{f} \ar@{.>}[ur] & k}

8

Proof. By definition, f: N → k is strongly nilpotent if and only if the dual map f^{∗}: k → N^{∗} is strongly nilpotent. So the equivalence of (i) and (ii) follows by noting that kE/Rad^{2}(kE) is isomorphic to the dual of Soc^{2}(kE).

Given any extension 0 → k → U → k → 0, there is a map kE/Rad^{2}(kE) → U whose composite with U → k is the map kE/Rad^{2}(kE) → k. Thus (ii) implies (iii). To see that (iii) implies (ii), let 0 → k → U_{i} → k → 0 (1 ⩽ i ⩽ r) be extensions representing a basis for Ext_{kE}^{1}(k, k). Then kE/Rad^{2}(kE) → k is the pullback of the maps U_{i} → k. So finding a lift of f: N → k to N → kE/Rad^{2}(kE) is equivalent to finding a lift to N → U_{i} for 1 ⩽ i ⩽ r. □

Similarly, we have the following.

Lemma 6.5. Let E be an elementary abelian p-group and f: k → M be a map of kEmodules. Then f is strongly nilpotent if and only if for all extensions 0 → k → U → k → 0 in Ext_{kE}^{1}(k, k), f factors through U as in the following diagram:

\xymatrix{k \ar[r]^{f} \ar[d] & M \\ U \ar@{.>}[ur]}

Proof. The proof is dual to the proof of the equivalence of (ii) and (iii) in Lemma 6.4. □

Lemma 6.6. Suppose that E is an elementary abelian p-group. Let f: N → M be a map of kE-modules. If f is strongly nilpotent, then f is tensor nilpotent. More precisely, f^{⊗r(p−1)} is null.

Proof. Let X_{i} = g_{i} − 1 ∈ kE for 1 ≤ i ≤ r. Then strong nilpotence means that for each i there exists f_{i}: N → M such that f = X_{i}f_{i} and X_{j}f_{i} = 0 for j ≠ i. So we have that

f^{⊗r(p−1)} = X_{1}^{p−1} … X_{r}^{p−1}(f_{1}^{⊗(p−1)} ⊗ ⋯ ⊗ f_{r}^{⊗(p−1)})

is null. □

Theorem 6.7. For a general finite group G, there is a constant C depending only on G such that if f: N → M is a strongly nilpotent map of kG-modules then f^{⊗C} is null.

Proof. This follows from Theorem 2.4 and Lemma 6.6. □

Remark 6.8. One of the advantages of strong nilpotence is that it is less dependent on the Hopf algebra structure. In particular, the condition for a map f: k → M to be strongly nilpotent does not mention the Hopf algebra structure. The same is true of maps f: N → k. For more general maps f: N → M, the situation is less clear.

7. Ghosts

In this section we present some background on ghost maps. In this section all modules should be assumed to be finitely generated. Other papers studying ghosts include Kelly [17], Christensen [9], Hovey and Lockridge [14, 15], Oppermann and Šťovíček [19] and Beligiannis [1].

Definition 7.1. Let M be a kG-module. For X and Y finite dimensional kG-modules, we say that a map of kG-modules f: X → Y is an M-ghost if f_{∗}: Ex̂t_{kG}^{∗}(M, X) → Ex̂t_{kG}^{∗}(M, Y) is the zero map.

Although this definition makes sense for arbitrary modules M, we shall assume during this section that M is finitely generated.

Lemma 7.2. Let M be a finitely generated kG-module and let f: X → Y be a homomorphism of finitely generated kG-modules. The following are equivalent

(i) For all finitely generated kG-modules U, f: X → Y is an M ⊗ U-ghost.

(ii) For all finitely generated kG-modules U, f: X → Y is an M^{∗} ⊗ U-ghost.

(iii) For all finitely generated kG-modules V, f ⊗ Id_{V} : X ⊗ V → Y ⊗ V is an M-ghost. (iv) For all finitely generated kG-modules V, f ⊗ Id_{V} : X ⊗ V → Y ⊗ V is an M^{∗}-ghost.

(v) The map Id_{M} ⊗ f: M ⊗ X → M ⊗ Y is null.

(vi) The map Id_{M^{∗}} ⊗ f: M^{∗} ⊗ X → M^{∗} ⊗ Y is null.

Proof. To see that (i) and (iii) are equivalent, take U and V to be duals of each other. Then the following diagram commutes.

\xymatrix{Ex̂t_{kG}^{∗}(M ⊗ U, X) \ar[r]^{f_{∗}} \ar[d]^{≅} & Ex̂t_{kG}^{∗}(M ⊗ U, Y) \ar[d]^{≅} \\ Ex̂t_{kG}^{∗}(M, X ⊗ V) \ar[r]^{(f⊗Id_{V} )_{∗}} & Ex̂t_{kG}^{∗}(M, Y ⊗ V).}

Similarly, (ii) and (iv) are equivalent.

To see that (i) implies (v), take U = M^{∗} ⊗ X in (i) to see that f induces the zero map as the upper horizontal map in the following diagram:

\xymatrix{Ex̂t_{kG}^{∗}(M ⊗ M^{∗} ⊗ X, X) \ar[d]^{≅} \ar[r]^{f_{∗}} & Ex̂t_{kG}^{∗}(M ⊗ M^{∗} ⊗ X, Y) \ar[d]^{≅} \\ Ex̂t_{kG}^{∗}(M ⊗ X, M ⊗ X) \ar[r]^{(Id_{M} ⊗f)_{∗}} & Ex̂t_{kG}^{∗}(M ⊗ X, M ⊗ Y)}

The image of the identity element in Ex̂t_{kG}^{0}(M ⊗ X, M ⊗ X) under the lower horizontal map is

Id_{M} ⊗ f: M ⊗ X → M ⊗ Y,

which is therefore null. Similarly, (ii) implies (vi).

To see that (vi) implies (i), if Id_{M^{∗}} ⊗ f is null then f induces the zero map

\xymatrix{Ex̂t_{kG}^{∗}(U, M^{∗} ⊗ X) \ar[r]^{(Id_{M^{∗}}⊗f)_{∗}} \ar[d]^{≅} & Ex̂t_{kG}^{∗}(U, M^{∗} ⊗ Y) \ar[d]^{≅} \\ Ex̂t_{kG}^{∗}(M ⊗ U, X) \ar[r]^{f_{∗}} & Ex̂t_{kG}^{∗}(M ⊗ U, Y)}

and so f is an M ⊗ U-ghost. Similarly, (v) implies (ii). □

Definition 7.3. We say that a map of finitely generated kG-modules f: X → Y is a tensor M-ghost if the equivalent conditions of Lemma 7.2 are satisfied. It is apparent from the lemma that f is a tensor M-ghost if and only if it is a tensor M^{∗}-ghost.

Lemma 7.4. A map of finitely generated kG-modules f: X → Y is a tensor k-ghost if and only if it is null.

Proof. See statement (v) of Lemma 7.2. □

Lemma 7.5. Let ι : k → M ⊗ M^{∗} be the adjoint to the identity map M → M, and complete to a triangle

\xymatrix{N \ar[r]^{f} & k \ar[r]^{ι} & M ⊗ M^{∗} \ar[r] & Ω^{−1}(N)}

in stmod(kG). Then f : N → k is a tensor M-ghost. This map is null if and only if the dimension of M is not divisible by p.

Proof. The map 1 ⊗ ι : M^{∗} ≅ M^{∗} ⊗ k → M^{∗} ⊗ M ⊗ M^{∗} is split injective. That is, the trace map Tr_{M} : M^{∗} ⊗ M → k, given by Tr_{M}(λ ⊗ m) = λ(m) has the property that Tr_{M} ⊗ 1 : M^{∗} ⊗ M ⊗ M^{∗} → k ⊗ M^{∗} ≅ M^{∗} splits 1 ⊗ ι, meaning (Tr_{M} ⊗ 1)(1 ⊗ ι) is the identity on M^{∗}. Hence in the triangle

\xymatrix{M^{∗} ⊗ N \ar[r]^{1⊗f} & M^{∗} \ar[r]^{1⊗ι} & M^{∗} ⊗ M ⊗ M^{∗} \ar[r] & M^{∗} ⊗ Ω^{−1}(N)}

the map 1 ⊗ f is null.

For the last statement, the map N → k is null if and only if k → M ⊗M^{∗} is split injective. By Theorem 2.1 of Benson and Carlson [2], this happens if and only if the dimension of M is not divisible by p. □

Lemma 7.6. (i) Suppose that M_{1} → M_{2} → M_{3} ⊕ Z is a triangle in stmod(kG). If f: N_{1} → N_{2} is an M_{1}-ghost and f′: N_{2} → N_{3} is an M_{2}-ghost then f′ ∘ f: N_{1} → N_{3} is an M_{3}-ghost.

(ii) Suppose that M_{1}⊗U_{1} → M_{2}⊗U_{2} → M_{3}⊕Z is a triangle in stmod(kG). If f: N_{1} → N_{2} is a tensor M_{1}-ghost and f′: N_{2} → N_{3} is a tensor M_{2}-ghost then f′ ∘f: N_{1} → N_{3} is a tensor M_{3}-ghost.

Proof. For a kG-module U consider the diagram

[figure: Ex̂t_{kG}^{i}(M_{2} ⊗ U_{2} ⊗ U, N_{1}); Ex̂t_{kG}^{i}(M_{2} ⊗ U_{2} ⊗ U, N_{2}); 0; ⟳; Ex̂t_{kG}^{i}(M_{2} ⊗ U_{2} ⊗ U, N_{3}); Ex̂t_{kG}^{i}((M_{3} ⊕ Z) ⊗ U, N_{1}); f_{∗}; Ex̂t_{kG}^{i}((M_{3} ⊕ Z) ⊗ U, N_{2}); f′_{∗}; Ex̂t_{kG}^{i}((M_{3} ⊕ Z) ⊗ U, N_{3}); ⟳; Ex̂t_{kG}^{i+1}(M_{1} ⊗ U_{1} ⊗ U, N_{1}); 0; Ex̂t_{kG}^{i+1}(M_{1} ⊗ U_{1} ⊗ U, N_{2}); Ex̂t_{kG}^{i+1}(M_{1} ⊗ U_{1} ⊗ U, N_{3})]

with exact columns and commuting squares. Take U_{1} = U_{2} = U = k in part (i), and U_{1}, U_{2} as given and U arbitrary in part (ii). The hypotheses imply that the arrows marked as zero are indeed zero. This implies the existence of the dotted arrows making the indicated triangles commute. Then f′_{∗} ∘ f_{∗} is a composite of four maps, the middle two of which compose to zero, so f′_{∗} ∘ f_{∗} = 0. □

Definition 7.7. The (tensor) M-ghost number for G is the smallest n such that every composite of n (tensor) M-ghosts in stmod(kG) is null.

Remark 7.8. Notice that the tensor k-ghost number is always zero. Every tensor M-ghost is an M-ghost, and hence the tensor M-ghost number is never bigger than the M-ghost number. On the other hand if X is a nonprojective module in a nonprincipal block of kG, then the identity map 1_{X} : X → X is a k-ghost but not a tensor k-ghost. So if such a module exists then the k-ghost number is infinite though the tensor k-ghost number is zero.

It follows from Remark 8.6 and Theorem 8.10, below, that the tensor M-ghost number is finite if and only if the tensor products of M with other modules generate stmod(kG).

8. Generation

In this section we consider the generation of the stable category of finitely generated kG-modules, and introduce a variation on the classical generation number. This is an old concept and we refer to the paper by Rouquier [21] for references to the literature. The basic definition is the following.

Definition 8.1. We say that a finitely generated kG-module N is generated in n steps from a collection of modules {M_{α}} if there is a triangle

N_{1} → N_{2} → N ⊕ Z

in stmod(kG), for some Z, where N_{1} is generated in n − 1 steps from {M_{α}} and N_{2} is a direct sum of shifts of modules M_{α} in the collection. To begin the induction, N is generated in one step from the M_{α} if it is stably isomorphic to a summand of a direct sum of shifts of the modules M_{α}.

We offer the following variation on this idea.

Definition 8.2. We say that a finitely generated kG-module N is tensor generated in n steps from a collection of modules {M_{α}} if it is generated in n steps by the collection of modules M_{α} ⊗ X with M_{α} in the original collection and X arbitrary. In other words, inductively there is a triangle

N′ ⊗ X′ → M ⊗ X → N ⊕ Z

with N′ tensor generated in n − 1 steps from M. We don’t need to mention shifts because Ω^{i}(M) is stably isomorphic to M ⊗ Ω^{i}(k).

Lemma 8.3. If N is tensor generated in n steps from {M_{α}} then so is every module of the form N ⊗ Y.

Proof. In the recipe for generating N in n steps, replace each M_{α} ⊗ X with M_{α} ⊗ X ⊗ Y and each map with its tensor product with the identity on Y. □

Proposition 8.4. (i) If N is generated in n steps from M then every composite of n Mghosts is an N-ghost.

(ii) If N is tensor generated in n steps from M then every composite of n tensor M-ghosts is a tensor N-ghost.

Proof. This follows from Lemma 7.6 and induction on n. □

To make use of this proposition, we make the following definition. Recall that the usual M-generation number is the number d such that every object in stmod(kG) can be generated in d steps from M. The following is a variation on this notion.

Definition 8.5. The tensor M-generation number is the number of steps it takes to generate k from modules of the form M ⊗ X (or infinity if k cannot be generated from such modules).

Remark 8.6. It follows from the main theorem of Benson, Carlson and Rickard [4] that the tensor M-generation number is finite if and only if the support variety V_{G}(M) is equal to V_{G}(k). Notice also by Lemma 8.3, that if k can be tensor generated from M in n steps, then any object in stmod(kG) can be tensor generated from M in n-steps. It should also be emphasized that while the trivial module k does not necessarily generate the entire stable category (see [5]), it does tensor generate stmod(kG).

Corollary 8.7. If the tensor M-generation number is at most n, then the tensor M-ghost number is at most n.

Proof. This follows from Proposition 8.4 together with Lemma 7.4. □

Proposition 8.8. If the tensor M-generation number is n then every module in stmod(kG) can be tensor generated in n steps from M.

Proof. This follows from the definition and Lemma 8.3. □

Proposition 8.9. Let k → M ⊗ M^{∗} be the adjoint to the identity map M → M, and complete to a triangle

N ⟶^{f} k → M ⊗ M^{∗}

in stmod(kG). If N^{⊗n} → k is null then the tensor M-generation number is at most n.

Proof. Construct a Koszul complex with homology N^{⊗n}. That is, consider the complex C : k → M with k in degree zero and take the tensor product C^{⊗n}. In the stable category we have the sequence of maps:

\xymatrix{N^{⊗n} \ar[r] & k \ar[rr] \ar[dr] &  & (M ⊗ M^{∗})^{⊕n} \ar[r] & ⋯ \ar[r] & (M ⊗ M^{∗})^{⊗n} \\  &  & Y \ar[ur]}

If N^{⊗n} → k is null then k → Y splits, and so k is isomorphic to a summand of Y. But Y is built in n steps from the modules in the Koszul complex, which all have the form M ⊗X. □

With this information we can prove one of our main theorems (See Theorem 1.2).

Theorem 8.10. Let M be a finitely generated kG-module, let k → M ⊗ M^{∗} be the adjoint to the identity map M → M, and complete to a triangle

N ⟶^{f} k → M ⊗ M^{∗}

in stmod(kG). Then the following statements are equivalent.

(i) The tensor M-generation number is at most n.

(ii) The map f^{⊗n}: N^{⊗n} → k is null.

(iii) The tensor M-ghost number is at most n.

These statements hold for some n ≥ 1 if and only if the support variety V_{G}(M) is equal to V_{G}(k).

Proof. The implication (i) ⇒ (iii) is Corollary 8.7. The implication (ii) ⇒ (i) is Proposition 8.9. The implication (iii) ⇒ (ii) follows from the fact that f^{⊗n} is a composite of n tensor M-ghosts

N^{⊗n} → … ⟶^{f⊗1⊗1} N ⊗ N ⟶^{f⊗1} N ⟶^{f} k

by Lemma 7.5.

For the final statement, see Remark 8.6. □

Remark 8.11. Part of the motivation for this paper is a different proof, shown to us by Paul Balmer, also depending on the main theorem of [4], that if the support variety of M is equal to V_{G}(k) then the map f is tensor nilpotent. We also note that a somewhat different proof of the equivalence of (i) and (iii) in the theorem may be given by applying Theorem 3.5 of Christensen [9] to the projective class of summands of modules of the form M ⊗ X.

9. Bounded generation

Recall Question 1.3 which asks if for every kG-module M which tensor generates stmod(kG) there is a bound on the tensor M-generation number, depending only on G and not on M? In this section we consider this question.

Lemma 9.1. If G is a finite p-group, let ℓ be the radical length of kG. If the tensor Mgeneration number is n then the M-generation number is at most ℓn.

Proof. Given any kG-module X, the radical filtration of X has length and most ℓ, and the filtered quotients are direct sums of copies of k. So M ⊗ X can be generated in at most ℓ steps from M. □

Remark 9.2. As a result of Theorem 2.4, in order to answer Question 1.3, it suffices to consider the case that G is an elementary abelian p-group.

One might attempt to answer Question 1.3, trying to prove that there is a bound on the tensor powers that appear for tensor nilpotent maps, and then applying Proposition 8.9. In Section 5, we gave a family of examples which show that this approach cannot succeed. However, a modification of this approach might succeed: if we can show that the map f in the triangle N ⟶^{f} k → M ⊗ M^{∗} is strongly nilpotent then we could use Theorems 6.7 and 8.10 to conclude a positive answer to Question 1.3. So let us investigate what this would mean. We begin by describing the trace map.

Definition 9.3. Let G be a finite group and M a finitely generated kG-module. There is a trace map

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

which may be described as follows. Given an element ζ ∈ Ext_{kG}^{n}(M, M), we represent it by a cocycle ζ̂: M → Ω^{−n}(M). Noticing that Ω^{−n}(M) is stably isomorphic to Ω^{−n}(k) ⊗ M, we see that adjoint to this is a map M ⊗ M^{∗} → Ω^{−n}(k). Composing with the natural map k → M ⊗ M^{∗} adjoint to the identity, we obtain a map k → Ω^{−n}(k). The element of Ext_{kG}^{n}(k, k) represented by this map is the trace Tr_{M}(ζ).

In other words, the map is the usual adjunction Ext_{kG}^{∗}(M, M) → Ext_{kG}^{∗}(k, Hom_{k}(M, M)) followed by the map induced by the ordinary trace map Hom_{k}(M, M) → k. The trace map is not a homomorphism of graded rings, but it is a homomorphism of graded Ext_{kG}^{∗}(k, k)- modules. If the dimension of M is not divisible by p, so that the ordinary trace map is right split, then Tr_{M} is also right split.

Example 9.4. Suppose that E is elementary abelian, and M does not generate stmod(kE). Then V_{E}(M) is a proper subset of V_{E}(k), and so there exists a non-nilpotent element x ∈ Ext_{kE}^{∗}(k, k) which annihilates Ext_{kE}^{∗}(M, M). It therefore annihilates the image of the trace map. But for an elementary abelian p-group, non-nilpotent elements are non zero-divisors, so the image of the trace map is zero.

On the other hand, if the dimension of M is not divisible by p, then the natural maps k → M ⊗ M^{∗} → k have non-zero composite, and so the trace map is surjective. Hence, the interesting case is the one in which M is an indecomposable module which generates stmod(kE) and has dimension divisible by p.

Theorem 9.5. Given a kE-module M, form the triangle N ⟶^{f} k → M ⊗M^{∗} as usual. Then the following are equivalent:

(i) f is strongly nilpotent.

(ii) The trace map Tr_{M}: Ext_{kE}^{1}(M, M) → Ext_{kE}^{1}(k, k) is surjective.

Proof. By Lemma 6.4, the statement that f is strongly nilpotent is equivalent to the statement that for all extensions 0 → k → U → k → 0 in Ext_{kE}^{1}(k, k) the map f factors through a (dotted) map N → U as in the following diagram.

\xymatrix{ & k \ar[d] \\  & U \ar[d] \\ N \ar[r]_{f} \ar@{.>}[ur] & k \ar[d] \ar[r] & M ⊗ M^{∗} \ar@{-->}[dl] \\  & Ω^{−1}(k)}

This is equivalent to the statement that the composite N → k → Ω^{−1}(k) is null, which is again equivalent to the statement that the map k → Ω^{−1}(k) extends to a (dashed) map M ⊗ M^{∗} → Ω^{−1}(k). Such a map represents an element of Ext_{kE}^{1}(M, M) whose trace is the given element of Ext_{kE}^{1}(k, k). □

Finally, we are able to formulate a conjecture.

Conjecture 9.6. Let E be an elementary abelian p-group. Suppose that M generates stmod(kE), i.e., V_{E}(M) = V_{E}(k). Then the trace map

ζ_{M}: Ext_{kE}^{1}(M, M) → Ext_{kE}^{1}(k, k)

is surjective.

Theorem 9.7. If Conjecture 9.6 holds for all elementary abelian p-groups E, then Question 1.3 has a positive answer for all finite groups G.

Proof. By Remark 9.2, we may assume that G = E is elementary abelian. Suppose that M generates stmod(kE), and let N ⟶^{f} k → M ⊗ M^{∗} be the usual triangle. By Theorem 9.5, if Conjecture 9.6 holds then f: N → k is strongly nilpotent. From this, it follows using Theorem 6.7 that there is a bound, depending only on G, on the degree of tensor nilpotence of f. Finally, by Theorem 8.10, it follows that this is a bound for the M-generation number, and so Question 1.3 has a positive answer. □

10. Testing the conjecture

In this section we note several examples for which the conjecture is true. We begin with some preliminaries necessary for one of the checks.

Proposition 10.1. Let M and N be finitely generated kG-modules. Then the following diagram commutes.

\xymatrix{Ex̂t_{kG}^{∗}(N, M) ⊗ Ex̂t_{kG}^{∗}(M, N) \ar[rr]^{≅} \ar[d] &  & Ex̂t_{kG}^{∗}(M^{∗}, N^{∗}) ⊗ Ex̂t_{kG}^{∗}(N^{∗}, M^{∗}) \ar[d] \\ Ex̂t_{kG}^{∗}(M, M) \ar[dr]_{_{Tr_{M}}} &  & Ex̂t_{kG}^{∗}(N^{∗}, N^{∗}) \ar@{-->}[dl]^{_{Tr_{N^{∗}}}} \\  & Ex̂t_{kG}^{∗}(k, k)}

Here, the vertical arrows are given by Yoneda composition, and the diagonal maps are the trace maps for M and N^{∗}. The horizontal isomorphism is given by taking adjoints, moving M and N across to N^{∗} and M^{∗}.

Proof. Let ζ ∈ Ex̂t_{kG}^{∗}(M, M) be the image of α ⊗ β, with α ∈ Ex̂t_{kG}^{i}(N, M) and β ∈ Ex̂t_{kG}^{n−i}(M, N). Then ζ̂: M → Ω^{−n}(M) is represented by the composite

M ⟶^{β̂} Ω^{−i}(N) ⟶^{α̂} Ω^{−n}(M) ≅ Ω^{−n}(k) ⊗ M.

Write α′ for the element of Ex̂t_{kG}^{∗}(M^{∗}, N^{∗}) adjoint to α, β′ for the element of Ex̂t_{kG}^{∗}(N^{∗}, M^{∗}) adjoint to β, and ζ′ ∈ Ex̂t_{kG}^{∗}(N^{∗}, N^{∗}) for the image of α′ ⊗ β′. Finally, write β′′ for the element of Ex̂t_{kG}^{∗}(k, N ⊗ M^{∗}) adjoint to β. Consider the commutative diagram

\xymatrix{ & M ⊗ M^{∗} \ar[r]^{β̂⊗Id_{M^{∗}}} & Ω^{−i}(N) ⊗ M^{∗} \ar@{-->}[dr]^{≅} \\ k \ar[ur] \ar[dr] \ar[rrr]^{β̂′′} &  &  & Ω^{−i}(N ⊗ M^{∗}) \ar[r]^{α̂′} & Ω^{−n}(k) \\  & N ⊗ N^{∗} \ar[r]^{Id_{N} ⊗β̂′} & N ⊗ Ω^{−i}(M^{∗}) \ar@{-->}[ur]_{≅}}

Following along the top of this diagram gives a map k → Ω^{−n}(k) representing Tr_{M}(ζ), while following along the bottom of the diagram gives a map representing Tr_{N^{∗}}(ζ′). These are therefore equal. □

Remark 10.2. The reader may wonder why we don’t just reverse the order of the terms in the left hand vertical map of Proposition 10.1, and do away with the duals. The reason is that this introduces some signs. Namely, the two ways round the square would give answers which were not equal, but rather related by taking a Yoneda extension and dualising it. It turns out that this is an antiautomorphism on Ext_{kG}^{∗}(k, k) which is given in degree n by multiplication by (−1)^{n(n+1)/2}. Nonetheless, the effect is that up to a sign, the lower horizontal map can be thought of as composing maps between shifts of N and M rather than N^{∗} and M^{∗} if we wish.

Corollary 10.3. Let M be a finitely generated kG-module. Then the following diagram commutes,

\xymatrix{Ex̂t_{kG}^{∗}(k, M) ⊗ Ex̂t_{kG}^{∗}(M, k) \ar[r] \ar[d]^{≅} & Ex̂t_{kG}^{∗}(M, M) \ar[d]^{_{Tr_{M}}} \\ Ex̂t_{kG}^{∗}(M^{∗}, k) ⊗ Ex̂t_{kG}^{∗}(k, M^{∗}) \ar[r] & Ex̂t_{kG}^{∗}(k, k)}

Here, the horizontal arrows are given by Yoneda composition, the right hand vertical map is the trace map for M, and the left hand vertical isomorphism is given by taking adjoints.

Proof. This follows by setting N = k in Proposition 10.1. □

Note that Ex̂t_{kG}^{−1}(k, k) ≅ k and the composite

Ĥ^{n}(G, M) ⊗ Ĥ^{n−1}(G, M^{∗}) = Ex̂t_{kG}^{n}(k, M) ⊗ Ex̂t_{kG}^{n−1}(M, k) ⟶ Ex̂t^{−1}(k, k)

is the usual form for Tate duality.

Remark 10.4. Applying Remark 10.2, we can rephrase Corollary 10.3 as follows. If α ∈ Ex̂t_{kG}^{∗}(k, M) and β ∈ Ex̂t_{kG}^{∗}(M, k), then the trace of βα ∈ Ex̂t_{kG}^{∗}(M, M) is, up to a sign, equal to the trace ofαβ ∈ Ex̂t_{kG}^{∗}(k, k).

Letting G = E be an elementary abelian p-group in Corollary 10.3 we see using Remark 10.4 that if x ∈ Ext_{kE}^{1}(k, k), and we can find maps k → M and M → Ω^{−1}(k) whose composite is a map k → Ω^{−1}(k) representing x, then x is in the image of Tr_{M}. What’s more, we can replace M by any syzygy of M before finding such maps, and x will still be in the image of Tr_{M}.

One way of visualising such maps is that \underline{Hom}_{kE}(M, Ω^{−1}(k)) is isomorphic to Ext_{kE}^{1}(M, k). So we are looking for extensions of M by k whose restriction to one dimensional submodules of M give the required extensions of k by k.

Example 10.5. Let E = ℤ/3 × ℤ/3 and k have characteristic three. Let M be the three dimensional module module Soc^{2}(kE). Then we cannot find maps k → M → Ω^{−1}(k) with non-zero composite. However, we have Ω^{−1}(M) = kE/Soc^{2}(kE), and there are maps k → Ω^{−1}(M) → Ω^{−1}(k) with any desired one dimensional image.

M = [figure: •; •; •] Ω^{−1}(M) = [figure: •; •; •; •; •; •] Ω^{−1}(k) = [figure: •; •; •; •; •; •; •; •]

It follows that in this example, the trace map

Tr_{M}: Ext_{kE}^{1}(M, M) → Ext_{kE}^{1}(k, k)

is surjective. So for this module M, the tensor M-generation number is at most r(p − 1), which is equal to 4.

If we take the homomorphism from k to Ω^{−1}(M) to have the middle basis element of the socle as image, then the required extensions of Ω^{−1}(M) by k for the two (dashed) generators of Ext_{kE}^{1}(k, k) are as follows:

[figure: •; •; •; •; •; •; •; •; •; •; •; •; •; •]

Alternatively, we could use just one of these, and two different maps from k to Ω^{−1}(M).

Remark 10.6. There is a variant of this method, as follows. By Proposition 10.1, in order to show that x ∈ Ext_{kE}^{1}(k, k) is in the image of Tr_{M}, it suffices to find a module N, and maps N → M → Ω^{−1}(N) whose composite has image x under Tr_{N}.

Example 10.7. Let p = 3 and E = ℤ/3 × ℤ/3. Consider the module M given by the following diagram:

[figure: •; •; •; •; •; •; •; •; •; •; •; •]

Then it is easy to check that there are no maps k → M and M → Ω^{−1}(k) with non-null composite. Nor are there maps Ω(k) → M and M → k with non-null composite. So it is difficult to tell whether the method of Corollary 10.3 may be applied to some syzygy of M. However, we checked using the computer algebra package MAGMA (see [6]) that the trace map Tr_{M} is surjective in degree one, so this is not a counterexample to Conjecture 9.6.

The algorithm for the computer check is straightforward. First one constructs the trace map M^{∗} ⊗M → k and the module Ω(k). A standard function computes a basis for the space Hom_{kG}(Ω(k), M^{∗} ⊗ M). If the rank of the space of these maps composed with the trace map is equal to the dimension of Ext^{1}(k, k) then Conjecture 9.6 is verified.

The module M is an example of a W module as defined in Section 3 of Carlson, Friedlander and Suslin [8]. In their notation it is W_{5,3}. In general the modules W_{n,p} for ℤ/p × ℤ/p in characteristic p, with n ≥ 2p − 1, behave the same way. We tested a number of these modules, including for example W_{9,5} in characteristic five, and in the cases we tested the trace map was surjective in degree one. Also tested were some quotients of W -modules and some quotients of submodules of W -modules. Every module M that we checked had the property that V_{G}(M) = V_{G}(k), and in every case the conjecture was confirmed.

Perhaps equally interesting is the question of whether the image of the trace map

Ext_{kE}^{∗}(M, M) → Ext_{kE}^{∗}(k, k)

includes the entire exterior algebra generated by the elements of degree one when p is odd. This also was true in every one of the many cases that we checked with the computer. It was also confirmed for several examples over an elementary abelian group of order 27 = 3^{3}. In this case modules were constructed by taking random elements in the second Jacobson radical of a free module as generators and then factoring out the module generated by some random element of the second socle. Interestingly, in all examples for the group of rank 3, it was the case that the trace map

Ext_{kE}^{n}(M, M) → Ext_{kE}^{n}(k, k)

was surjective both in degree n = 1 and n = 3; in the latter case not just hitting the product of the three exterior generators, but hitting all ten dimensions.

References

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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