THE DECOMPOSITION OF THE TRIVIAL MODULE

IN THE COMPLEXITY QUOTIENT CATEGORY

Jon F. Carlson^{∗}

Department of Mathematics University of Georgia Athens, GA 30602 (email: jfc@sloth.math.uga.edu)

1. Introduction.

Let G be a finite group and let k be an algebraically closed field of characteristic p > 0. The purpose of this paper is to show that, except for a certain complexity factor, the module theory for kG is controlled at the level of the centralizers of the elementary abelian p-subgroups. To make sense of this statement, let Stmod-kG be the stable category of kG-modules modulo projectives, and let C denote the full subcategory of all kG-modules with less than maximal complexity. Then in the quotient category Stmod-kG/C = Q a certain multiple of the trivial module is isomorphic to a direct summand of a direct sum of multiples of the modules induced from the trivial module on the centralizers of the maximal elementary abelian p-subgroups. The statement about control follows from Frobenius reciprocity, i.e. for any kG-module M

M ⊗ (k_{C}^{↑G}) ≅ (M_{C} ⊗ k)^{↑G} ≅ M_{C}^{↑G}

where M_{C} is the restriction of M to C and k_{C} is the trivial kC-module. Thus we have that in Q, for suitable positive integers m and m_{E}

M^{m} is a direct summand of ∑ (M_{C_{G}(E)}^{↑G})^{m_{E}}

where the sum is over a complete set of representatives of the conjugacy classes of maximal elementary abelian p-subgroups E of G.

The “complexity factor” C is understood by means of the maximal ideal spectrum V_{G}(k) of the cohomology ring H^{∗}(G, k). Any finitely generated kG-module has an associated subvariety V_{G}(M) corresponding to the annihilator in H^{∗}(G, k) of the cohomology of M. The complexity of M is the dimension of V_{G}(M), and theorems of Quillen and others tell us that the complexity is at most equal to the p-rank of G. The complexity of a module

Typeset by AMS-TEX

∗Partly supported by a grant from NSF

can also be interpreted as the degree of the polynomial rate of growth of the terms of a minimal projective resolution.

The quotient category construction is essentially a localization process. The category Stmod-kG is triangulated and the (likewise triangulated) subcategory C is thick in StmodkG. In the quotient, Q, the objects are the same as those in Stmod-kG, but the morphisms are obtained by inverting any stable homomorphism with the property that the third object in the triangle of the homorphism is in C. A main result of [6] is that the group of morphisms from M to N in Q is the zero grading of a homogeneous localization of Ext_{kG}^{∗}(M, N). In particular, this means that the endomorphism ring of the trivial module decomposes into blocks corresponding to the components of the variety V_{G}(k). So it was natural to expect a similar decomposition of the trivial module itself. The nature of the decomposition was more mysterious, and, for the one case in which the decomposition had been achieved, the methods were ad hoc and not generalizable. Instead the proper viewpoint seems to be to look at the transfer maps in cohomology as in [4]. Some background on varieties and transfers is presented in Section 2. The necessary results on transfers are demonstrated in Section 3. Sections 4 and 5 contain the proof of the main theorem, first for the local case of a single maximal elementary abelian p-subgroup, and then in the global case with everything considered.

The final section is concerned with some applications. A more substantial application to modules with vanishing cohomology will appear in [7]. There it is shown that if p > 2 and M is a kG-module in the principal block with H^{∗}(G, M) = 0, then the complexity of M is less than maximal, i.e. M ∈ C. The main point is that for p odd, the centralizers of the maximal elementary abelian p-subgroups are p-nilpotent and hence, in the principal block, cohomology can be computed as for a p-group. In Section 6, we give three observations on the implications of the main theorems. We begin by interpreting the main theorems in the context of the quotient category Q. In each of the three cases the application is clearly indicated by this viewpoint. The actual proofs are only slightly more difficult.

We have phrased the main results of this paper in terms of exact sequences of kGmodules. However, in some sense, they are best viewed as statements about triangles in the stable category. The problem with the triangles is that it seems often necessary to write them in terms of exact sequences in order to obtain any sort of proofs. In order to represent an element α ∈ \underline{Hom}_{kG}(M, N) as a homomorphism in a short exact sequence we may be required to take a direct sum of a projective module with either M or N so that the representing map is either surjective or injective as desired. Here PHom_{kG}(M, N) is the set of homomorphism from M to N which factor through projective modules, and \underline{Hom}_{kG}(M, N) = Hom_{kG}(M, N)/PHom_{kG}(M, N) is the set of morphism from M to N in Stmod-kG. To appreciate the relationship between the triangles and the exact sequences we invite the reader to prove the following exercise. Part of this is a variation on Schanuel’s Lemma.

Lemma 1.1. Let M and N be kG-module and let α ∈ \underline{Hom}_{kG}(M, N) be a stable homomorphism. Suppose that P_{1} and P_{2} are projective modules so that we have exact sequences

0 ⟶ L_{i} ⟶ P_{i} ⊕ M ⟶^{α_{i}} N ⟶ 0

where each α_{i} represents α. Then L_{1} and L_{2} are isomorphic modulo projectives, i.e. there exist projective modules Q_{1} and Q_{2} so that L_{1} ⊕ Q_{1} ≅ L_{2} ⊕ Q_{2}. Moreover there is a uniquely determined projective module P_{1} such that L_{1}, as above, has no projective summands.

On the other side, if P′_{1} and P′_{2} are projective modules such that there are exact sequences

0 → M ⟶^{α′_{1}} N ⊕ P′_{i} → L′_{i} → 0

with α′_{i} representing α, then L′_{1} and L′_{2} are isomorphic modulo projectives. Finally for i, j ∈ {1, 2} there is a projective module Q_{ij} such that there is an exact sequence

0 → L_{i} → Q_{ij} → L′_{j} → 0.

In other words the objects L_{1} and L_{2} are isomorphic in Stmod-kG, and they represent the third object in the triangle containing M, N and α. Of course, much of this depends on the fact that kG is a self-injective ring.

2. Varieties and complexity.

Throughout the paper G denotes a finite group and k a field of characteristic p > 0. We shall assume that k is algebraically closed, though this assumtion probably is not necessary for some of the main results and certainly is not necessary for some of the applications. Suppose that (P, ∂) is a minimal projective resolution of the trivial kG-module, which we also denote by k, then the kernel of differential ∂_{n−1} is called Ω^{n}(k) = ∂_{n}(P_{n}). So we have an exact sequence

0 → Ω^{n}(k) → P_{n−1} ⟶^{∂_{n−1}} ⋯ ⟶^{∂_{1}} P_{0} ⟶^{ϵ} k → 0.

A cohomology class ζ ∈ H^{n}(G, k) is represented by a cocycle ζ′ : P_{n} → k. As ζ′ · ∂_{n} = 0 there is an induced map ζ̂ : Ω^{n}(k) ≅ P_{n}/∂_{n}(P_{n+1}) → k. In this way we may identify H^{n}(G, k) with \underline{Hom}_{kG}(Ω^{n}(k), k). Actually, because k is a simple kG-module and Ω^{n}(k) has no projective submodules, \underline{Hom}_{kG}(Ω^{n}(k), k) = Hom_{kG}(Ω^{n}(k), k).

The ring H^{∗}(G, k) is graded-commutative and finitely generated as a k-algebra. Hence it has a maximal ideal spectrum, V_{G}(k), which is a finite dimensional, homogeneous affine variety.

All kG-modules in this paper are finitely generated. If M and N are kG-modules then the cohomology Ext_{kG}^{∗}(M, N) is finitely generated as a module over H^{∗}(G, k). Suppose that J(M) is the annihilator of Ext_{kG}^{∗}(M, M) in H^{∗}(G, k), and V_{G}(M) ⊆ V_{G}(k) is the corresponding subvariety of all maximal ideals which contain J(M). The varieties of modules have several nice properties which will be useful. The reader is referred to the books by Benson [1] or Evens [9] for details and proof.

Lemma 2.1. (i) A kG-module M is projective if and only if V_{G}(M) = {0}. (The zero point in V_{G}(k) is the homogeneous maximal ideal of all elements of positive degree.)

(ii) If M and N are kG-module then V_{G}(M ⊗ N) = V_{G}(M) ∩ V_{G}(N). (The tensor product M ⊗ N is over k with diagonal G-action).

(iii) Suppose that 0 → M_{1} → M_{2} → M_{3} → 0 is an exact sequence of kG-modules. Then

V_{G}(M_{i}) ⊆ V_{G}(M_{j}) ∪ V_{G}(M_{k})

provided {1, 2, 3} = {i, j, k}.

If G ≅ (ℤ/p)^{r} is an elementary abelian p-group of rank r then V_{G}(k) ≅ k^{r}. This is because H^{∗}(G, k) modulo its radical is a polynomial ring of dimension r and k is, as assumed, algebraically closed. For a general group G, let A(G) be the set of all elementary abelian p-subgroups of G and let M(G) ⊆ A(G) be the maximal elements. The structure of variety V_{G}(k) is expressed in Quillen’s Dimension Theorem [10].

Theorem 2.2. V_{G}(k) = ∪V_{E} where the union is over a set of representatives of the conjugary classes of elements E ∈ M(G). The closed subvarieties V_{E} are the irreducible components of V_{G}(k) and for E ∈ M(G)

V_{E} = res_{G,E}^{∗}(V_{E}(k))

where res_{G,E}^{∗} is the map on varieties induced by the restriction res_{G,E} : H^{∗}(G, k) → H^{∗}(E, k). Hence V_{E} is the set of all maximal ideals which contain the kernel of res_{G,E}. The map res_{G,E} is finite-to-one so that

dimV_{E} = dimV_{E}(k) = p-rank(E).

The maps res_{G,E}^{∗} can be described in more detail. For notation, let W_{E} ⊆ V_{E}(k) be the closed subvariety corresponding to the ideal of all elements which vanish on restriction to all proper subgroups of E. In fact, W_{E} is the union of all 𝔽_{p}-rational subspaces of dimension r − 1 in V_{E}(k) = k^{r}. Let W_{G,E} = res_{G,E}^{∗}(W_{E}). Now N_{G}(E) acts on H^{∗}(E, k) and hence also on V_{E}(k). It turns out that V_{E} \ W_{G,E} is isomorphic to the set of N_{G}(E)- orbits of V_{E}(k)\W_{E} [11]. Also C_{G}(E) acts trivially on V_{E}(k). We shall need to know the following result, which is a consequence of this type of argument.

Lemma 2.3. If E ∈ M(G) is central, i.e. G = C_{G}(E). Then res_{G,E}^{∗} : V_{E}(k) → V_{G}(k) is an isomorphism. (Notice that with this hypothesis M(G) = {E}).

¿From the standpoint of modules, notice that V_{E} = V_{G}(k_{E}^{↑G}) = V_{G}(k_{C}^{↑G}) for C = C_{G}(E), E ∈ M(G). This follows from the last lemma and the fact that the usual Eckmann-Shapiro isomorphism

Ext_{kG}^{∗} (k_{E}^{↑G}, k_{E}^{↑G}) ≅ Ext_{kE}^{∗} (k, k_{E}^{↑G})

is compatable with the restriction to E. Likewise

W_{G,E} = ⋃_{F <E} V_{G}(k_{F}^{↑G}), W_{E} = ⋃_{F <E} V_{E} (k_{F}^{↑E}) .

For any kG-module M, the dimension of the variety V_{G}(M) is known as the complexity of M. The complexity can also be described as the polynomial rate of growth of the terms of a minimal projective resolution of M. It is also equal to the order of the pole at 1 of the Poincaré series for the cohomology Ext_{kG}^{∗}(M, M).

Finally we need to recall some facts about the rank varieties of modules over elementary abelian p-groups. Let E be an elementary abelian subgroup of G of order | E |= p^{r} and suppose that E has generators z_{1}, z_{2}, ⋯, z_{r}. If α = (α_{1}, ⋯, α_{r}) ∈ k^{r}, α ≠ 0, then the

unit u_{α} = 1 + ∑_{i=1}^{r} α_{i}(z_{i} − 1) has order p. Moreover kE is free as a k⟨u_{α}⟩-module, and so

it makes sense to talk about restriction maps to the group S = ⟨u_{α}⟩. Such a subgroup S ⊆ kE is called a shifted cyclic subgroup of kE (or of kG if we think of E ⊆ G). For any kE-module M the rank variety of M over E is the set

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

The Theorem of Avrunin and Scott guarantees that,

V_{E}^{r}(M) ≅ V_{E}(M).

The map between the two is the identity in the case of p = 2, but involves a shift by the Frobenius automorphism otherwise. In particular, we can measure the complexity of a kE-module by computing the dimension of the rank variety of M. We shall need the following results on shifted subgroups. The first is very standard and we refer the reader to the books [1] or [9] or to [3].

Lemma 2.4. If m ⊆ H^{∗}(G, k) is a maximal ideal then there is a maximal elementary abelian p-subgroup E ⊆ G and a cyclic shifted subgroup S ⊆ kE such that the kernel of restriction to S is in m. In other words, the homomorphism ϕ : H^{∗}(G, k) → k with kernel m factors through the restriction to H^{∗}(S, k).

Lemma 2.5. Suppose that m ∈ V_{G}(k) corresponds to a cyclic shifted subgroup S ⊆ kE ⊆ kG. Assume that m is not the ideal of all elements of positive degree. Let ζ ∈ H^{n}(G, k) and suppose that either ζ is not nilpotent or, if p > 2, that the degree of ζ is even. Then ζ ∉ m if and only if res_{G,S}(ζ) ≠ 0.

Proof. Now because S is a cyclic p-group H^{∗}(S, k) ≅ k[γ], a polynomial ring in γ, modulo its radical. If p = 2 then γ has degree 1 and the radical of H^{∗}(S, k) is zero. If p > 2, the elements of odd degree in H^{∗}(S, k) and also in H^{∗}(G, k) are nilpotent and hence are in the radical of the cohomology rings. Now we are assuming that m contains the kernel of the restriction to S. Therefore m is the pullback under res_{G,S} of the kernel of a map θ : H^{∗}(S, k) → k which sends γ to θ(γ) = a ∈ k. Because m is not the ideal of all elements of positive degree, a ≠ 0. By hypothesis res_{G,S}(ζ) = bγ^{m} for b ∈ k and some m. So θ(res_{G,S}(ζ)) = ba^{m} is zero if and only if b = 0.

Lemma 2.6. Suppose that n > 0 is an even integer if the prime p is odd. Let E be an elementary abelian p-subgroup of G and let S ⊆ kE be a cyclic shifted subgroup. There exists a kS-homomorphism τ : k → Ω^{n}(k) with the property that a cohomology element ζ ∈ H^{n}(G, k) has nonzero restriction to S if and only if ζ̂τ ≠ 0 for any cocycle ζ̂: Ω^{n}(k) → k which represents ζ.

Proof. Because S is cyclic k_{S} has a minimal projective kS-resolution (Q_{∗}, ∂′) in which every Q_{i} ≅ kS and also Ω^{n}(k_{S}) ≅ k provided n is even. Now a cocycle σ : Ω^{n}(k_{S}) → k represents a nonzero class in H^{n}(S, k) if and only if σ ≠ 0. Let (P, ∂) be a kG-projective resolution of k. The restriction map from G to S is given by composition with a kSchain map μ : Q → P which lifts the identity on k. But μ induces a homomorphism τ : Ω^{n}(k_{S}) ≅ k → Ω^{n}(k). As we have said, a cocycle ζ̂: Ω^{n}(k) → k represents a class with nonzero restriction to S if and only if ζ̂τ ≠ 0.

Suppose now that E is a maximal elementary abelian p-subgroup of G, having rank r. Then E is isomorphic to the additive subgroup of an 𝔽_{p}-vector space of dimension r. From this viewpoint the conjugation action of N = N_{G}(E) on E is 𝔽_{p}-linear.

Definition 2.7. For E, N as above let D = D_{G}(E) be the set of all elements of N whose action on E is by scalar multiplication. That is, if x ∈ D then there exists a nonzero a ∈ 𝔽_{p} such that xyx^{−1} = y^{a} for all y ∈ E.

The following results are easily derived from the definition.

Lemma 2.8. With the above notation, C_{G}(E) ⊆ D_{G}(E)◁N_{G}(E). Moreover D_{G}(E)/C_{G}(E) is a cyclic group of order dividing p − 1.

The action of N = N_{G}(E) on E induces an action on H^{∗}(E, k) and also on V_{E}(k) = k^{r}. In particular, the action of N on V_{E}(k) is k-linear with the elements of D_{G}(E) acting by scalar multiplication. For x ∈ N, a ∈ k, let

W (x, a) = {m ∈ V_{E}(k)|xm = am}.

That is, W (x, a) is the eigenspace corresponding to the eigenvalue a of the action of x on V_{E}(k). Thus W (x, a) is zero for all but a finite number of values of a and dim W (x, a) = dim V_{G}(k) for some a, if and only if x ∈ D_{G}(E). The following will be useful in the next section.

Lemma 2.9. Suppose x ∈ N = N_{G}(E) where E is a maximal elementary abelian psubgroup of G. Suppose that C = C_{G}(E) and the element xC ≠ C has order a power of p in N/C. Then W (x, a) = {0} unless a = 1, while W (x, 1) ⊆ W_{E}.

Proof. Let n be the order of the coset xC as an element of the group N/C. Then clearly W (x, a) = {0} inless a is an n^{th} root of unity in k. Thus the first statements follow because if n is a power of p then 1 is the only n^{th} root of unity. For the second statement, notice that the action of x on V_{G}(k) is 𝔽_{p}-rational and hence the eigenspace of the eigenvalue 1 ∈ 𝔽_{p} is an 𝔽_{p}-rational subspace. Thus W (x, 1) ⊆ W_{E}.

Defintion 2.10. For E a maximal elementary abelian p-subgroup of G, let

W_{E} = ∪W (x, a)

where the union is taken over all a ∈ k and all x ∈ N_{G}(E) with x ∉ D_{G}(E). Let

V_{G} = ∪res_{G,E}^{∗}(W_{E})

where the union is overall E ∈ M(G).

The condition that x ∉ D_{G}(E) assures that W_{E} is a proper closed subvariety of V_{E}(k). Hence also V_{G} contains no component of V_{G}(k). The property of W_{E} which will be most useful to us is the following.

Proposition 2.11. Suppose that m ∈ V_{E}(k) and that m ∉ W_{E}. Then there exists a homogeneous element γ ∈ H^{∗}(D, k), D = D_{G}(E) with the property that for any x ∈ N_{G}(E), x ∉ D, then

res_{D,E}(γ) ∉ m but res_{D,E}(γ) ∈ xm.

Proof. The action of D on V_{E}(k) is a scalar action. Hence two points m, m′ are in the same D-orbit only if they lie on the same line through the origin in V_{E}(k). Now V_{D}(k) = V_{E}(k)/(D/C), the orbit space of the action of D/C. Homogeneous elements in the cohomology H^{∗}(D, k) will separate points on different lines of V_{D}(k) (i.e. points in the projective variety of V_{D}(k)), but not points on the same line. Therefore for each coset xD in N/D there exists a homogeneous element γ_{x} ∈ H^{∗}(D, k) such that γ_{x} ∉ ℓ, γ_{x} ∈ xℓ where ℓ = res_{D,E}^{∗}(m). This is because, by the definition of W_{E}, xℓ is not on the same line as ℓ in V_{D}(E). But now let γ = ∏ γ_{x} where the product is over the nonidentity cosets of N modulo D. It is straight forward to check that γ has the desired property.

3. Transfers and points in the variety.

Suppose that H is a subgroup of G. Let T r_{H}^{G} : H^{∗}(H, k) → H^{∗}(G, k) be the transfer or corestriction map. Briefly, it can be defined as follows. Suppose that (P_{∗}, ∂) is a projective kG-resolution of k. Then by restriction it is a projective kH-resolution and any element γ of H^{n}(H, k) is represented as a cocycle γ′ : P_{n} → k. Then μ = T r_{H}^{G}(γ) is represented by μ′ : P_{n} → k where μ′(x) = ∑_{G/H} γ′(g^{−1}x), for x ∈ P_{n}, the sum being over a set of

representatives, g, of the left cosets of H in G. It follows that, as in the introduction to the last section if γ̂ : Ω^{n}(k) → k is a kH-homomorphism representing γ then μ̂ : Ω^{n}(k) → k, given by μ̂(x) = ∑_{G/H} μ(g^{−1}x) represents T r_{H}^{G}(γ). The transfer map satisfies Frobenius

reciprocity so that its image T r_{H}^{G}(H^{∗}(H, k)) is an ideal in H^{∗}(G, k). For this paper a property of transfers, in which we are most interested, is the following.

Lemma 3.1. Suppose that γ ∈ H^{n}(H, k). Then the transfer μ = T r_{H}^{G}(γ) is represented by a cocycle μ̂ : Ω^{n}(k) → k which factors through the induced module k_{H}^{↑G}.

Proof. Now k_{H}^{↑G} = kG ⊗_{kH} k with G action on the left. Define ϵ : k_{H}^{↑G} → k to be the augmentation given by ϵ( ∑_{G/H} g ⊗ a_{g}) = ∑_{G/H} a_{g}. Suppose that γ̂ : Ω^{n}(k) → k is a kH-

homomorphism representing γ. Define τ : Ω^{n}(k) → k_{H}^{G} by τ(x) = ∑_{G/H} g ⊗ γ(g^{−1}x). It is

easily checked that τ is a kG-homomorphism and that μ̂ = ϵτ represents μ.

The main purpose of this section is to establish the existence of transfer maps which avoid certain maximal ideals of H^{∗}(G, k). In particular the maximal ideals which are not contained in W_{G,E} for any E ∈ M(G) do not contain the ideal of transfers from the centralizer of E, provided the Sylow p-subgroup of G is not abelian. The statement is similar to the main theorem of [4] or the generalization by Benson in [2]. For the sake of completeness we include a proof similar to that of [4]. Following that we prove a variation on the transfer theorem. The variation is precisely what is required for the next section. The techniques of the proofs are crucial for the next two sections.

If m ⊆ H^{∗}(G, k) is a maximal ideal then we say that m is minimally supported on an elementary abelian subgroup E of G provided m ∈ V_{E} ⊆ V_{G}(k) but m ∉ W_{G,E}. Otherwise said, m contains the kernel of the restriction to E but not the kernel of the restriction to any proper subgroup of E.

Theorem 3.2. Suppose that m ∈ V_{G}(k) is minimally supported on a maximal elementary abelian p-subgroup E ∈ M(G). Let C = C_{G}(E). Then T r_{C}^{G}(H^{∗}(C, k)) ⊈ m.

Proof. Suppose that ℓ ∈ V_{C}(k) and j ∈ V_{E}(k) are maximal ideals such that m = res_{G,C}^{−1}(ℓ) and ℓ = res_{C,E}^{−1}(j). Then ℓ ∉ W_{C,E}, j ∉ W_{E}, and so both ℓ and j are minimally supported on E. Let N = N_{G}(E). Now N/C acts on H^{∗}(C, k) and hence also on V_{C}(k) ≅ V_{E}(k).

We claim that C_{N}(ℓ) = C. For suppose that x ∈ N, x ∉ C. Notice first that x does not act trivially on V_{C}(k) because it does not act trivially on E (see Lemma 2.3 or [11]). On the other hand, the set of fixed points under the action of x on V_{C}(k) is the eigenspace corresponding to the eigenvalue one. Hence it is an 𝔽_{p}-rational subspace. It follows that the fixed point set is in W_{C,E}. Therefore xℓ ≠ ℓ, for any x ∈ N, x ∉ C.

Let S = S_{ℓ} = {x ∈ N|xℓ = aℓ for some a ∈ k}. Notice that C ⊆ S_{ℓ} and, by Lemma 2.9, |S : C| is not divisible by p. Now let ℓ′ = res_{S,C}^{∗}(ℓ) ∈ V_{S}(k). Then for any element y ∈ N, y ∉ S the points ℓ′ and yℓ′ are on different lines through the origin (represent different points in the projective variety) of V_{S}(k). So for each coset y ∈ N/S, y ≠ 1, choose a homogeneous element γ ∈ H^{∗}(S, k) with the properties that γ_{y} ∉ ℓ′ but γ_{y} ∈ yℓ′. Then let

γ = Πγ_{y}

with the product taken over all y ∈ N/S, y ≠ 1. It follows that γ ∉ ℓ′ but γ ∈ yℓ′ for all y ≠ 1. Then let ζ = T r_{S}^{G}(γ). We claim that ζ ∉ m or equivalently that res_{G,E}(ζ) ∉ j. To see this we need only check the Mackey formula:

res_{G,E}(ζ) = res_{G,E}(T r_{S}^{G}(γ))

= ∑_{ExS} T r_{E∩xSx^{−1}}^{E}(res_{xSx^{−1},E∩xSx^{−1}}(xγ)). (3.3)

Here the sum is over the E − S double cosets ExS in G. Now if x ∉ N then E ∩ xSx^{−1} is a proper subgroup of E. But because E is an abelian p-group the image of any transfer map T r_{E′}^{E} from a proper subgroup E′ is always in the radical of H^{∗}(E, k) and hence is also in j. On the other hand if x ∈ N and x ∉ S then res_{S,E}(xγ) ∈ j because xγ ∈ ℓ. We have then that modulo j

res_{G,E}(ζ) ≡ res_{S,E}(γ) ≢ 0

because γ ∉ ℓ. Therefore ζ ∉ m.

But because |S : C| is relatively prime to p,

ζ = T r_{S}^{G}(γ) = T r_{E}^{G} ((1/|S : C|) res_{S,C}(γ)) .

Proposition 3.4. Suppose that E ∈ M(G) and let m ∈ V_{E}(k), m ∉ W_{E}. There exists a homogeneous element γ ∈ H^{∗}(D, k), D = D_{G}(E), such that res_{D,E}(γ) ∉ m but xγ ∈ m for all x ∈ N = N_{G}(E), x ∉ D (Propostion 2.11). If ζ = T r_{D}^{G}(γ), then ζ ∉ res_{G,E}^{∗}(m) or, equivalently, res_{G,E}(γ) ∉ m.

Proof. Let γ be chosen as in (2.11). Then by the Mackey formula ((33) with D in place of S) we have that

res_{G,E}(ζ) ≡ ∑_{N/D} xγ (mod(radH^{∗}(E, k))).

That is, we again have that, for x ∉ N, E ∩ xDx^{−1} is a proper subgroup of E since E is precisely the set of all elements of order dividing p in D. But then, modulo the maximal ideal m, res_{G,E}(ζ) ≡ γ ≢ 0.

4. The construction for one component.

In this section we concentrate on a single component V_{E} of the variety, V_{G}(k), corresponding to a fixed maximal elementary abelian p-subgroup E of G. Basically we show that for suitable m and n, there is a surjection of (Ω^{n}(k))^{m} onto k_{D_{G}(E)}^{↑G} whose kernel has a variety which does not contain all of V_{E}. Because we are dealing with modules it seems natural to measure our varieties as rank varieties. However the arguments are such that the interested reader should be able to convert to cohomological varieties.

We begin by fixing a maximal elementary abelian p-subgroup E of G. Let p^{r} =| E |. Let D = D_{G}(E) (see (2.7)), and N = N_{G}(E). We wish to consider the endomorphism ring of the induced module k_{D}^{↑G} = kG ⊗_{kD} k. Notice that

Hom_{kG}(k_{D}^{↑G}, k_{D}^{↑G}) ≅ Hom_{kD}(k, k_{D}^{↑G} ↓_{D}) = (k_{D}^{↑G})^{D}.

Here M^{D} denotes the D-fixed points on M. The space of D-fixed points on k_{D}^{↑G} is spanned by the set of elements of the form X ⊗ 1 where X = x_{1} + ⋯ + x_{t} for x_{1}, ⋯, x_{t} a complete set of representatives of the left cosets of D in the double coset Dx_{1}D. In particular, if x_{1} ∈ N then X = x_{1} as Dx_{1}D = x_{1}D. In any case the corresponding element σ_{X} of Hom_{kG}(k_{D}^{↑G}, k_{D}^{↑G}) is determined by the fact that σ_{X}(α ⊗ 1) = αX ⊗ 1. So we have the following.

Lemma 4.1. For any x ∈ N there is a homomorphism σ_{x} : k_{D}^{↑G} → k_{D}^{↑G} given by σ_{x}(y ⊗ 1) = yx ⊗ 1 for all y ∈ G. Moreover if U = ∑_{x∈N} x ⊗ k, T = ∑_{x∉N} x ⊗ k are

the subspaces so that k_{D}^{↑G} ↓_{D}≅ U ⊕ T, then the projection k_{D}^{↑G} → U commutes with σ_{x} for all x ∈ N.

We are now prepared to state the main theorems of the section.

Theorem 4.2. Suppose that E ⊆ M(G) and that m = |N : D|. Let m ∈ V_{E}(k) be a maximal ideal with m ∉ W_{E} ∪ W_{E} (see (2.10)). Then for some n > 0 and all t > 0 there exists a projective module P and an exact sequence

0 ⟶ L ⟶ (Ω^{nt}(k))^{m} ⊕ P ⟶ k_{D}^{↑G} ⟶ 0

such that m ∉ V_{E}(L_{E}).

Proof. Let ℓ = res_{D,E}^{∗}(m) ∈ V_{D}(k). By Proposition 3.4, for some n > 0 we can choose an element γ ∈ H^{n}(D, k) such that γ ∉ ℓ, xγ ∈ ℓ for all x ∈ N, x ∉ D, and ζ = T r_{D}^{G}(γ) ∉ res_{G,E}^{∗}(m). This last is the same as saying that res_{G,E}(ζ) ∉ m.

Let γ̂ : Ω^{n}(k) → k be a kD-cocycle representing γ. Here γ̂ is only a kD-homomorphism, although Ω^{n}(k) = Ω_{G}^{n}(k) is a kG-module. Then ζ̂ = ∑_{x∈G/D} xγ̂ : Ω^{n}(k) → k represents ζ.

By Lemma 3.1, ζ̂ can be written as the composition

Ω^{n}(k) ⟶^{μ} k_{D}^{↑G} ⟶^{ϵ} k

where ϵ is the usual augmentation, and μ is given by the formula

μ(a) = ∑_{G/D} x ⊗ γ(x^{−1}a).

We write elements of (Ω^{n}(k))^{m} as sequences (m-tuples) (a_{i})_{i=1}^{m} where a_{i} ∈ Ω^{n}(k). Let 1 = x_{1}, ⋯, x_{m} be a complete set of representatives of the left cosets of D in N. Let σ_{i} = σ_{x_{i}} as in Lemma 4.1. With this notation we define η : (Ω^{n}(k))^{m} → k_{D}^{↑G} by

η((a_{i})_{i=1}^{m}) = ∑_{i=1}^{m} σ_{i}μ(a_{i}).

Clearly η is a kG-homomorphism.

Let ρ : k_{D}^{↑G} ↓_{D}→ U be the projection with kernel T. First we prove the following.

Lemma 4.3. The sequence

0 ⟶ M ⟶ (Ω^{n}(k))^{m} ⟶^{ρη} U ⟶ 0

has the property that m ∉ V_{E}(M_{E}).

Proof. Let S be a cyclic shifted subgroup of kE such that the kernel of the restriction to S in H^{∗}(E, k) is contained in m. Notice that because γ ∉ ℓ and res_{G,E}(ζ) ∉ m we must have that neither res_{D,S}(γ) nor res_{G,S}(ζ) is nilpotent in H^{∗}(S, k). Let τ : k → Ω^{n}(k) be the kS-split homomorphism satisfying the condition of Lemma 2.6. Notice that S acts trivially on U and hence U_{S} ≅ k^{m}. Define θ : U → (Ω^{n}(k))^{m} by

θ(x_{i} ⊗ 1) = (a_{j})_{j=1}^{m}

where a_{j} = 0 for j ≠ i and a_{i} = τ(1). Then for any i = 1, ⋯, m

ρηθ(x_{i} ⊗ 1) = ρ ∑_{j=1}^{m} σ_{j}(μ(θ(x_{i} ⊗ 1)))

= ρσ_{i} ∑_{G/D} xγ̂(x^{−1}τ(1))

= ρσ_{i}(1 ⊗ 1) = ρ(x_{i} ⊗ 1) = x_{i} ⊗ 1.

Hence θ is a kS-homomorphism which splits ρη. But observe that Ω^{n}(k) ↓_{S}≅ k⊕ (proj), and hence as a kS-module M is projective. It follows immediately that m ∉ V_{E}(M_{E}), proving the lemma.

Returning to the proof of the theorem we notice that T_{E} ≅ ∑_{ExD,x∉N} k_{E∩xDx^{−1}}^{↑E} by the

Mackey formula. Hence the variety of T_{E} is V_{E}(T_{E}) = ∑_{x∉N} V_{E}(k_{E∩xDx^{−1}}^{↑E}) ⊆ W_{E}. Now we

have a commutative diagram of kE-modules

0  0
↓  ↓
0 ⟶ Y ⟶  M ⊕ P  ⟶ T ⟶ 0
∥  ↓  ↓
0 ⟶ Y ⟶ (Ω^{n}(k))^{m} ⊕ P ⟶^{η′} k_{D}^{↑G} ⟶ 0
↓^{ρη′}  ↓^{ρ}
U  U
↓  ↓
0  0

with exact rows and columns. Here P is a projective module which we add to the middle term in order that there will exist a surjective homomorphism η′ which coincides with η on (Ω^{n}(k))^{m}. Now V_{E}(Y) ⊆ V_{E}(M) ∪ V_{E}(T). Because m ∉ V_{E}(M) by the last lemma and m ∉ W_{E} we must have that m ∉ V_{E}(Y). The theorem is proved by setting L = Y.

To prove the theorem for tn rather than n, it is only necessary to replace γ by γ^{t}. All of the statements about varieties still hold.

Corollary 4.4. Let E ∈ M(G), m = |N : D|. Let m ∈ V_{G}(k), m ∉ W_{G,E} ∪ res_{G,E}^{∗}(W_{E}). Then for some n > 0 and all t > 0 there exists a projective module P and an exact sequence

0 ⟶ L ⟶ (Ω^{nt}(k))^{m} ⊕ P ⟶ k_{D}^{↑G} ⟶ 0

such that m ∉ V_{G}(L).

Proof. For any such sequence V_{G}(L) ⊆ V_{E}. So if m ∉ V_{E} then we are done by Theorem 4.2. Suppose m ∈ V_{E}. Then m = res_{G,E}^{∗}(ℓ) for ℓ ∈ V_{E}(k), ℓ ∉ W_{E}. By Theorem 4.2 there is such a sequence with ℓ ∉ V_{E}(L_{E}). But then m ∉ V_{G}(L).

Remark. In the statement of Theorem 4.2 it is actually only necessary to assume that m ∉ Ŵ_{E} ∪ W_{E} where

Ŵ_{E} = ∪V_{E} (k_{E∩xDx^{−1}}^{↑E})

where the union is over all x ∈ G, x ∉ N. In some cases, such as when N = G, Ŵ_{E} is a much smaller set then W_{E}. However the restriction that m ∉ W_{E} seems to be more essential (see Section 6).

5. The general case.

In this section we show that it is possible to construct a sequence which satisfies some of the properties of that in Theorem 4.2 for all maximal elementary abelian p-subgroups E at the same time. For notation let C = {E_{1}, ⋯, E_{s}} be a complete set of representatives of the conjugacy classes of elements of M(G), the maximal elementary abelian p-subgroups.

Recall that V_{G}(k) = ⋃_{i=1}^{s} V_{i} where V_{i} = V_{E_{i}}. Thus V_{1}, ⋯, V_{s} are the components of V_{G}(k),

V_{i} = res_{G,E_{i}}^{∗}(V_{E_{i}}(k)). The following is our main theorem.

Theorem 5.1. Let m be the least common multiple of the numbers |N_{G}(E) : D_{G}(E)| for E ∈ C. Let m_{i} = m/|N_{G}(E_{i}) : D_{G}(E_{i})|. Then for some n > 0 and all t > 0, there is a projective module P and an exact sequence

0 ⟶ L ⟶ (Ω^{nt}(k))^{m} ⊕ P ⟶^{θ_{i}} ∑_{i=1}^{s} (k_{D_{G}(E_{i})}^{↑G})^{m_{i}} ⟶ 0

such that V_{G}(L) ∩ V_{i} < V_{i} for all i = 1, ⋯, s.

Proof. Notice that |N_{G}(E_{i}) : D_{G}(E_{i})| = m/m_{i}. By the theorem of the previous section, for each i = 1, ⋯, s we have an exact sequence

0 ⟶ L_{i} ⟶ (Ω^{n_{i}t_{i}}(k))^{m/m_{i}} ⊕ P_{i} ⟶^{η_{i}} k_{D_{G}(E_{i})}^{↑G} ⟶ 0

with V_{G}(L_{i}) ∩ V_{i} < V_{i}. Such an exact sequence exists for some n_{i} > 0 (depending on E_{i}) and all t_{i} > 0. Here P_{i} is a projective kG-module which is added in to ensure that η_{i} is surjective. The actual P_{i} may depend on i and t_{i}. By summing m_{i} copies of the above sequence we get another sequence

0 ⟶ (L_{i})^{m_{i}} ⟶ (Ω^{n_{i}t_{i}}(k))^{m} ⊕ P̂_{i} ⟶^{η̂_{i}} (k_{D_{G}(E_{i})}^{↑G})^{m_{i}} ⟶ 0.

Also let n be the least common multiple of numbers n_{1}, ⋯, n_{s}. Then we can choose t_{i} so that n_{i}t_{i} = n. Then let

θ : (Ω^{n}(k))^{m} ⟶ ∑_{i=1}^{s}(k_{D_{G}(E_{i})}^{↑G})^{m_{i}}

be given by θ(a) = (θ_{1}(a), ⋯, θ_{s}(a)) where θ_{i} is the restriction of η̂_{i} to (Ω^{n}(k))^{m}.

Now take the direct sum of (Ω^{n}(k))^{m} with a projective module P so that there is an exact sequence

0 ⟶ L ⟶ (Ω^{n}(k))^{m} ⊕ P ⟶^{θ′} ∑_{i=1}^{s}(k_{D_{G}(E_{i})}^{↑G})^{m_{i}} ⟶ 0

such that θ′ coincides with θ on (Ω^{n}(k))^{m}. Now consider the commutative diagram:

0  0
↓  ↓
0 ⟶ L ⟶  W_{i}  ⟶ ∑_{j≠i}(k_{D_{G}(E_{j})}^{↑G})^{m_{j}} ⟶ 0
∥  ↓  ↓
0 ⟶ L ⟶ (Ω^{n}(k))^{m} ⊕ P ⟶^{θ′} ∑_{j=1}^{s}(k_{D_{G}(E_{j})}^{↑G})^{m_{j}} ⟶ 0
θ′_{i}↓  ↓^{ρ}
(k_{D_{G}(E_{i})}^{↑G})^{m_{i}}  (k_{D_{G}(E_{i})}^{↑G})^{m_{i}}
↓  ↓
0  0

where ρ is the projection and W_{i} is the kernel of θ′_{i} = ρθ′. Notice that θ′_{i} coincides with θ_{i} on (Ω^{n}(k))^{m}. Hence θ′_{i} and η̂_{i} differ by a map which factors through a projective. So by Lemma 1.1, V_{G}(W_{i}) = V_{G}(L_{i}). Also for j < i, the Mackey formula tells us that

k_{D_{G}(E_{j})}^{↑G} ↓_{E_{i}} ≅ ∑_{E_{i}xD_{G}(E_{j})}(k_{E_{i}∩xD_{G}(E_{j})x^{−1}})^{↑E_{i}}.

Because E_{i}∩xD_{G}(E_{j})x^{−1} is a proper subgroup of E_{i}, we have that V_{E_{i}}(k_{D_{G}(E_{j})}^{↑G}) < V_{E_{i}}(k),

and correspondingly V_{G}(k_{D_{G}(E_{j})}^{↑G}) ∩ V_{i} < V_{i}. Therefore we have that

V_{G}(L) ∩ V_{i} ⊆ (V_{G}(W_{i}) ∩ V_{i}) ∪ (V_{G}(∑_{j≠i} k_{D_{G}(E_{j})}^{↑G}) ∩ V_{i}) < V_{i}.

The proof of the theorem is complete when we notice that n, which was the least common multiple of the n_{i}’s could be replaced by any multiple of itself with no other alterations required for the argument.

6. Varieties and variations.

The purpose of this section is to consider the necessity of some of the conditions in the last two sections. First we notice that, in come sense, D_{G}(E) can be replace by the centralizer C_{G}(E). Of course, if p = 2 or if G is a p-group then D_{G}(E) = C_{G}(E). In addition, it is always true that k_{D}^{↑G} is a direct summand of k_{C}^{↑G}. Our point, however, is to show generally that there exist sequences similar to those in Theorem 4.2 and 5.1 with C replacing D provided we allow the middle term of the sequence to be a sum of syzygies of k in different degrees. In another direction we explore with an example the necessity of the condition that m ∉ W_{E} in Theorem 4.2. Even though we do not prove a general theorem it appears than the phenomenon exposed by the example is generic rather that exceptional. This might be an interesting area for future investigation.

First consider the specific case that G has a normal subgroup P = ⟨x|x^{p} = 1⟩ such that G/P is cyclic of order q dividing p − 1. Assume also that y ∈ G, y ∉ P has order q and that yxy^{−1} = x^{a} for a ∈ 𝔽_{p} a primitive q^{th} root of 1. Thus P = C_{G}(P) and G = D_{G}(P). Assume also that p > 2.

Lemma 6.1. H^{∗}(P, k) = k[ζ, η]/η^{2} where y · ζ = aζ and y · η = aη.

Proof. The form of the polynomial ring is well known. The only question here is the action of y. Notice that k_{P}^{↑G} = M_{0}⊕⋯⊕M_{q−1} where for each i = 0, ⋯, q−1, M_{i} = k·m_{i} is a onedimension k-vector space, generated by m_{i}, such that ym_{i} = a^{−i}m_{i}. Thus M_{0} ≅ k. The projective cover of M_{i} is P_{i} = M_{i} ↓_{<y>}^{↑G}≅ k_{<y>}^{↑G} ⊗ M_{i}. Now notice that in the conjugation action of y on kP we have that

y(x − 1)y^{−1} = x^{a} − 1 ≡ a(x − 1) mod (RadkP)^{2}.

So, it can be checked that there is a kG-projective resolution of k of the form

⋯ ⟶ P_{2} ⟶ P_{2} ⟶^{∂_{3}} P_{1} ⟶^{∂_{2}} P_{1} ⟶^{∂_{1}} P_{0} ⟶^{ϵ} k ⟶ 0

where ∂_{2i−1}(1 ⊗ m_{i}) = α ⊗ m_{i−1}

∂_{2i}(1 ⊗ m_{i}) = (∑_{j=0}^{p−1} x^{j}) ⊗ m_{i}

for α ∈ Rad kP an eigenvector with eigenvalue a for the action of y, i.e. α ≡ x − 1 mod (RadkP)^{2} and with the indices on the m_{i}’s taken modulo q. So, for example, if ζ : P_{1} → k is a cocycle of degree 2, then

(yζ)(1 ⊗ m_{i}) = ζ(1 ⊗ am_{i}) = (aζ)(1 ⊗ m_{i}).

The action on η is similarly derived.

Now suppose that E ∈ M(G) and that C = C_{G}(E), D = D_{G}(E). Then E = ⟨x_{i}, ⋯, x_{r}⟩ and D/C is generated by an element yC such that yxy^{−1} = x^{a}, for a ∈ 𝔽_{p}, a primitive q^{th} root of unity, q = |D/C|. Then

H^{∗}(E, k) = k[ζ_{1}, ⋯, ζ_{r}] ⊗ Λ(η_{1}, ⋯, η_{r})

where deg(ζ_{i}) = 2, deg(η_{i}) = 1 and Λ denotes the exterior algebra. This is a consequence of the fact that

H^{∗}(E, k) = H^{∗}(⟨x_{1}⟩, k) ⊗ ⋯ ⊗ H^{∗}(⟨x_{r}⟩, k)

with the usual anti-commutativity of multiplication. Then by the Lemma we have that yζ_{i} = aζ_{i} and yη_{i} = aη_{i}. Now let k_{C}^{↑D} = M_{0} ⊕ ⋯ ⊕ M_{q−1} where M_{i} = km_{i} for m_{i} ∈ M_{i} and ym_{i} = am_{i}.

Proposition 6.2. Suppose G, k, M_{i} are as above. Let m be a nonzero point in V_{E}(k) ≅ k^{r}. Then for b sufficiently large there exists a sequence of kD-modules

0 ⟶ L ⟶ Ω^{2ip^{b}}(k) ⟶ M_{i} ⟶ 0

such that m ∉ V_{E}(L ↓_{E}).

Proof. If p = 2 then M_{i} = k because C = D. In this case the theorem is true because the map Ω^{n}(k) → k can be taken to represent any cohomology element θ ∈ H^{∗}(D, k) with m ∉ V_{E}(res_{G,E}(θ)).

Hence we may assume that p > 2. For p odd, the centralizers of the maximal elementary abelian p-subgroups are p-nilpotent. Moreover O_{p′}(D) = O_{p′}(C) is the kernel of the principal kD- and kC-blocks. So we may assume that O_{p′}(C) = {1}. Hence D is given by an extension

1 ⟶ P ⟶ D ⟶ ⟨y⟩ ⟶ 1

where P = C is a p-group and y has order q dividing p − 1.

The principle of the proof from this point is that H^{∗}(P, k) has a system of parameters consisting of elements in degrees which are twice powers of p. This is a consequence (by induction) of the norm-map proof of the finite generation of cohomology (see [1] or [9]). So, in particular, there is an element ζ ∈ H^{2p^{b}}(P, k), such that m ∉ V_{E}(res_{P,E}(ζ)). We claim that we may assume that yζ = aζ. Notice first that p^{b} ≡ 1 (mod q) and hence

y(res_{P,E}(ζ)) ≡ a res_{P,E}(ζ) (modRadkE)

Therefore res_{G,E}(yζ −aζ) ∈ RadkE. So, because E is the only maximal elementary abelian p-subgroup of D, yζ − aζ must be nilpotent. Therefore (yζ − aζ)^{p^{c}} = y(ζ^{p^{c}}) − a(ζ^{p^{c}}) = 0 for c sufficiently large. Therefore, the claim follows if we replace ζ by ζ^{p^{c}}. Now let μ = ζ^{i} ∈ H^{2ip^{b}}(G, k). We have that yμ = a^{i}μ.

Now consider a kP -cocycle μ̂ : Ω^{n}(k) → k representing μ. Again we may assume that yμ̂ = a^{i}μ̂. Then the homomorphism

θ : Ω^{n}(k) → M_{i}

given by θ(u) = μ̂(u)m_{i} is a kD-homomorphism. However the kernel θ coincides with the kernel of μ. This is the module L_{μ} and we have that V_{P} (L_{μ}) = V_{P} (μ) does not contain res_{P,E}^{∗}(m). Hence L has the desired property.

Corollary 6.3. With the same hypothesis, if m ∈ V_{E}(k) is any nonzero point then for b sufficiently large there exists an exact sequence of kD-modules

0 ⟶ L ⟶ ∑_{i=1}^{q} Ω^{2ip^{b}}(k) ⟶ k_{C}^{↑D} ⟶ 0

such that m ∉ V_{E}(L_{E}).

Proof. The point is to add up the sequences in (6.2). For notation, we are thinking of M_{q} as M_{0} = k and then we need only remember that

k_{C}^{↑D} = M_{1} ⊕ ⋯ ⊕ M_{q}.

Now we can generalize the last result to get the main theorem of this section.

Theorem 6.4. Suppose that E ∈ M(G) and that m = |N : D|, N = N_{G}(E), D = D_{G}(E), C = C_{G}(E). Let m ∈ V_{E}(k), m ∉ W_{E} ∪ W_{E}. Then there exists an integer n > 0, such that for all t > 0 and any sufficiently large b, there is a projective module P and an exact sequence

0 ⟶ L ⟶ ∑_{i=1}^{q} (Ω^{nt+2ip^{b}}(k))^{m} ⊕ P ⟶ k_{C}^{↑G} ⟶ 0

such that m ∉ V_{E}(L_{E}).

Proof. Consider the diagram

0  0
↓  ↓
0 ⟶ X ⟶  L  ⟶  Y  ⟶ 0
∥  ↓  ↓
0 ⟶ X ⟶ ∑_{i=1}^{q}(Ω^{nt+2ip^{b}}(k))^{m} ⊕ P ⟶ ∑_{i=1}^{q} Ω^{2ip^{b}}(k_{D}^{↑G}) ⟶ 0
↓  ↓
k_{C}^{↑G}  k_{C}^{↑G}
↓  ↓
0  0

Here the right-hand column is the sequence of Corollary 6.3, induced to G. The middle row is a sum of translates, by Ω^{2ip^{b}}, i = 1, ⋯, q, of the sequence in Theorem 4.2. Then we have that V_{E}(L_{E}) ⊆ V_{E}(X_{E}) ∪ V_{E}(Y_{E}). So the middle column is the sequence of the theorem.

Remark. In this context, it is also possible to create an exact sequence which combines all of the maximal elementary abelian p-subgroups as in Theorem 5.1. That is, the right-hand

term in the sequence would be ∑_{i=1}^{t}(k_{C_{G}(E_{i})}^{↑G})^{m_{i}}, the middle term would be a sum of syzygies

of k and the left term would be a module whose variety contained no component of V_{G}(k). The proof is very similar to that of Theorem 5.1 and we leave this exercise to the reader.

We end this section with an example. Suppose that G = A_{4} the alternating group on four letters. Let p = 2 and let E be the Sylow 2-subgroup which is elementary abelian. Let y ∈ G be an element of order 3. It is easy to check that H^{∗}(E, k) = k[ζ_{1}, ζ_{2}] where ζ_{1}, ζ_{2} are in degree one and can be chosen so that y · ζ_{1} = ωζ_{1}, yζ_{2} = ω^{2}ζ_{2} for ω ∈ k a primitive cube root of unity. Now

k_{E}^{↑G} ≅ k ⊕ M_{1} ⊕ M_{2}

where M_{1} and M_{2} are one-dimensional modules generated by m_{1} and m_{2} with ym_{i} = ω^{i}m_{i}. Now any kP -homomorphism θ : Ω^{n}(k) → M_{i} can be expressed as a polynomial θ = f(ζ_{1}, ζ_{2}) in ζ_{1} and ζ_{2}. Of course, f must be homogeneous of degree n. So

θ = f(ζ_{1}, ζ_{2}) = ∑_{i=0}^{n} a_{i}ζ_{1}^{i}ζ_{2}^{n−i}

for some a_{0}, ⋯, a_{n} ∈ k. Now θ is a kG-homomorphism if and only if y · θ = ω^{i}θ. Hence we can prove the following.

Propostion. If θ : Ω^{n}(k) → M_{i} is a nonzero kG-homomorphism with kernel L, then either V_{E}(ζ_{1}) ⊆ V_{E}(L) or V_{E}(ζ_{2}) ⊆ V_{E}(L). In particular W_{E} = V_{E}(ζ_{1}) ∪ V_{E}(ζ_{2}). If, for any t, θ : (Ω^{n}(k))^{t} → k_{E}^{↑G} is surjective with kernel L then W_{E} ⊆ V_{E}(L).

Proof. By a rank variety argument, V_{E}(L) = V_{E}(k) unless t = 3. But when t = 3, we can split θ into three homomorphisms (θ_{i} : Ω^{n}(k) → M_{i}, i = 0, 1, 2, M_{0} = k). Likewise L = ∑ L_{i} where L_{i} is the kernel of θ_{i}. Now notice that for i = 1, 2, the requirement that yθ_{i} = ω^{i}θ_{i} means that for θ_{i} = ∑ a_{j}ζ_{1}^{i}ζ_{2}^{n−j}, either a_{0} = 0 or a_{n} = 0. Hence θ_{i} is a multiple of either ζ_{1} or ζ_{2}. The proposition is an easy consequence of the fact that if θ_{1} is a multiple of ζ_{1} then θ_{2} must be a multiple of ζ_{2} and vice versa.

7. Some Applications.

As in the introduction let Stmod-kG be the stable category of all kG-modules modulo projectives. Let C be the full subcategory of modules M with complexity less than the p-rank of G. Let Q be the quotient category Stmod-kG/C. For details of the quotient construction we refer the reader to [6]. In brief terms, the objects in Q are the same as those of Stmod-kG while, for morphisms, we invert any morphism in the stable category for which the third object in the triangle of that morphism is in C. Hence if we are given an exact sequence

0 ⟶ A ⟶ B ⟶^{β} C ⟶ 0

and if A ∈ C, then the class of β in Stmod-kG, becomes an isomorphism in Q. Thus the class of the homomorphism θ in Theorem 5.1 is an isomorphism in Q

The other thing which we need to recall from [6] is that every object in Q is periodic. That is, for any kG-module M, there is a positive integer t such that Ω^{t}(M) ≅ M in Q. In particular, there is a positive integer t such that Ω^{t}(k) ≅ k. So in Theorem 5.1, the integer ℓ can be chosen so that nℓ is a multiple t. This yields the following interpretation of Theorem 5.1.

Theorem 7.1. Let m be the least common multiple of the numbers |N_{G}(E) : D_{G}(E)| for E ∈ M(G). For each E in M(G) let m_{E} = m/|N_{G}(E) : D_{G}(E)|. Then in the quotient category Q

k^{m} ≅ ∑ (k_{D_{G}(E)}^{↑G})^{m_{E}}

where the sum is over a complete set of representatives of the conjugacy class of elements E ∈ M(G).

Suppose now that p = 2. Notice that C_{G}(E) = D_{G}(E) for any E. In [6] it was shown that for the group G = (ℤ/2)^{2} of order 4, Q has only a single indecomposable object, k, and every other object is isomorphic to a direct sum of copies of k. In general it makes sense to form the representation ring or Green ring 𝔄(G) of the quotient category Q. This is the abelian group generated by the symbols [M] corresponding the isomorphism classes of objects M in Q. The group is subject to the relations [M ⊕ N] = [M] + [N] for two objects M and N in Q. The multiplication on 𝔄(G) is given by [M][N] = [M ⊗_{k} N]. Because of the fact that there is no Krull-Schmidt theorem in Q, we have no reason to believe that 𝔄(G) should be a free abelian group. However if we allow rational coefficients, i.e., look at 𝔄_{ℚ}(G) = ℚ ⊗_{ℤ} 𝔄(G) then we have a ℚ-vector space whose basis can be taken as a subset of the isomorphism classes of indecomposable objects.

Proposition 7.2. Suppose that p = 2 and G is a group of 2-rank 2. Suppose that the maximal elementary abelian 2-groups are self centralizing. Then the dimension of 𝔄_{ℚ}(G) is equal to the number of conjugacy classes of maximal elementary abelian 2-groups.

The proof is simple Frobenius reciprocity. We are assuming that C = C_{G}(E) = E ≅ (ℤ/2)^{2} for all E ∈ M(G). So if M is a kG-module, then

M ⊗ k_{C}^{↑G} ≅ (M_{C})^{↑G}

is isomorphic in Q to a direct sum of copies of k_{C}^{↑G}. The rest follows from Theorem 7.1.

The main application of Proposition 7.2 would be to groups with dihedral or semidihedral Sylow 2-subgroup. In [6] the Proposition was proved ad hoc for dihedral groups. In that case 𝔄(G) is a free abelian group on two generators, even though there are three indecomposable objects in Q. They are the induced modules from the rank 2 elementary abelian groups and the trivial module. They satisfy the relation of Theorem 7.1.

There is another observation which can be made from the standpoint of Frobenius reciprocity and block theory. Notice that if M is a kG-module then for any n > 0

Ω^{n}(k) ⊗ M ≅ Ω^{n}(M) ⊕ P

where P is some projective module. In particular Ω^{n}(M) and M are in the same block. So if the sequence of theorem 5.1 is tensored with M then the resulting middle term has a non-projective module only in the same block as M. The term of the tensor product on the left may have non-projective modules in several blocks, but the tensor product has smaller than maximal variety. On the right hand term we use Frobenius reciprocity. The technical statement of the result comes out as follows.

Proposition 7.3. Suppose that E ∈ M(G) is a maximal elementary abelian p-subgroup of G. Suppose that M is kG-module and that U is a component (direct summand) of M_{C_{G}(E)}^{↑G}. If U is not in the same block as M, then V_{G}(U) ⊆ W_{G,E} ∪ V_{E}. Here we recall that W_{G,E} = res_{G,E}^{∗}(W_{E}) and V_{E} = res_{G,E}(W_{E}).

Proof. Suppose that m ∈ V_{G}(k), m ∉ W_{G,E}∪V_{E}. Then by Theorem 6.4 there is a sequence

0 ⟶ L ⟶ ∑_{i=1}^{q} (Ω^{n_{i}}(k))^{m} ⊕ P ⟶ k_{C}^{↑G} ⟶ 0

for some integers n_{i}, some projective module P and with the property that m ∉ V_{G}(L). We tensor this sequence with M to get a sequence

0 ⟶ L ⊗ M ⟶ ∑_{i=1}^{q} Ω^{n_{i}}(M) ⊕ P′ ⟶ M_{C}^{↑G} ⟶ 0

for some projective module P′. This sequence is a direct sum of sequences, one for each block of kG. By hypothesis the direct summand of M_{C}^{↑G} in the block of U has the form U ⊕ U′ for some U′. The direct summand of Ω^{n_{i}}(M) ⊕ P′ in the block of U is a projective module Q. So we have an exact sequence

0 ⟶ Ω(U ⊕ U′) ⟶ Q ⟶ U ⊕ U′ ⟶ 0.

Now V_{G}(U) ⊆ V_{G}(U ⊕ U′) = V_{G}(Ω(U ⊕ U′)). Because Ω(U ⊕ U′) is a direct summand of L ⊗ M, V_{G}(U) ⊆ V_{G}(L ⊗ M) ⊆ V_{G}(L) by Lemma 2.1. Because m ∉ V_{G}(L) we must have that m ∉ V_{G}(U).

Finally we show that the results of the paper can be used to give a new proof of main theorem of [5]. The cohomology ring Ext_{kG}^{∗}(M, M) of a kG-module M can be noncommutative in a way that is far more essential than the graded (non)commutativity of H^{∗}(G, k). In [5] it was shown how induction and transfer maps can play a part in this structure. In fact, the proofs of Theorems 4.2 and 5.1 use the same ideas as that of the main theorem of [5]. However the point of view is greatly changed.

Proposition 7.4. Suppose that G is a p-group and that E ∈ M(G) and that m = |N_{G}(E) : C_{G}(E)|. Then the cohomology ring Ext_{kG}^{∗}(k_{C}^{↑G}, k_{C}^{↑G}) has a two sided ideal 𝔍 such that

Ext_{kG}^{∗}(k_{C}^{↑G}, k_{C}^{↑G})/𝔍 ≅ Mat_{m}(k)

the ring of m × m matrices over k.

Proof. To get an efficient proof it seems best to modify the quotient category construction of [6]. We begin by choosing a maximal ideal m ∈ V_{E} = V_{G}(k_{C}^{↑G}) but m ∉ W_{G,E}. Let C_{m} be the full subcategory of Stmod-kG of all modules L such that m ∉ V_{G}(L). Let Q_{m} = Stmod-kG/C_{m} be the quotient category. It is easily checked that C_{m} is a thick, triangulated subcategory. So the quotient is defined and also triangulated. With minor modifications of the arguments of [6], it can be shown that, for M and N kG-modules,

Hom_{Q_{m}}(M, N) = [Ext_{kG}^{∗}(M, N) · S^{−1}]_{0}

where S = {ζ · Id_{M} ∈ H^{∗}(G, k)|ζ homogeneous and ζ ∉ m}. Likewise

Ext_{Q_{m}}^{∗}(M, N) = ∑_{j≥0} [Ext_{kG}^{∗}(M, N)S^{−1}]_{j}

where [ ]_{j} indicates the j^{th} grading. For any kG-module M we have a ring homomorphism ψ_{m} : Ext_{kG}^{∗}(M, M) → Ext_{Q_{m}}(M, M).

By Corollary 4.4, we have that k^{m} ≅ k_{C}^{↑G} in Q_{m}. Consequently in Q_{m} we have that

Hom_{Q_{m}} (k_{C}^{↑G}, k_{C}^{↑G}) ≅ Mat_{m} (Hom_{Q_{m}}(k, k))

and also

Ext_{Q_{m}}^{∗} (k_{C}^{↑G}, k_{C}^{↑G}) ≅ Mat_{m} (Ext_{Q_{m}}^{∗}(k, k)) .

Now suppose that θ : Ext_{kG}^{∗}(k, k) → k is a homomorphism with kernel m. Then we get an induced map θ′ : Ext_{Q_{m}}^{∗}(k, k) → k given by θ′(γζ^{−1}) = θ(γ)θ(ζ)^{−1}. This is well defined because ζ ∉ m and hence 0 ≠ θ(ζ) ∈ k. The composition

Ext_{kG}^{∗}(k, k) ⟶^{ψ_{k}} Ext_{Q_{m}}^{∗}(k, k) ⟶^{θ′} k

is equal to θ.

Now θ also induces a ring homomorphism

θ̂: Mat_{m}(Ext_{Q_{m}}^{∗}(k, k)) → Mat_{m}(k)

which is also surjective. Let m′ be the kernel of θ̂ pulled back to a two sided ideal in Ext_{Q_{m}}^{∗}(k_{C}^{↑G}, k_{C}^{↑G}). The proof of the proposition is completed by showing that the composition

Ext_{kG}^{∗}(k_{C}^{↑G}, k_{C}^{↑G}) → Ext_{Q_{m}}^{∗}(k_{C}^{↑G}, k_{C}^{↑G}) → Ext_{Q_{m}}^{∗}(k_{C}^{↑G}, k_{C}^{↑G})/m′

is surjective. However this is clear. The reason simply is that in the quotient category construction the set S = {ζ · Id_{M}|ζ homogeneous, ζ ∉ m} of elements which are inverted are in the center of the ring Ext_{kG}^{∗}(M, M). Thus it is impossible for the second map to be surjective without the composition also being surjective.

Remark. The decompositions in this paper are based on taking multiplicities of modules in the quotient categories. A very different, and more general, sort of decomposition can be obtained in the form of M ⊕ Ω(M) being the direct sum in the quotient category of modules each of which has only a single component of maximal dimension in its variety [8]. Such a decomposition exists for modules of all complexities, but it has not yet been as useful as the inductive statement given in this paper.

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