THE NUCLEUS, AND EXTENSIONS BETWEEN MODULES FOR A FINITE GROUP

D. J. BENSON

Abstract. Let G be a finite group and k be a field of characteristic p dividing |G|. If we are given two finitely generated kG-modules M and N, how do we tell whether there are extensions (i.e., whether Ext_{kG}^{i}(M, N) ≠ 0 for some i > 0)? Certainly, if the varieties of M and N do not overlap, then there are no extensions. The converse is false. For modules in the principal block, if there are no extensions then the overlap between the varieties has to lie in a certain subvariety of the cohomology variety, called the nucleus. This variety can be described explicitly in terms of the subgroup structure; it is the union of the images of the varieties of subgroups whose centralizers are not p-nilpotent.

A number of attempts have been made to understand the corresponding problem for nonprincipal blocks. The situation is certainly harder than in the case of the principal block. An analysis of some examples leads to some very explicit questions about the finitely generated modules for some finite dimensional algebras.

A typical example which we do not properly understand leads to the following question. Let A be the k-algebra generated by elements x and y, with relations x^{p} = 0, y^{p} = 0 (p = char(k) a odd prime), xy + yx = 0. If M and N are finitely generated indecomposable A-modules whose restriction to the subalgebra generated by x is not free, is it necessarily true that Ext_{A}^{i}(M, N) ≠ 0 for some i > 0?

1. Support varieties and the nucleus

Let G be a finite group and k a field of characteristic p dividing |G|. The cohomology ring H^{∗}(G, k) = Ext_{kG}^{∗}(k, k) is a finitely generated graded commutative k-algebra (Evens [8] and Venkov [13]), and its maximal ideal spectrum Max H^{∗}(G, k) is an affine variety whose dimension is equal to the p-rank of G (Quillen [11, 12]).

If M is a finitely generated kG-module, then tensoring with M gives rise to a natural map

H^{∗}(G, k) = Ext_{kG}^{∗}(k, k) ⟶^{−⊗_{k}M} Ext_{kG}^{∗}(M, M).

The kernel I_{G}(M) defines a closed homogeneous subvariety V_{G}(M) of V_{G}, called the support variety of M. This concept has been the object of a great deal of interest in the last twenty years or so, and has been generalized in

The author is partly supported by a grant from the NSF.

various directions, including infinitely generated modules, large classes of infinite groups, and p-restricted Lie algebras.

Some of the properties of support varieties are as follows (see for example [1]).

(i) V_{G}(M ⊕ N) = V_{G}(M) ∪ V_{G}(N).

(ii) V_{G}(M ⊗_{k} N) = V_{G}(M) ∩ V_{G}(N).

(iii) V_{G}(M) = {0} if and only if M is projective.

(iv) V_{G}(M^{∗}) = V_{G}(M).

(v) The dimension of V_{G}(M) is an invariant called the complexity of M, which determines the polynomial rate of growth of the minimal resolution of M as a kG-module.

(vi) Given any closed homogeneous subvariety V of V_{G}, there exists a finitely generated kG-module M such that V_{G}(M) = V.

So for example, V_{G}(M)∩V_{G}(N) = {0} if and only if M ⊗_{k} N is projective, so that this gives a way of testing whether tensor products of modules are projective. Similarly, if V_{G}(M)∩V_{G}(N) = {0} then Hom_{k}(M, N) ≅ M^{∗}⊗_{k} N is a projective kG-module and so

Ext_{kG}^{i}(M, N) ≅ H^{i}(G, Hom_{k}(M, N)) = 0

for all i > 0. The converse is true if G is a p-group, but not in general. If Ext_{kG}^{i}(M, N) = 0 for all i > 0 it does not follow that V_{G}(M) ∩ V_{G}(N) = {0}.

[Remark. For the sake of convenience, we use Tate cohomology instead of excluding i = 0 from ordinary cohomology, and we note that Ext_{kG}^{i}(M, N) = 0 for all i > 0 is equivalent to the statement that Êxt_{kG}^{∗}(M, N) = 0 (see theorem 1.1 of [4]).]

The most obvious reason for this failure of the converse is block theory. A block of kG is by definition an indecomposable two sided ideal direct factor of kG as an algebra. We have a block decomposition

kG = B_{0} × B_{1} × ⋯ × B_{s}

and a corresponding decomposition of the identity element

1 = e_{0} + ⋯ + e_{s}

as a sum of centrally primitive idempotents. For any kG-module M, we have a corresponding decomposition

M = e_{0}M ⊕ ⋯ ⊕ e_{s}M.

If M is indecomposable, then there is a unique value of i for which M = e_{i}M, and we say that M belongs to the block B_{i}.

The principal block B_{0} is defined to be the one to which the trivial kGmodule k belongs. The remaining blocks are called nonprincipal blocks. If M is in a nonprincipal block, then Ĥ^{∗}(G, M) = Êxt_{kG}^{∗}(k, M) = 0, even though V_{G}(M) = V_{G}(k) ∩ V_{G}(M) need not be equal to {0}.

But even within the principal block, there can be finitely generated modules M and N such that V_{G}(M) ∩ V_{G}(N) ≠ {0} but Êxt_{kG}^{∗}(M, N) = 0. For example, let G = ℤ/3 × Σ_{3}, where Σ_{3} is the symmetric group of degree three (i.e., the dihedral group of order six), and let k be a field of characteristic three. Let M be the trivial module, and let N be the three dimensional projective cover of the nontrivial one dimensional kΣ_{3}-module, regarded as a kG-module with the first factor of ℤ/3 acting trivially. Then N is periodic with period two as a kG-module, and Êxt_{kG}^{∗}(M, N) = 0. Both M and N are in the principal block, and the intersection of their varieties is equal to the image of the variety of the first factor ℤ/3 in the variety V_{G}. This example is described in more detail following Proposition 1.1.

In order to investigate this situation, the idea of the nucleus was introduced by Benson, Carlson and Robinson [4]. The nucleus Y_{G} ⊆ V_{G} is defined to be the union of the images of the maps res_{G,H}^{∗}: V_{H} → V_{G} where H runs over the subgroups of G with the property that C_{G}(H) is not p-nilpotent (a finite group is said to be p-nilpotent if it has a normal subgroup whose order is coprime to p and whose index is a power of p).

The representation theoretic nucleus Θ_{G} ⊆ V_{G} was defined in the same paper to be the union of the varieties V_{G}(M) where M is a finitely generated module in the principal block such that Ĥ^{∗}(G, M) = 0. It was conjectured in [4] and proved in [2] that these two concepts coincide, namely that Y_{G} = Θ_{G}. The relevance to our discussion is that the representation theoretic nucleus also has the following formulation.

Proposition 1.1. The representation theoretic nucleus is equal to the union of the varieties of the form V_{G}(M) ∩ V_{G}(N) as M and N run over finitely generated modules in the principal block such that Êxt_{kG}^{∗}(M, N) = 0.

Proof. Since the proof of this does not appear in the literature, and requires some machinery, it is deferred to Appendix A in order to preserve the flow of the discussion.

In fact, the proof of Proposition 1.1 can be used to show that Θ_{G} is exactly the union of the lines ℓ through the origin in V_{G} with the property that there exist finitely generated modules M and N in the principal block such that V_{G}(M) = V_{G}(N) = ℓ and Êxt_{kG}^{∗}(M, N) = 0.

Example. Let us see how this Proposition and the remark following it work in the example described above. As before, let G be the group ℤ/3 × Σ_{3}, and let k be an algebraically closed field of characteristic three. Then there are two simple kG-modules, namely the trivial module k and the nontrivial one dimensional module ε coming from the sign representation of Σ_{3}. The projective covers P_{k} and P_{ε} of these modules have the following structure.

[figure: k; k; ε; k; ε; k; ε; k; k; P_{k}; ε; ε; k; ε; k; ε; k; ε; ε; P_{ε}]

We take

N = [figure: ε; k; ε] ,  ΩN = [figure: ε; ε; k; k; ε; ε] ,  Ω^{2}N ≅ N.

Then N is periodic with period two, and since Tate cohomology can be calculated by taking the invariants in the cokernels of a complete resolution, it follows that Ĥ^{∗}(G, N) = Êxt_{kG}^{∗}(k, N) = 0.

The variety V_{G} is isomorphic to affine two-space 𝔸^{2}(k), and the subvariety V_{G}(N) = V_{G}(k) ∩ V_{G}(N) is the copy of 𝔸^{1}(k) corresponding to the image of res^{∗} : V_{ℤ/3×1} → V_{G}. This is also equal to the nucleus Y_{G}, because there is only one nontrivial subgroup whose centralizer is not 3-nilpotent, namely ℤ/3 × 1.

It is somewhat more difficult to write down a module M with V_{G}(M) = Y_{G} and Êxt_{kG}^{∗}(M, N) = 0, as in the above remark following Proposition 1.1. But there is a systematic way of finding such a module, using the modules L_{ζ} as described in Appendix A, and the result in this case is the module

M = [figure: ε; k; k; ε; k; ε; k; ε; ε; k; ε; k; ε; k; k; ε; ε; k] .

2. Nonprincipal blocks and Linckelmann’s varieties

The theory of the nucleus for the principal block is so satisfactory that it seems like a good idea to try to find a corresponding theory for nonprincipal blocks. The first steps in this direction have been taken by Carlson [6] and Linckelmann [9].

Linckelmann [9] has found the what seems to be the correct formulation of the cohomology ring and support varieties for a nonprincipal block of kG. The problem to be addressed is that there is no object playing the role that the trivial module plays for the principal block. So Linckelmann makes the observation that the stable element method for calculating the cohomology of a finite group can be used as a definition for a nonprincipal block. A defect group D of a block B is a minimal subgroup with the property that the multiplication map B ⊗_{kD} B → B splits as a homomorphism of B-B-bimodules. The defect groups form a conjugacy class of p-subgroups of G. The cohomology ring H^{∗}(B) is defined to be the subring of H^{∗}(D, k) consisting of elements which are stable with respect to fusions which keep track of the corresponding block idempotents in centralizers in a suitable sense.

The variety V_{B} is defined to be the maximal ideal spectrum of the ring H^{∗}(B). It is again a homogeneous affine variety. There is a natural map H^{∗}(G, k) → H^{∗}(B), and the induced map V_{B} → V_{G} displays V_{B} as a finite cover of the image of res_{G,D}^{∗}: V_{D} → V_{G}.

If M is a finitely generated module in B, then Linckelmann uses Hochschild cohomology to define a natural map

H^{∗}(B) → Ext_{B}^{∗}(M, M) = Ext_{kG}^{∗}(M, M),

and the kernel of this map defines a subvariety V_{B}(M) of V_{B}. This definition enjoys similar properties to the varieties V_{G}(M) described previously. So for example, M is projective if and only if V_{B}(M) = {0}. The image of V_{B}(M) under V_{B} → V_{G} is equal to V_{G}(M), but V_{B}(M) is not usually the inverse image of V_{G}(M), so there is definitely more information. In fact, given any closed homogeneous subvariety V of V_{B}, there exists a finitely generated module M in B such that V_{B}(M) = V. Linckelmann proves that his variety theory is invariant under “splendid stable equivalences” in the sense of Rickard.

3. The nucleus for a nonprincipal block

In terms of Linckelmann’s varieties, there are analogs of the nucleus and of the representation theoretic nucleus. In [9], Linckelmann defines the nucleus Y_{B} ⊆ V_{B} to be the union of the images of V_{Q} → V_{B}, as Q runs over the subgroups of the defect group D such that the corresponding block of kC_{G}(Q) is not a “nilpotent block” in the sense of Broué and Puig [5, 10].

In the light of Proposition 1.1, a reasonable definition of a representation theoretic nucleus Θ_{B} ⊆ V_{B} would be the union of the varieties of the form V_{B}(M) ∩ V_{B}(N) where M and N are finitely generated modules in B such that Êxt_{B}^{∗}(M, N) = 0.

It is not at all obvious whether these two varieties Y_{B} and Θ_{B} are equal. Neither inclusion seems obvious. In the next section, we describe an example of a nonprincipal block of a finite group algebra for which we do not know whether the two varieties agree.

It turns out that a principal block is nilpotent if and only if it has just one simple module. The same is not at all true of nonprincipal blocks, and this is really the source of the problem with understanding the nucleus for nonprincipal blocks. The example we describe in the next section is an example of a non-nilpotent block with just one simple module.

It is tempting to think that this may lead to an alternative candidate for the nucleus. Instead of asking that the corresponding block of the centralizer is non-nilpotent, why don’t we ask that it have more than one simple module? The problem with this property is that going down to (smaller) centralizers of larger subgroups may take us from a block with just one simple to a block with more than one simple, and this causes trouble.

4. An example

Let p be an odd prime, and let G be the semidirect product

G = (ℤ/p × ℤ/p)⋊Q_{8},

where the central element z of order two in the quaternion group Q_{8} acts trivially, and the quotient acts as inversion on the two factors. In other words, writing g and h for the generators of ℤ/p × ℤ/p, and i and j for the generators of Q_{8}, we have igi^{−1} = g, jgj^{−1} = g^{−1}, ihi^{−1} = h^{−1}, jhj^{−1} = h.

Let k be a field of characteristic p. Then the group algebra kG decomposes into two blocks B_{0} ×B_{1}, and the corresponding idempotents are \frac{1}{2}(1+z) and \frac{1}{2}(1 − z). The block B_{0} has four simple modules, each of dimension one, and B_{1} has one simple module, which has dimension two. We shall write B for B_{1}. It is an example of a non-nilpotent block with only one simple module.

The cohomology ring H^{∗}(G, k) is the tensor product of a polynomial ring on two generators in degree four with an exterior algebra on two generators in degree three. So the cohomology variety V_{G} is equal to affine two space 𝔸^{2}(k). The defect group of each of B_{0} and B is equal to the Sylow p-subgroup ⟨g, h⟩, and the fusion is the same in the two blocks. So the natural map V_{B} → V_{G} is an isomorphism in this case. The nucleus Y_{G} is the union of the two affine lines corresponding to the inclusions of ⟨g⟩ and ⟨h⟩ into G, since these are the subgroups whose centralizers (of the form ℤ/p × (ℤ/p⋊ℤ/4)) are not pnilpotent. The nucleus Y_{B} for the nonprincipal block B corresponds to Y_{G} under the natural map V_{B} → V_{G}.

It turns out that B is isomorphic to the algebra of 2 × 2 matrices over the algebra

A = k⟨X, Y ⟩/(X^{p}, Y^{p}, XY + Y X),

which can be thought of as a sort of twisted version of the group algebra of the Sylow p-subgroup. In Appendix B, we give details of the isomorphism.

The representation theoretic nucleus Θ_{B} is either {0} or the nucleus Y_{B}. In order to decide which is true, we need to answer the following question.

Question 1. Are there examples of finitely generated A-modules M and N such that neither M↓_{⟨X⟩} nor N↓_{⟨X⟩} is a free k⟨X⟩/(X^{p})-module, but

Êxt_{A}^{∗}(M, N) = 0?

Standard techniques in finite group representation theory can be used to reduce this to the following question.

Question 2. Let H be the group ℤ/p × D_{2p}. Write ε for the nontrivial one dimensional kH-module, and write α for the automorphism of H which inverts the first direct factor ℤ/p and centralizes the second direct factor D_{2p}. Do there exist finitely generated kH-modules M and N such that neither M nor N restricts to a free kℤ/p-module (where ℤ/p is the first direct factor of H) and such that both Êxt_{kH}^{∗}(M, N) = 0 and Êxt_{kH}^{∗}(M, ε ⊗_{k} α^{∗}(N)) = 0?

For a proof of the equivalence of Questions 1 and 2, see Appendix B.

Appendix A: Proof of Proposition 1.1

It is clear that the representation theoretic nucleus is contained in the subset described in the proposition, because we can take M to be the trivial kG-module k. Conversely, suppose that ℓ is a line through the origin in V_{G} but not in the representation theoretic nucleus, and that V_{G}(M)∩V_{G}(N) ⊇ ℓ. Choose nonzero elements ζ_{1}, …, ζ_{t} of degrees n_{1}, …, n_{t} in H^{∗}(G, k) such that V_{G}⟨ζ_{1} … ζ_{t}⟩ = ℓ, and set L = L_{ζ_{1}}⊗_{k}⋯⊗_{k}L_{ζ_{t}}. Here, L_{ζ_{i}} is the module defined by representing ζ_{i} by a homomorphism ζ̂_{i}: Ω^{n_{i}}(k) → k and taking the kernel. Then V_{G}(L) = ℓ and

V_{G}(L ⊗_{k} M) = V_{G}(L) ∩ V_{G}(M) = ℓ.

By theorem 4.5 of [3], it follows that L ⊗_{k} M is in the thick subcategory ⟨k⟩ of the stable module category stmod(kG) generated by the trivial module k. So

Hom_{k}(L ⊗_{k} M, N) = L^{∗} ⊗_{k} M^{∗} ⊗_{k} N

is in the principal block B_{0}. So by corollary 5.3 of [2], it follows that Êxt_{kG}^{∗}(L ⊗_{k} M, N) ≠ 0.

We now set L_{i} = L_{ζ_{i+1}} ⊗_{k} ⋯ ⊗_{k} L_{ζ_{t}} for i = 0, …, t − 1, and L_{t} = k, and we prove by induction on i that Êxt_{kG}^{∗}(L_{i} ⊗_{k} M, N) ≠ 0. For i = 0, this is what we have proved above, so this begins the induction. For i = t, this is the statement Êxt_{kG}^{∗}(M, N) ≠ 0, which is required to finish the proof of the Proposition.

For the inductive step, we have a short exact sequence

0 → L_{ζ_{i}} → Ω^{n_{i}}(k) ⟶^{ζ̂_{i}} k → 0.

This induces a long exact sequence

… → Êxt_{kG}^{∗}(L_{i−1} ⊗_{k} M, N) → Êxt_{kG}^{∗+n_{i}}(L_{i} ⊗_{k} M, N)

→ Êxt_{kG}^{∗}(L_{i} ⊗_{k} M, N) → Êxt_{kG}^{∗+1}(L_{i−1} ⊗_{k} M, N) …

Since Êxt_{kG}^{∗}(L_{i−1} ⊗_{k} M, N) ≠ 0, it follows that Êxt_{kG}^{∗}(L_{i} ⊗_{k} M, N) ≠ 0. By induction on i, this completes the proof of the Proposition.

Appendix B: Details of the example

In this Appendix, we give details of the example described in Section 4. Recall that G = (ℤ/p × ℤ/p)⋊Q_{8} is generated by commuting elements g and h of order p, and elements i and j generating a quaternion group of order eight, with action given by igi^{−1} = g, jgj^{−1} = g^{−1}, ihi^{−1} = h^{−1}, jhj^{−1} = h. We begin with the isomorphism between the block \frac{1}{2}(1 − z)kG and Mat_{2}(A), where

A = k⟨X, Y ⟩/(X^{p}, Y^{p}, XY + Y X).

We have a homomorphism from kG to Mat_{2}(A) given by

i ↦ I = ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) , j ↦ J = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) ,

g ↦ exp(IX), h ↦ exp(JY).

Here, for the entries in the matrices involving i, it is to be interpreted as a square root of −1 in k. The exponential of a matrix is defined using the power series

exp(x) = 1 + x + \frac{1}{2}x^{2} + …

Since X^{p} = Y^{p} = 0, the terms in which p is in the denominator are just taken to be zero. Since I and J anticommute, and X and Y anticommute, we see that IX commutes with JY, and hence exp(IX) commutes with exp(JY). Since conjugation by I negates J and fixes I, it inverts exp(JY) and fixes exp(IX). These and similar statements show that the above assignment gives a homomorphism kG → Mat_{2}(A). To check that it is surjective, we note that kQ_{8} is mapped surjectively onto Mat_{2}(k), log(g) maps to IX, and log(h) maps to JY. Here, the logarithm is defined by the power series

log(1 + x) = x − \frac{1}{2}x^{2} + \frac{1}{3}x^{3} − …

where x is taken to be g − 1 or h − 1, and the relations (g − 1)^{p} = 0 and (h − 1)^{p} = 0 mean that the series terminates before division by p gives problems, as before. Since \frac{1}{2}(1 + z) is in the kernel, we obtain a surjective homomorphism from B = \frac{1}{2}(1 − z)kG to Mat_{2}(A). Since these algebras both have dimension 4p^{2} over k, this homomorphism is an isomorphism.

Next, we prove that Questions 1 and 2 are equivalent. Let K be the subgroup of G generated by g, h and i. Let v be the one dimensional representation of G on which K acts trivially and j acts as the matrix (−1). Choose a representative Ω^{2}k → v of a nontrivial element of the one dimensional space H^{2}(G, v), and write L for the kernel, so that we have a short exact sequence

0 → L → Ω^{2}k → v → 0.

Then V_{G}(L) is the image of the map V_{⟨h⟩} → V_{G}.

Let M and N be finitely generated kG-modules whose restrictions to ⟨g⟩ are not free, and such that Êxt_{kG}^{∗}(M, N) = 0. Then V_{G}(M) ∩ V_{G}(N) ⊆ Y_{G}. We can also assume without loss of generality that M and N are indecomposable. A theorem of Carlson [7] states that the variety of an indecomposable module is (projectively) connected, so we have V_{G}(M) = V_{G}(N) is the image of V_{⟨g⟩} → V_{G}. So V_{G}(L)∩V_{G}(M) = {0}, and L⊗_{k} M is projective. The short exact sequence

0 → L ⊗_{k} M → Ω^{2}k ⊗_{k} M → v ⊗_{k} M → 0

shows that Ω^{2}M is stably isomorphic to v ⊗_{k} M. It follows that

Êxt_{kG}^{∗}(v ⊗_{k} M, N) = 0.

Now

M↓_{K}↑^{G} ≅ k_{K}↑^{G} ⊗_{k} M ≅ (k ⊕ v) ⊗_{k} M ≅ M ⊕ v ⊗_{k} M,

and so

Êxt_{kK}^{∗}(M↓_{K}, N↓_{K}) ≅ Êxt_{kG}^{∗}(M↓_{K}↑^{G}, N) = 0.

If a kK-module is a restriction from G, then it is invariant under conjugation by j ∈ G \ K. Conversely, if M_{1} is any kK-module, then M_{1}↑^{G}↓_{H} ≅ M_{1} ⊕ j^{∗}(M_{1}), so M_{1} ⊕ j^{∗}(M_{1}) is a restriction from G. So we are reduced to the question of whether there exist finitely generated kK-modules M_{1} and N_{1}, on which z ∈ Z(K) acts as multiplication by −1, such that the restriction of M_{1} and N_{1} to ⟨g⟩ are not free, and such that both Êxt_{kK}^{∗}(M_{1}, N_{1}) = 0 and Êxt_{kK}^{∗}(M_{1}, j^{∗}(N_{1})) = 0.

Let w be a one dimensional representation of K on which z acts as multiplication by −1. Then the irreducible representations of K are k, w, w^{2} = ε and w^{3} = w^{∗} = j^{∗}(w). Setting M_{2} = w ⊗_{k} M_{1} and N_{2} = w ⊗_{k} N_{1}, we have

w ⊗_{k} j^{∗}(M_{1}) = w ⊗_{k} j^{∗}(w^{∗} ⊗_{k} M_{2})
≅ w ⊗_{k} j^{∗}(w^{∗}) ⊗_{k} j^{∗}(M_{2}) ≅ ε ⊗_{k} j^{∗}(M_{2})

and so the conditions on M_{2} and N_{2} are that Êxt_{kK}^{∗}(M_{2}, N_{2}) = 0 and Êxt_{kK}^{∗}(M_{2}, ε ⊗_{k} j^{∗}(N_{2})) = 0. The modules M_{2} and N_{2} have z in the kernel of the action, and passing to the quotient by ⟨z⟩ does not affect cohomology. So we are now dealing with modules for K/⟨z⟩ ≅ ℤ/p × D_{2p}. It is not hard to check that each of the steps above is reversible, and so the equivalence of Questions 1 and 2 is established.

Acknowledgement. I would like to thank Jon Alperin, Jon Carlson, Markus Linckelmann and Jeremy Rickard for their interest in this work and for the ideas they have contributed.

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

E-mail address: djb@byrd.math.uga.edu