INFINITE DIMENSIONAL MODULES FOR A FINITE GROUP

D. J. BENSON

Dedicated to Walter Feit on the occasion of his 65th birthday

This paper is a transcription of the lecture I gave at the Ohio State University Conference on Representation Theory of Finite Groups. My intention was to talk about the ideas involved in a small corner of my recent joint work with Jon Carlson and Jeremy Rickard [2, 3] on infinitely generated modules, and try to explain the role of generic points of varieties in this context.

As we move into the second century of finite group representation theory, we still find that the vast majority what is being done is concerned with finitely generated modules. It seems to me that the reason for this is largely that we have very few techniques that work in a wider context, say for example the context of arbitrary modules for the group algebra of a finite group over a field. Linear transformations on infinite dimensional spaces don’t necessarily have any eigenvalues. There are modules with no indecomposable summands. The Krull–Schmidt theorem fails quite badly, so there are no vertices and sources. There is a module M for ℤ_{2} × ℤ_{2} in characteristic two, which satisfies M ≅ M ⊕ M ⊕ M but not M ≅ M ⊕ M.

In the light of these pathologies, it is tempting just to give up and return to the relatively safe world of finitely generated modules, especially as there are still many interesting unanswered questions in this context. However, it turns out that even if we are only interested in finitely generated modules, there are recent theorems whose proofs use infinitely generated modules in an essential way; for example, Rickard’s (as yet unpublished) classification of the thick subcategories of the stable finitely generated module category for a p-group, and my recent proof [1] of the conjectures formulated in [4] about finitely generated modules with no cohomology.

1. A Vector Space Lemma

The following lemma serves as a replacement for the theory of eigenvalues, and is really the starting point for the recent developments I’m going to discuss.

Lemma 1.1. Let k be an algebraically closed field, and k(t) be a simple transcendental extension of k. Let V be a nonzero (and possibly infinite dimensional) k-vector space, and let f be linear transformations from V to itself. Then for some λ in k(t), the linear map

1 ⊗ f − λ ⊗ Identity : k(t) ⊗_{k} V → k(t) ⊗_{k} V

is not an isomorphism.

In other words, the reason why there need be no eigenvalues, even over an algebraically closed field, is because the field isn’t big enough. Over some extension field, there is always an eigenvalue, if interpreted suitably.

The appearance of transcendental extensions in this lemma gives rise to the relevance of “generic points” for infinite dimensional representations, in a way that never becomes relevant for finite dimensional representations.

The proof of the lemma is very straightforward. We regard V as a k[x]-module in the normal way by letting x act as the linear transformation f. If f − λ.Identity is an isomorphism for all λ ∈ k, then the action of k[x] extends to an action of the field of fractions k(x). Since k(x) is a field, any nonzero k(x)-module has a summand isomorphic to k(x) itself. But multiplication by 1 ⊗ x − t ⊗ 1 is not an isomorphism on k(t) ⊗_{k} k(x).

The form in which the lemma gets used is the following: if f, g : V → W are two maps with the property that after tensoring with k(t), all nontrivial linear combinations of f and g give isomorphisms from k(t) ⊗_{k} V to k(t) ⊗_{k} W, then the vector spaces V and W are both zero. To reduce to the previous form of the lemma, use g to identify V with W.

2. Dade’s Lemma

The way we use the vector space lemma of the last section is via an infinite dimensional version of Dade’s lemma. The original lemma (Dade [6]) says the following. Let k be a field of characteristic p, and let E = ⟨g_{1}, …, g_{r}⟩ be an elementary abelian group of order p^{r}. Let X_{i} be the element g_{i} − 1 of the group algebra kE, so that X_{i}^{p} = 0 and X_{i} is in the Jacobson radical J(kE). Let V_{E}^{r}(k) denote the quotient space J(kE)/J^{2}(kE), and let x_{1}, …, x_{r} be the images of X_{1}, …, X_{r} in V_{E}^{r}(k). It is not hard to show that they form a basis for this quotient space. Let y_{1}, …, y_{r} be the linear functions V_{E}^{r}(k) → k given by y_{i}(x_{j}) = 1 if i = j and zero otherwise. Then regarding V_{E}^{r}(k) as affine space, its coordinate ring is the polynomial ring k[y_{1}, …, y_{r}]. If

α = λ_{1}x_{1} + ⋯ + λ_{r}x_{r} ∈ V_{E}^{r}(k)

is not equal to zero, then we set

u_{α} = 1 + λ_{1}X_{1} + ⋯ + λ_{r}X_{r},

an element of order p in the group algebra kE.

Theorem 2.1 (Dade’s lemma). Suppose that k is algebraically closed. If M is a finitely generated kE-module such that the restriction of M to ⟨u_{α}⟩ is free for each point 0 ≠ α ∈ V_{E}^{r}(k), then M is a free kE-module.

The hypothesis that M is finitely generated is certainly necessary here. There are examples of infinitely generated modules for ℤ/2 × ℤ/2 which are not projective, but whose restriction to each ⟨u_{α}⟩ is free (an example is sketched in Section 6). But in some sense, this is because the field is not big enough, because after enlarging the field, one finds that there are values of α for which the restriction to ⟨u_{α}⟩ is not free. The correct version of Dade’s lemma for modules which are not necessarily finitely generated was formulated in [3]:

Theorem 2.2. Let K be an algebraically closed extension of k of transcendence degreee at least r − 1, and set V_{E}^{r}(K) = J(KE)/J^{2}(KE). If M is aa kE-module such that (K ⊗_{k} M) ↓_{⟨u_{α}⟩} is free for all nonzero α ∈ V_{E}^{r}(K), then M is a free kE-module.

The idea of the proof is to reduce to the rank two case, and then choose two linearly independent elements

η, η′ ∈ H^{1}(E, 𝔽_{p}) ≅ J(𝔽_{p}E)/J^{2}(𝔽_{p}E),

so that the Bocksteins β(η) and β(η′) are algebraically independent elements of H^{2}(E, 𝔽_{p}). They induce maps

f, g : Êxt_{kE}^{0}(k, M) → Êxt_{kE}^{2}(k, M)

with the property that for all λ, μ ∈ K, not both zero,

λf + μg : Êxt_{KE}^{0}(K, K ⊗_{k} M) → Êxt_{KE}^{2}(K, K ⊗_{k} M)

is an isomorphism. It now follows from the vector space lemma discussed earlier, that Êxt_{kE}^{0}(k, M) = 0, which implies that M is free.

3. The rank variety

For M a finitely generated kE-module, with k algebraically closed, Carlson’s definition [5] of the rank variety of M is

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

This is a closed homogeneous subvariety of V_{E}^{r}(k), and Dade’s lemma may be interpreted as saying that M is free if and only if V_{E}^{r}(M) = {0}. In fact, the dimension of V_{E}^{r}(M) determines the polynomial rate of growth of the minimal free resolution of M as a kE-module, which is called the complexity of the module. Thus for example the dimension of V_{E}^{r}(M) is equal to one if and only if the minimal resolution of M is periodic (after the first term, which may be too big because of the free summands of M).

For infinitely generated modules, the theory is somewhat different, because we must extend the field in order for Dade’s lemma to hold. The naïve thing to do is just to look at V_{E}^{r}(K ⊗_{k} M), where K is a “large enough” transcendental field extension of k. In general this is not a closed subset of V_{E}^{r}(K), so what sorts of subsets occur this way? To answer this question, we next discuss the theory of generic points.

4. Generic points

For this section, we suppose that k is algebraically closed, and that K is an extension of k of transcendence degree at least r.

Recall that if V ⊆ V_{E}^{r}(k) is a closed irreducible subvariety, then the set

𝔭 = {f ∈ k[y_{1}, …, y_{r}] | f vanishes on V }

is a prime ideal in k[y_{1}, …, y_{r}], and

k[V] = k[y_{1}, …, y_{r}]/𝔭

is the coordinate ring of V. It is an integral domain, and its field of fractions k(V) is the function field of V. It is generated as an extension field of k by the images ȳ_{1}, …, ȳ_{r} of the elements y_{1}, …, y_{r}. The transcendence degree of k(V) over k is equal to the Krull dimension of k[V], and is by definition the dimension of the variety V.

Since K is an algebraically closed extension field of k of transcendence degree at least as big as that of k(V), it follows that the inclusion of k into K extends (not by any means uniquely) to an embedding of k(V) into K. Let t_{1}, …, t_{r} be the images of ȳ_{1}, …, ȳ_{r} under such an embedding. The generic point of V is defined to be the element

ȳ_{1}x_{1} + ⋯ + ȳ_{r}x_{r} ∈ V_{E}^{r}(k(V)).

Note that this point is well defined, independently of the chosen basis for V_{E}^{r}(k). A point which is of the form

t_{1}x_{1} + ⋯ + t_{r}x_{r} ∈ V_{E}^{r}(K)

for some embedding of k(V) into K as above, is said to be a generic point of V over K.

If t_{1}x_{1} + ⋯ + t_{r}x_{r} is any point in V_{E}^{r}(K), set 𝔭 ⊆ k[y_{1}, …, y_{r}] equal to the ideal consisting of all polynomial relations satisfied by t_{1}, …, t_{r} over k. Since K is a field, this ideal is necessarily prime. Let V be the associated subvariety of V_{E}^{r}(k). Then the point t_{1}x_{1} + … + t_{r}x_{r} is a generic point of V over K. So every point defined over K is generic for some uniquely determined closed irreducible subvariety defined over k.

If we look at a line through the origin in V_{E}^{r}(K), the points in that line may be generic for possibly different subvarieties. However, there is a uniquely determined homogeneous subvariety (i.e., one which is a union of straight lines through the origin) among them. To see this, if t_{1}x_{1}+⋯+t_{r}x_{r} is generic for some inhomogeneous subvariety V, then the dimension of V is less than r, so there is an element λ ∈ K which is algebraically independent of t_{1}, …, t_{r}. Then the point λt_{1}x_{1} +⋯+λt_{r}x_{r} is generic for the homogeneous hull of V, namely the smallest homogeneous subvariety containing V.

Now if M is a kE-module (not necessarily finitely generated), then the question of whether (K ⊗_{k} M)↓_{⟨u_{α}⟩} is free only depends on the line through α in V_{E}^{r}(K), and then only on the closed homogeneous irreducible subvariety of V_{E}^{r}(k) for which it is generic. So we define V_{E}^{r}(M) to be the collection of nonzero closed homogeneous irreducible subvarieties V of V_{E}^{r}(k) with the property that if α is the generic point of V then (k(V) ⊗_{k} M)↓_{⟨u_{α}⟩} is not free. It is easy to see that each of V_{E}^{r}(K ⊗_{k} M) and V_{E}^{r}(M) determines the other.

5. Properties of V_{E}^{r}(M)

The following list of properties of V_{E}^{r}(M) when M is not necessarily finitely generated parallels the list of properties of V_{E}^{r}(M) in the finitely generated case.

(i) V_{E}^{r}(M) = ∅ if and only if M is projective.

(ii) More generally, the dimension of V is at most c for all V ∈ V_{E}^{r}(M) if and only if M may be expressed as a filtered colimit of finitely generated modules of complexity at most c.

(iii) V_{E}^{r}(M ⊕ N) = V_{E}^{r}(M) ∪ V_{E}^{r}(N).

(iv) V_{E}^{r}(M ⊗_{k} N) = V_{E}^{r}(M) ∩ V_{E}^{r}(N).

(v) Every subset of V_{E}^{r}(k) = Proj k[y_{1}, …, y_{r}] is equal to V_{E}^{r}(M) for a suitable module M.

(vi) If M happens to be finitely generated, then V_{E}^{r}(M) is just the collection of all closed homogeneous irreducible subvarieties of V_{E}^{r}(M).

6. An Example

Let k be an algebraically closed field of characteristic two, and let E = ℤ/2×ℤ/2 = ⟨g_{1}, g_{2}⟩. Set X_{1} = g_{1} −1 and X_{2} = g_{2} −1. Let M_{1} be the kE-module with generators m_{1}, m_{2}, … and relations X_{1}m_{1} = 0, and X_{2}m_{i} = X_{1}m_{i+1} for each i ≥ 1. Let M_{2} be the kE-module with generators m′_{1}, m′_{2}, … and relations X_{2}m′_{i} = X_{1}m′_{i+1} for each i ≥ 1. The modules M_{1} and M_{2} are superficially similar, but M_{1} has complexity one, while M_{2} has complexity two. The set V_{E}^{r}(M_{1}) consists of just a single line through the origin, while

V_{E}^{r}(M_{2}) = V_{E}^{r}(k) \ V_{E}^{r}(M_{1}).

Let ϕ : k → M_{2} be the homomorphism taking the basis element of k to the element X_{1}m′_{1}. Then the cokernel of ϕ is isomorphic to M_{1}.

We can modify this example by replacing the actions of X_{1} and X_{2} by aX_{1} + bX_{2} and cX_{1} +dX_{2}, where ( \begin{matrix}a & b \\ c & d \end{matrix} ) is a nonsingular matrix. This has the effect of moving the distinguished line through the origin by the corresponding linear transformation on V_{E}^{r}(k). If the resulting line corresponds to the point (λ : μ) ∈ ℙ^{1}(k), let us write M_{(λ:μ)} for the module obtained from M_{2} in this way. Up to isomorphism, it only depends on the point (λ : μ) ∈ ℙ^{1}(k). Let ϕ_{(λ:μ)} be the corresponding homomorphism from k to M_{(λ:μ)}. Then we may put these together to form a map

k → ⨁_{(λ:μ)∈ℙ^{1}(k)} M_{(λ:μ)}.

Let M be the cokernel of this map. Then M is projective on restriction to ⟨u_{α}⟩ for every nonzero α ∈ V_{E}^{r}(k), but M itself is not projective. In some sense, what we have done is to take affine 2-space, and remove all the lines through the origin. However, what we haven’t done is to remove lines through the origin defined over a larger field. So the generic point of the affine plane is still in the variety. In fact, we have

V_{E}^{r}(M) = {V_{E}^{r}(k)}.

References

[1] D. J. Benson. Cohomology of modules in the principal block of a finite group. Submitted to the New York Journal of Mathematics.

[2] D. J. Benson, J. F. Carlson and J. Rickard. Complexity and varieties for infinitely generated modules. To appear in Math. Proc. Camb. Phil. Soc.

[3] D. J. Benson, J. F. Carlson and J. Rickard. Complexity and varieties for infinitely generated modules, II. Submitted to Math. Proc. Camb. Phil. Soc.

[4] D. J. Benson, J. F. Carlson and G. R. Robinson. On the vanishing of group cohomology. J. Algebra 131 (1990), 40–73.

[5] J. F. Carlson. The varieties and cohomology ring of a module. J. Algebra 85 (1983), 104– 143.

[6] E. C. Dade. Endo-permutation modules over p-groups, II. Annals of Math. 108 (1978), 317– 346.

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

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