COMPLEXITY AND VARIETIES FOR INFINITELY GENERATED MODULES, II.

D. J. BENSON, JON F. CARLSON, AND J. RICKARD

1. Introduction

It has now been almost twenty years since Alperin introduced the idea of the complexity of a finitely generated kG-module, when G is a finite group and k is a field of characteristic p > 0. In proving one of the first major results in the area [1], Alperin and Evens demonstrated the connection of the study of complexity for modules to the group cohomology. That connection eventually led to the categorization of modules according to their associated varieties in the maximal ideal spectrum of the cohomology ring H^{∗}(G, k). In all of the work that has followed, two principles have proved to be extremely important. The first is that the associated variety of a module is directly related to the structure of the module through the rank variety which is defined by the matrix representation of the module. The second major result is the tensor product theorem which says that the variety associated to a tensor product M ⊗_{k} N is the intersection of the varieties associated to the modules M and N. In this paper we generalize these results to infinitely generated kG-modules.

Our investigation of homological properties of infinitely generated kG-modules was started with the study of complexity in [5]. In that paper we defined the complexity and took our first stab at the problem of defining associated support varieties. As we pointed out, there are numerous problems to be encountered. For one thing, cohomology is not finitely generated, and hence the definition of the associated variety as the variety of the annihilator of the cohomology does not work well. Instead, what we define is an

Research of the first author partly supported by EPSRC grant GR/K33903. Research of the first two authors partly supported by grants from the NSF.

2 D. J. BENSON, JON F. CARLSON, AND J. RICKARD

extended notion of the rank variety of a module. The definition is valid when the group is an elementary abelian p-group, and when the module is finitely generated then the new rank variety is essentially the same as the old. The main problem with the definition is that the new rank variety is not really a variety, but only a collection of sets which are closed in the Zariski topology. All indications are that this particular feature is unavoidable. On the other hand the new rank variety does satisfy the tensor product theorem and can be extended to groups which are not elementary abelian.

The next three sections of the paper consist of preliminaries aimed at the proof of the new version of “Dade’s Lemma” and the characterization of the rank variety in terms of generic points. As we saw in [5], it is possible to construct a module whose rank variety in the old sense is simply affine r-space with all of the points removed. At the same time the module in the example is far from projective. Consequently it is necessary to extend the field in order to distinguish the module from a projective module. This is exactly the point: the new rank variety can be given as the old rank variety taken over a large extension K of k. Then Dade’s Lemma says that the module is projective if and only if its variety is trivial. On the other hand a point in the variety over K must be generic for some irreducible subvariety over k. Hence, with some preliminaries we are able to identify the new rank variety with a subset of the collection of all irreducible closed subvarieties of affine r-space.

In Section 5 we begin the development of the cohomological variety by showing how the cohomology can be adapted to give another proof of Dade’s Lemma. As a preliminary to later sections we also discuss a few of the details of the problems that are encountered in the communication between the cohomological and the rank varieties. Section 5 provides a proof of the Tensor Product Theorem for rank varieties. Although it is not directly related to the calculation of the cohomological varieties it is an important tool in the development of some of the properties. In particular in Section 10 it is used in showing that the rank variety is isomorphic (actually only isogenous) to the cohomological variety in the case in which the group is elementary abelian and in showing that the Tensor Product Theorem

COMPLEXITY AND VARIETIES FOR INFINITELY GENERATED MODULES, II. 3

holds for the cohomological variety in general. The latter is accomplished by restriction to the elementary abelian p-subgroups.

The actual definition of the cohomological variety is a two tiered process. First we must recall, in Section 8, the development of the idempotent modules given in [18]. Our summary of the facts about these modules is very brief, but it is here that we really use the cohomology to give the name “cohomological” to the variety. The fact is that the idempotent modules of interest are defined by the colimits of triangles defined by cohomology and enjoy a universal property with respect to the subcategory of finitely generated modules defined by subvarieties of the spectrum of the cohomology ring. The cohomological variety is defined in Section 10 in terms of a tensor product property involving the idempotent modules. The rank varieties of the idempotent modules, given in Section 9, play an important part in the theory. We end the paper by showing the relationship of this cohomological variety with an earlier idea defined in [5].

We should mention that the results of this paper have already seen some significant applications. The third author has used the extended version of Dade’s Lemma in his characterization of the thick subcategories of the stable category of finitely generated kGmodules in the case where G is a finite p-group. The first author has used some of the results to settle most of the questions raised in [6] on modules with vanishing cohomology. In addition, the second author in collaboration with Peng has used relative cohomology to develop idempotent systems of finitely generated modules which represent ideals in the cohomology rings and not just subvarieties of the spectrum [10].

A portion of this paper was written while all three authors were attending the London Mathematical Society Durham Symposium on “Geometry and cohomology in group theory”. We would like to thank the organizers of the meeting and the EPSRC for financial support.

2. Generic points

We begin with a discussion of generic points in this and the next section. The information is in some sense classical (see for example Zariski and Samuel [19], Chapters VI

4 D. J. BENSON, JON F. CARLSON, AND J. RICKARD

and VII for a discussion considerably more general than ours), but it is hard to find the material in the form in which we wish to use it.

Let k be an algebraically closed field of characteristic p, and let E = ⟨g_{1}, …, g_{r}⟩ ≅ (ℤ/p)^{r} be an elementary abelian p-group of rank r. Let V_{E}^{r}(k) denote the affine r-space J(kE)/J^{2}(kE). Then V_{E}^{r}(k) has as coordinate vectors the images x_{i} of the elements X_{i} = g_{i} − 1 ∈ J(kE) (1 ≤ i ≤ r). We write y_{i} for the linear function on V_{E}^{r}(k) which takes the value one on x_{i} and zero on the remaining x_{j} (j ≠ i). Then the coordinate ring of V_{E}^{r}(k) is the polynomial ring

k[V_{E}^{r}(k)] = k[y_{1}, …, y_{r}].

In order to define rank varieties for infinitely generated modules, we must first introduce a field extension of k large enough to define appropriate generic points. We denote by K an algebraically closed field of large transcendence degree over k. We shall see that such a field is big enough so that the variety V_{E}^{r}(K) contains generic points for every closed irreducible subvariety V of V_{E}^{r}(k) of dimension less than or equal to the transcendence degree of K over k.

Let V be a closed irreducible subvariety of V_{E}^{r}(k), with corresponding prime ideal 𝔭 ⊆ k[V_{E}^{r}(k)]. Then the coordinate ring of V is k[V] = k[V_{E}^{r}(k)]/𝔭, and the function field k(V) is the field of fractions of k[V]. Writing ȳ_{i} for the image of y_{i} in k(V), we say that the point defined over the extension field k(V) of k:

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

is “the” generic point for V. It is a canonically defined element of V_{E}^{r}(k(V)), which is independent of the choice of basis x_{1}, …, x_{r}.

If K is algebraically closed and has transcendence degree over k at least dim(V), then the function field k(V) may be embedded as a subfield of K containing k (and in such a way that the embedding is an isomorphism on k). Let t_{1}, …, t_{r} be the images in K of ȳ_{1}, …, ȳ_{r} under such an embedding. Then there is a corresponding generic point

α_{V} = t_{1}x_{1} + ⋯ + t_{r}x_{r}.

COMPLEXITY AND VARIETIES FOR INFINITELY GENERATED MODULES, II. 5

Conversely, given any point in V_{E}^{r}(K), say α = t_{1}x_{1} + ⋯ + t_{r}x_{r}, let 𝔭 be the ideal in k[y_{1}, …, y_{r}] consisting of the polynomial relations satisfied by t_{1}, …, t_{r} over k. Since k is a field, the ideal 𝔭 is prime, and so there is a corresponding closed irreducible subvariety V ⊆ V_{E}^{r}(k) whose coordinate ring is k[V] = k[V_{E}^{r}(k)]/𝔭. Thus there is a uniquely determined homomorphism from k(V) into K with the property that the corresponding extension of scalars V_{E}^{r}(k(V)) ↪ V_{E}^{r}(K) sends the generic point of V_{E}^{r}(k(V)) to α; namely the homomorphism sending each ȳ_{i} to the corresponding t_{i}. In this situation, we say that α is “a” generic point for V.

Proposition 2.1. Each point in V_{E}^{r}(K) is generic for a uniquely determined closed irreducible subvariety of V_{E}^{r}(k). If K is algebraically closed of transcendence degree d over k, then V_{E}^{r}(K) contains generic points for all closed irreducible subvarieties of dimension at most d in V_{E}^{r}(k).

Proof. This may be found in §5^{bis} of Chapter VI of Zariski and Samuel [19].

3. Generic points for projective varieties

As in the last section, we let

k[V_{E}^{r}(k)] = k[y_{1}, …, y_{r}].

Regarding this as a graded ring with each y_{i} in degree one, it is the graded coordinate ring of the projective space V̄_{E}^{r}(k) of dimension r − 1 over k, whose points correspond to the lines through the origin in V_{E}^{r}(k). A closed homogeneous irreducible subvariety V corresponds to a closed irreducible subvariety V̄ of V̄_{E}^{r}(k), and the corresponding prime ideal in k[V_{E}^{r}(k)] is homogeneous.

If V̄ is a closed irreducible subvariety of V̄_{E}^{r}(k), corresponding to a homogeneous prime ideal 𝔭 in k[V_{E}^{r}(k)], then k[V] = k[V_{E}^{r}(k)]/𝔭 is a graded integral domain. Localizing at the multiplicative subset consisting of all nonzero positive degree homogeneous elements, we obtain a ℤ-graded ring whose degree zero elements form a field denoted k(V̄). This is called the function field of V̄. The homogeneous elements ȳ_{i} are not in k(V̄), but rather

6 D. J. BENSON, JON F. CARLSON, AND J. RICKARD

all the ratios ȳ_{i}/ȳ_{j} are in k(V̄) whenever ȳ_{j} is nonzero in k[V]. It follows that there is a well defined line

(ȳ_{1} : ȳ_{2} : ⋯: ȳ_{r}) ∈ V̄_{E}^{r}(k(V̄)).

For example, if ȳ_{j} ≠ 0 then the point (ȳ_{1}/ȳ_{j}, …, 1, …, ȳ_{r}/ȳ_{j}) is in this line. We say that this point in V̄_{E}^{r}(k(V̄)) is “the” generic point for V̄. It is a canonically defined element, independent of the choice of basis x_{1}, …, x_{r}.

If K is an algebraically closed field, having transcendence degree over k at least dim(V̄) = dim V − 1, then the function field k(V̄) may be embedded as a subfield of K containing k. Moreover, it may be embedded in such a way that the embedding is an isomorphism on k. Let t_{1}, …, t_{r} be the images of ȳ_{1}/ȳ_{j}, …, 1, …, ȳ_{r}/ȳ_{j} in K, for some choice of j so that ȳ_{j} is nonzero in k[V]. Then there is a corresponding generic point

(t_{1} : ⋯: t_{r}) ∈ V̄_{E}^{r}(K).

It doesn’t depend on the choice of j, because the ratios of the t_{i}’s don’t depend on this choice.

Conversely, given any point (t_{1} : ⋯: t_{r}) ∈ V̄_{E}^{r}(K), let 𝔭 be the ideal in k[y_{1}, …, y_{r}] generated by the homogeneous polynomial relations between the t_{i}’s over k. This ideal only depends on the ratios of the t_{i}’s.

Proposition 3.1. Provided K has transcendence degree at least r over k, each line through the origin in V_{E}^{r}(K) contains a generic point for a uniquely determined closed homogeneous irreducible subvariety of V_{E}^{r}(k). For any point on such a line, this is the smallest homogeneous subvariety containing the subvariety for which the point is generic.

Proof. See the discussion in §4^{bis} of Chapter VII of Zariski and Samuel [19]. If V is not homogeneous, then there is an element y ∈ K which is transcendental over the subfield of K generated by k and the (defined) ratios of the λ_{i}’s. The point yα = yλ_{1}x_{1}+⋯+yλ_{r}x_{r} is in the same line as α. Given any polynomial satisfied by yλ_{1}, …, yλ_{r} over k, we may separate out the different powers of y to see that each homogeneous part of the polynomial must also be satisfied. So yα is a generic point for the smallest closed homogeneous irreducible subvariety containing V.

Remark 3.2. For V_{E}^{r}(K) to have generic points for all closed irreducible subvarieties of V_{E}^{r}(k), it is necessary and sufficient for K to have transcendence degree at least r over k. For V_{E}^{r}(K) to have lines through the origin corresponding to all of the closed homogeneous irreducible subvarieties of V_{E}^{r}(k), it is only necessary for K to have transcendence degree at least r − 1 over k. The extra transcendence degree simply provides enough generic points on these lines.

4. A Kronecker quiver lemma

The following lemma may be regarded as an infinite dimensional substitute for eigenvalues, see Remark 4.2.

Lemma 4.1. Suppose that V and W are vector spaces over an algebraically closed field k, and that K is a nontrivial extension of k. Suppose that f and g are linear maps from V to W with the property that for every pair of scalars λ and μ in K, not both zero, the linear map λf + μg : K ⊗_{k} V → K ⊗_{k} W is an isomorphism. Then both V and W are the zero vector space.

Proof. Use g to identify V and W. Then f is a k-endomorphism of V with the property that for all μ ∈ K, the endomorphism 1 ⊗ f + μ ⊗ Id of K ⊗_{k} V is invertible. The action of f makes V into a k[t]-module, with t acting as f does. Since f + μ.Id is invertible for all μ ∈ k, and k is algebraically closed, V is a k(t)-module. Now in the ring K ⊗_{k} k(t), any element of the form 1 ⊗ t + μ ⊗ 1 with μ an element of K which is not in k, is not invertible. If V is nonzero then it has k(t) as a summand as a k(t)-module, so that 1 ⊗ f + μ ⊗ Id is not invertible. This contradicts the hypothesis, and so the lemma is proved.

Remark 4.2. Of course, if V and W are finite dimensional, then passing to an extension field K is not necessary. One just uses the theory of eigenvalues to see that some f + μg has a nonzero kernel.

8 D. J. BENSON, JON F. CARLSON, AND J. RICKARD

5. Rank varieties and Dade’s lemma

Let E = ⟨g_{1}, …, g_{r}⟩ be an elementary abelian p-group of rank r, and let K ⊇ k be algebraically closed fields of characteristic p. If M is a kE-module (not necessarily finitely generated), we define V_{E}^{r}(K ⊗_{k} M) as follows. Given a point 0 ≠ α = λ_{1}x_{1} + ⋯ + λ_{r}x_{r} in V_{E}^{r}(K), we set

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

where X_{i} = g_{i} − 1 ∈ kE ⊆ KE. Then u_{α} is a unit of order p in the group algebra KE, so we may ask whether (K ⊗_{k} M)↓_{⟨u_{α}⟩} is free. The set V_{E}^{r}(K ⊗_{k} M) is defined to consist of the origin, together with the set of points α ∈ V_{E}^{r}(K) such that (K ⊗_{k} M) ↓_{⟨u_{α}⟩} is not free.

We call the above defined subgroup ⟨u_{α}⟩ of the group algebra KE a cyclic shifted subgroup of E. More generally, if α_{1}, …, α_{s} are linearly independent elements of V_{E}^{r}(K), we define the subgroup ⟨u_{α_{1}}, …, u_{α_{s}}⟩ of KE to be the shifted subgroup of E corresponding to α_{1}, …, α_{s}.

In the case of a finitely generated module M, Dade’s lemma [13] says that M is projective if and only if V_{E}^{r}(M) = {0}. For an infinitely generated module, this is no longer true. But this is in some sense because V_{E}^{r}(k) doesn’t have enough points.

Lemma 5.1. Suppose that g is an element of order p in the center of a finite p-group G, and that M is a kG-module. Then M is projective if and only if M ↓_{⟨g⟩} is projective and M/(g − 1)M is a projective k(G/⟨g⟩)-module.

Proof. The “only if” part is clear. So suppose that M ↓_{⟨g⟩} is projective and that the quotient M/(g − 1)M is a projective k(G/⟨g⟩)-module. If m ∈ M is a G-fixed point, then there is an element m′ ∈ M such that m = (g − 1)^{p−1}m′ = ∑_{i=0}^{p−1} g^{i}m′. If h ∈ G then (g − 1)^{p−1}(h − 1)m′ = (h − 1)m = 0, and so there is an element m′′ ∈ M with (h−1)m′ = (g−1)m′′. Thus m′+(g−1)M is a G/⟨g⟩-fixed point in M/(g−1)M. So there is an element m′′′ ∈ M such that ∑_{h∈G/⟨g⟩} h(m′′′)−m′ ∈ (g−1)M. Here, the sum runs over a set of coset representatives of ⟨g⟩ in G. Thus we have that ∑_{h∈G} h(m′′′) = ∑_{i=0}^{p−1} g^{i}m′ = m. Hence every fixed point is a trace, and so M is a projective kG-module.

Theorem 5.2 (Dade’s lemma for infinitely generated modules).

Suppose that k is an algebraically closed field, and that K is an algebraically closed field of transcendence degree at least r − 1 over k. Let E be an elementary abelian p-group of rank r. Then a kE-module M is projective if and only if V_{E}^{r}(K ⊗_{k} M) = {0}.

Proof. We offer two proofs of this theorem, both of which rely on Lemma 4.1. The second proof is given in the next section. Our first proof is a modification of the one in Dade [13], in which his eigenvalue argument has been replaced by the use of Lemma 4.1. For the sake of easy reading, we reproduce his argument with the appropriate modifications.

If M is projective, then it is clear that V_{E}^{r}(K ⊗_{k} M) = {0}. So assume that V_{E}^{r}(K ⊗_{k} M) = {0}. We argue by induction on the rank r of E. In case r = 1, the result is obvious, so we begin with the case in which r = 2. We may assume that M has no projective summands. Let M_{1} = J(kE)^{p}M, M_{2} = J(kE)^{p−1}M, and M_{3} be the submodule of M consisting of the elements m such that J(kE)m ⊆ M_{2}. Then M ⊇ M_{3} ⊇ M_{2} ⊇ M_{1}, and J(kE) annihilates M_{3}/M_{2} and M_{2}/M_{1}. Let M̂ = K ⊗_{k} M, M̂_{1} = K ⊗_{k} M_{1} = J(KE)^{p}M̂, M̂_{2} = K ⊗_{k} M_{2} = J(KE)^{p−1}M̂, and M̂_{3} = K ⊗_{k} M_{3} consist of the elements m in M̂ such that J(KE)m ⊆ M̂_{2}. We claim that for any λ and μ in K not both zero, the map ξ induced by x = λx_{1} + μx_{2} from M̂_{3}/M̂_{2} to M̂_{2}/M̂_{1} is an isomorphism.

To see that it is injective, if m + M̂_{2} is in the kernel of ξ, then xm ∈ M̂_{1}. Now xJ(KE)^{p−1} = J(KE)^{p}, so xM̂_{2} = M̂_{1}, and xm = xm′ for some m′ ∈ M̂_{2}. Then x(m − m′) = 0, and since the restriction of M̂ to ⟨1 + x⟩ is free, it follows that m − m′ = x^{p−1}m′′ for some m′′ ∈ M̂. Thus m = m′ + x^{p−1}m′′ ∈ M̂_{2}.

To see that it is surjective, we must prove that xM̂_{3} = M̂_{2}. Choose an element y = λ′x_{1} + μ′x_{2} not in the line containing x. Then J(KE)^{p−1} is K-linearly spanned by y^{p−1} and xJ(KE)^{p−2}, and we have J(KE)^{p−2}M̂ ⊆ M̂_{3}, and so M̂_{2} = y^{p−1}M̂ + xM̂_{3}. So if xM̂_{3} is a proper subspace of M̂_{2}, then there is an element m ∈ M̂ such that y^{p−1}m ∉ xM̂_{3}.

10 D. J. BENSON, JON F. CARLSON, AND J. RICKARD

Since M̂ has no projective summands, we have x^{p−1}y^{p−1}m = 0. Because the restriction of M̂ to ⟨1 + x⟩ is free, it follows that there is an element m′ ∈ M̂ such that y^{p−1}m = xm′. Then xym′ = y^{p}m = 0, and again from the fact that the restriction of M̂ to ⟨1+x⟩ is free, it follows that there is an element m′′ ∈ M̂ such that x^{p−1}m′′ = ym′. So both xm′ and ym′ are in M̂_{2}, which means that m′ ∈ M̂_{3}. But then y^{p−1}m = xm′ ∈ xM̂_{3}, contradicting the choice of m.

Now we may apply Lemma 4.1 to the maps induced by x_{1} and x_{2} from M_{3}/M_{2} to M_{2}/M_{1} to deduce that M_{1} = M_{2} = M_{3}. Thus J(kE)^{p−1}M = J(kE)^{p}M, which implies that J(kE)^{p−1}M = 0. But then the fact that the restriction to ⟨1 + x_{1}⟩ is projective implies that M = 0. This completes the proof in the case r = 2.

If r ≥ 3, and the theorem is true for groups of rank at most r − 1, we argue as follows. Let M̄ = M/x_{r}M, regarded as a module over the subgroup E′ generated by g_{1}, …, g_{r−1}. For each choice of λ_{1}, …, λ_{r−1} ∈ K, not all zero, let K_{0} be the algebraic closure in K of the subfield generated by k together with the ratios between λ_{1}, …, λ_{r−1}. In other words, if 1 ≤ m ≤ r − 1 with λ_{m} ≠ 0, then K_{0} is the subfield generated by λ_{1}/λ_{m}, …, λ_{r−1}/λ_{m}. So K_{0} has transcendence degree at most r − 2 over k, and so K is strictly bigger than K_{0}. Applying the theorem to the rank two subgroup of K_{0}E generated by 1+(λ_{1}/λ_{m})x_{1} +⋯+(λ_{r−1}/λ_{m})x_{r−1} and g_{r}, we see that V_{E′}^{r}(K ⊗_{k} M̄) = {0}. It follows that M̄ is projective as a kE′-module. Since the restriction of M to the subgroup generated by g_{r} is also projective, we may apply Lemma 5.1 to deduce that M is a projective kEmodule. This completes our first proof of the theorem.

At this point we could define the rank variety of a kE-module M to be the subset V_{E}^{r}(K ⊗_{k} M) of V_{E}^{r}(K), where K is any algebraically closed extension of k having sufficiently large transcendence degree. Of course, sufficiently large means that V_{E}^{r}(K) should have generic points for all closed homogeneous irreducible subvarieties of V_{E}^{r}(k) (see Remark 3.2). The obvious defect of this definition is that it would seem to depend on the choice of the extension K. One way of remedying this defect is to replace each line through the origin in V_{E}^{r}(K) by the closed homogeneous irreducible subvariety for which it is generic. To make sense of this definition, we need the following.

Proposition 5.3. Let M be a kE-module and let V ⊆ V_{E}^{r}(k) be a closed homogeneous irreducible subvariety. Suppose that K and K′ are algebraically closed extensions of transcendence degree at least dim V, and that α ∈ V_{E}^{r}(K) and β ∈ V_{E}^{r}(K′) are generic points for V. Then α ∈ V_{E}^{r}(K ⊗_{k} M) if and only if β ∈ V_{E}^{r}(K′ ⊗_{k} M).

Proof. For notation, we refer the reader to the discussion in Section 2. Let X_{i} = g_{i}−1, where E = ⟨g_{1}, …, g_{r}⟩. For each i, let x_{i} be the image of X_{i} in J(kE)/J^{2}(kE), so that the generic point of V in V_{E}^{r}(k(V)) is

γ = ȳ_{1}x_{1} + ⋯ + ȳ_{r}x_{r}.

To avoid confusion, we identify V_{E}^{r}(k(V)) with k(V)^{r} and V_{E}^{r}(K) with K^{r}. In this notation, the generic point of V is γ = (ȳ_{1}, …, ȳ_{r}). Write α = (α_{1}, …, α_{r}) ∈ K^{r}, β = (β_{1}, …, β_{r}) ∈ (K′)^{r}. There exist uniquely determined field homomorphisms σ : k(V) → K and τ : k(V) → K′ such that for all i, σ(ȳ_{i}) = α_{i} and τ(ȳ_{i}) = β_{i}, and such that σ and τ extend the embedding of k into K.

Now a nonzero element α ∈ V_{E}^{r}(K) is in V_{E}^{r}(K ⊗_{k} M) if and only if K ⊗_{k} M is not free as a K⟨u_{α}⟩-module, where u_{α} = 1 + ∑_{i} α_{i} ⊗ X_{i} ∈ K ⊗_{k} kE. Let

u_{γ} = 1 + ∑_{i} ȳ_{i} ⊗ X_{i} ∈ k(V) ⊗_{k} kE.

We note that there are isomorphisms

K ⊗_{k(V)} k(V) ⊗_{k} kE → K ⊗_{k} kE ≅ KE

and

K ⊗_{k(V)} k(V) ⊗_{k} M → K ⊗_{k} M

given by a ⊗ b ⊗ u → a.σ(b) ⊗ u. The former isomorphism sends 1 ⊗ u_{γ} to u_{α}. Clearly k(V) ⊗_{k} M is free as a k(V)⟨u_{γ}⟩-module if and only if K ⊗_{k(V)} k(V) ⊗_{k} M is free as a K⟨1 ⊗ u_{γ}⟩-module. This is because the latter is obtained from the former by extension of scalars. So via the above isomorphism, K ⊗_{k} M is free as a K⟨u_{α}⟩-module if and only if k(V) ⊗_{k} M is free as a k(V)⟨u_{γ}⟩-module. Hence we are done, because we can make the same argument with α replaced by β and σ replaced by τ.

Definition 5.4. Let V_{E}^{r}(k) be the set of all nonzero closed homogeneous irreducible subvarieties of V_{E}^{r}(k). For any kE-module M, let V_{E}^{r}(M) be the set of all V ∈ V_{E}^{r}(k) such that for any algebraically closed extension K of transcendence degree at least dim V and any generic point α ∈ V_{E}^{r}(K) for V, we have α ∈ V_{E}^{r}(K ⊗_{k} M).

It follows from Proposition 3.1 that whenever K is an algebraically closed extension of transcendence degree at least r − 1 over k, then V_{E}^{r}(K ⊗_{k} M) can be recovered from V_{E}^{r}(M).

Remarks 5.5. (i) Recall that a closed homogeneous irreducible set in V_{E}^{r}(k) corresponds to a homogeneous prime ideal in k[y_{1}, …, y_{r}]. So V_{E}^{r}(M) is a subset of the projective scheme Proj k[y_{1}, …, y_{r}].

(ii) If M is finitely generated then

V_{E}^{r}(M) = {V ∈ V_{E}^{r}(k) | V ⊆ V_{E}^{r}(M)}.

Corollary 5.6. A kE-module M is projective if and only if V_{E}^{r}(M) = ∅.

Proof. This follows from Proposition 5.3 and Theorem 5.2.

6. Cohomology, Bocksteins, and a different view of Dade’s lemma

In this section we introduce some cohomological considerations which also play a role in the later sections of the paper. The explanation of the Bockstein and the map on varieties is a bit technical but also unavoidable. The proof of Dade’s Lemma that we obtain from this construction is in some sense a generalization of the cohomological proof in Carlson [7].

Again let E = ⟨g_{1}, …, g_{r}⟩ be an elementary abelian p-group of rank r. Recall that

H^{∗}(E, k) = \{ \begin{matrix}k[ζ_{1}, …, ζ_{r}] ⊗ Λ(η_{1}, …, η_{r}) if p > 2 \\ k[η_{1}, …, η_{r}] if p = 2. \end{matrix}

COMPLEXITY AND VARIETIES FOR INFINITELY GENERATED MODULES, II. 13

Here deg(ζ_{i}) = 2, deg(η_{i}) = 1 and Λ is the exterior algebra generated by η_{1}, …, η_{r}. The generators of the cohomology may be chosen so that res_{E,⟨g_{i}⟩}(η_{j}) = 0 and res_{E,⟨g_{i}⟩}(ζ_{j}) = 0 whenever j ≠ i, and also that res_{E,⟨g_{i}⟩}(η_{i}) and res_{E,⟨g_{i}⟩}(ζ_{i}) are the “canonical” generators in degrees one and two respectively.

In the case that k = 𝔽_{p}, there is a linear cohomology operation β : H^{n}(E, 𝔽_{p}) → H^{n+1}(E, 𝔽_{p}) called the Bockstein map. On degree one elements it is given by β(η_{i}) = ζ_{i}, where ζ_{i} is taken to mean η_{i}^{2} if p = 2. Some problems arise when we attempt to extend β to cohomology with coefficients in a larger field k. First notice that H^{1}(E, k) is naturally isomorphic to Hom(E, k) which we may identify as the k-linear dual of J(kE)/J^{2}(kE) = V_{E}^{r}(k). Likewise the k-linear dual of H^{2}(E, k) ∩ Im(β) can be identified with the maximal ideal spectrum of the polynomial ring k[ζ_{1}, …, ζ_{r}]. The latter is denoted V_{E}(k). Hence the Bockstein β should induce a map of k-linear duals β^{∗} : V_{E}(k) → V_{E}^{r}(k) which is certainly bijective. The problem is to make the map commute with restrictions to cyclic shifted subgroups.

The technicality needed here is the following. Suppose that ζ = f(ζ_{1}, …, ζ_{r}) is a polynomial of degree n in the degree two generators ζ_{1}, …, ζ_{r} (so that ζ has degree 2n). Let u_{α} = 1 + λ_{1}X_{1} + ⋯ + λ_{r}X_{r} generate a cyclic shifted subgroup in kE (see Section 5 for notation). Then

res_{E,⟨u_{α}⟩}(ζ) = f(λ_{1}^{p}, …, λ_{r}^{p})μ,

where μ is the canonical degree two generator of H^{2}(⟨u_{α}⟩, k) (see for example [7]). It follows that in order to make the Bockstein commute with restriction to cyclic shifted subgroups on kE, it is necessary to extend it semilinearly, using the Frobenius map. That is, we set

β(∑_{i} λ_{i}η_{i}) = ∑_{i} λ_{i}^{p}β(η_{i}).

The reason for doing this is that the restriction to a cyclic shifted subgroup can be viewed as an element of the dual space of H^{1}(E, k) or H^{2}(E, k) ∩ Im(β). So the map on

14 D. J. BENSON, JON F. CARLSON, AND J. RICKARD

the dual space is a bijective semilinear homomorphism

β^{∗} : V_{E}(k) → V_{E}^{r}(k)

given by

β^{∗}(α_{1}, …, α_{r}) = (α_{1}^{p}, …, α_{r}^{p}).

As a map of varieties it is an isogeny but not an isomorphism since its inverse is not a polynomial map. Summing up, we have the following.

Proposition 6.1 (Avrunin and Scott [3]).

The Bockstein homomorphism induces a semilinear isogeny β^{∗} : V_{E}(k) → V_{E}^{r}(k) which commutes with restrictions to shifted subgroups of E.

With the above information we can now present our second proof of Dade’s Lemma. This proof is slightly shorter but makes use of more sophisticated ideas.

Second proof of Theorem 5.2. We proceed by induction on the rank r of E. If r = 1, the theorem is obvious. So we shall assume that r > 1 and that the theorem is true for elementary abelian p-groups of rank r − 1. Let K′ be an algebraically closed field with k ⊆ K′ ⊆ K, and such that K′ has transcendence degree r − 2 over k.

Choose two linearly independent elements η and η′ in H^{1}(E, 𝔽_{p}), so that their Bocksteins β(η) and β(η′) may be regarded as algebraically independent elements of H^{2}(E, 𝔽_{p}) ⊆ H^{2}(E, K). The idea will be to show that for all λ and μ in K, not both zero, the maps f and g induced by β(η) and β(η′):

f, g : Êxt_{K′E}^{0}(K′, K′ ⊗_{k} M) → Êxt_{K′E}^{2}(K′, K′ ⊗_{k} M)

(Êxt = Tate Ext) have the property that for all λ and μ in K, not both zero, the map

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

is an isomorphism. (Note that Êxt behaves well under field extensions.) Then we deduce from Lemma 4.1 that Êxt_{K′E}^{0}(K′, K′ ⊗_{k} M) = 0, so that Êxt_{kE}^{0}(k, M) = 0, and hence M is projective.

COMPLEXITY AND VARIETIES FOR INFINITELY GENERATED MODULES, II. 15

To prove that λf + μg is an isomorphism (λ and μ not both zero), we argue as follows. Let E′ be a shifted subgroup of E of rank r − 1 over K, corresponding to the kernel of

λ^{1/p}η + μ^{1/p}η′ ∈ H^{1}(E, K) ≅ Hom(J(KE)/J^{2}(KE), K).

Then the element λβ(η) + μβ(η′) = β(λ^{1/p}η + μ^{1/p}η′) ∈ H^{2}(E, K) ≅ Ext_{KE}^{2}(K, K) is represented by the exact sequence

0 → K → K_{KE′}↑^{KE}→ K_{KE′}↑^{KE}→ K → 0.

We would like to say that tensoring this sequence with M gives us a sequence of the form

0 → K ⊗_{k} M → (K ⊗_{k} M)↓_{KE′}↑^{KE}→ (K ⊗_{k} M)↓_{KE′}↑^{KE}→ K ⊗_{k} M → 0.

For then, by the inductive hypothesis, (K′ ⊗_{k} M) ↓_{K′E′} is projective, and hence so is (K ⊗_{k} M)↓_{KE′}↑^{KE}. This implies that λβ(η) + μβ(η′) induces a stable isomorphism from Ω^{2}(K ⊗_{k} M) to K ⊗_{k} M, so that λf + μg is an isomorphism as required.

It is not quite true that tensoring the above sequence with M gives us a sequence of the desired form, because shifted subgroups do not behave in quite the same way as subgroups in this respect. However, we can argue as follows. The map λf + μg can be defined using Yoneda products instead of cup products. This means that we can change the Hopf algebra structure on KE without altering the effect of this map. So we change the group basis so that E′ is an actual subgroup of E rather than just a shifted subgroup. With this Hopf algebra structure, tensoring the sequence with M has the desired effect. This completes our second proof of the Theorem 5.2.

7. The tensor product theorem

The main problem with proving the tensor product theorem, namely that the rank variety of a tensor product of two modules is the intersection of their rank varieties, is that tensor product does not commute with restriction to cyclic shifted subgroups. To get around this, we begin with an external form of the tensor product theorem and then restrict to a diagonal subgroup to get an internal tensor product theorem.

16 D. J. BENSON, JON F. CARLSON, AND J. RICKARD

For finitely generated modules the tensor product theorem was proved in Carlson [7]. Avrunin and Scott verified the theorem for cohomological varieties when they proved Proposition 6.1. The proof that we give here follows the lines of a second proof in [3].

Let k ⊆ K be an extension of fields of characteristic p. We begin with the remark that for k-vector spaces M and N, there is a natural isomorphism

K ⊗_{k} (M ⊗_{k} N) ≅ (K ⊗_{k} M) ⊗_{K} (K ⊗_{k} N).

Remark 7.1. In contrast to the tensor product, in general K ⊗_{k} Hom_{k}(M, N) can be a proper subspace of Hom_{K}(K ⊗_{k} M, K ⊗_{k} N).

Theorem 7.2. Suppose that E and E′ are elementary abelian p-groups, M is a kEmodule, and N is a kE′-module. Then

V_{E×E′}^{r}(K ⊗_{k} (M ⊗_{k} N)) = V_{E}^{r}(K ⊗_{k} M) × V_{E′}^{r}(K ⊗_{k} N)

as subvarieties of V_{E×E′}^{r}(K) = V_{E}^{r}(K) × V_{E′}^{r}(K).

Proof. For points in either V_{E}^{r}(K) × {0} or {0} × V_{E′}^{r}(K), the statement is clear. If 0 ≠ α ∈ V_{E}^{r}(K) and 0 ≠ β ∈ V_{E′}^{r}(K), then we have elements u_{α} ∈ KE and u_{β} ∈ KE′, so that ⟨u_{α}⟩ × ⟨u_{β}⟩ is a rank two subgroup of K(E × E′) = KE ⊗_{K} KE′. Since restriction to cyclic shifted subgroups does commute with taking external tensor products, we have

(K ⊗_{k} (M ⊗_{k} N))↓_{⟨u_{α}⟩×⟨u_{β}⟩}≅ (K ⊗_{k} M)↓_{⟨u_{α}⟩} ⊗_{K} (K ⊗_{k} N)↓_{⟨u_{β}⟩} .

We need to restrict further to u_{(α,β)} ∈ K(⟨u_{α}⟩ × ⟨u_{β}⟩) = K⟨u_{α}⟩ ⊗_{K} K⟨u_{β}⟩. This generates a shifted subgroup of ⟨u_{α}⟩ × ⟨u_{β}⟩ corresponding to a point in V_{⟨u_{α}⟩×⟨u_{β}⟩} not in the subsets V_{⟨u_{α}⟩} or V_{⟨u_{β}⟩}.

For the group algebra of a cyclic group, every module (including infinite dimensional ones) is a direct sum of indecomposables of dimensions between one and p (see for example Köthe [15] or Auslander [2]). The variety of an external tensor product of such indecomposables is:

(i) the origin {0} if both have dimension p,

(ii) the line V_{⟨u_{α}⟩} or V_{⟨u_{β}⟩} if one has dimension p and the other does not, and

COMPLEXITY AND VARIETIES FOR INFINITELY GENERATED MODULES, II. 17

(iii) the whole of V_{⟨u_{α}⟩×⟨u_{β}⟩} if neither has dimension p (because then the dimension of the tensor product is not divisible by p).

It follows that the restriction of K ⊗_{k} (M ⊗_{k} N) ≅ (K ⊗_{k} M) ⊗_{K} (K ⊗_{k} N) to ⟨u_{(α,β)}⟩ is not projective if and only if both (K ⊗_{k} M)↓_{⟨u_{α}⟩} and (K ⊗_{k} N)↓_{⟨u_{β}⟩} have a summand of dimension less than p; namely if neither of them is projective. Thus (α, β) ∈ V_{E×E′}^{r}(K ⊗_{k} (M ⊗_{k} N)) if and only if both α ∈ V_{E}^{r}(K ⊗_{k} M) and β ∈ V_{E′}^{r}(K ⊗_{k} N).

Proposition 7.3. If E_{1} ≤ E_{2} are elementary abelian p-groups and M is a kE_{2}-module, then regarding V_{E_{1}}^{r}(K) as a subvariety of V_{E_{2}}^{r}(K), we have that

V_{E_{1}}^{r}(K ⊗_{k} M) = V_{E_{1}}^{r}(K) ∩ V_{E_{2}}^{r}(K ⊗_{k} M).

Proof. This is clear from the definitions, since whether a point in V_{E_{1}}^{r}(K) is in the rank variety does not depend on whether it is regarded as a point in V_{E_{1}}^{r}(K) or as a point in V_{E_{2}}^{r}(K).

Theorem 7.4. Let E be an elementary abelian p-group, and let M and N be kE-modules. Then

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

Proof. Let Δ : E → E × E be the diagonal embedding defined by Δ(g) = (g, g). Then the internal tensor product M ⊗_{k} N as a kE-module is equal to the restriction of the external tensor product via the diagonal embedding Δ. So we have

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

This completes the proof of the theorem.

Corollary 7.5. (See Definition 5.4) Let E be an elementary abelian p-group and let M and N be kE-modules. Then V_{E}^{r}(M ⊗_{k} N) = V_{E}^{r}(M) ∩ V_{E}^{r}(N).

In particular, M ⊗_{k} N is projective if and only if V_{E}^{r}(M) ∩ V_{E}^{r}(N) = ∅.

Proof. The first part follows from Theorem 7.4, and the second part then follows from Corollary 5.6.

Proposition 7.6. Let M be a kE-module, and let u, v ∈ kE be units of order p with u − v ∈ J^{2}(kE). Then M ↓_{⟨u⟩} is free if and only if M ↓_{⟨v⟩} is free.

Proof. The proof of this is exactly parallel to the proof of Lemma 6.4 of Carlson [7], but with the reference to Theorem 5.6 of [7] replaced by a reference to the above Theorem 7.4.

Question 7.7. What is the rank variety of M^{∗} = Hom_{k}(M, k), or more generally of Hom_{k}(M, N)? The difficulty with this question is related to Remark 7.1.

8. Idempotent modules

In preparation for a discussion of the cohomological variety of a module in the next section, we summarize the necessary material on idempotent modules in the stable category from Rickard [18].

If C is a thick subcategory of the stable category of finitely generated kG-modules stmod(kG), we denote by C^{⊕} the smallest full subcategory of the stable module category StMod(kG) that contains C and is closed under arbitrary direct sums. Then for each object X in StMod(kG), there is a distinguished triangle

E_{C}(X) → X → F_{C}(X) → Ω^{−1}E_{C}(X)

characterized by the following properties. The object E_{C}(X) lies in C^{⊕}, and

Hom_{StMod(kG)}(Y, F_{C}(X)) = 0

for every Y in C, and hence also for every Y in C^{⊕}. We express this by saying that F_{C}(X) is local with respect to C or with respect to C^{⊕}.

Corresponding to each collection W of closed homogeneous subvarieties V ⊆ V_{G}(k) closed under specialization (i.e., V ∈ W, W ⊆ V implies W ∈ W) and finite unions, there

COMPLEXITY AND VARIETIES FOR INFINITELY GENERATED MODULES, II. 19

is a thick subcategory C_{W} of stmod(kG) consisting of the modules whose variety is an element of W. We write

E(W) → k → F (W) → Ω^{−1}E(W)

for the corresponding triangle of the trivial module. Then the corresponding triangle of any module X is given by

E(W) ⊗ X → X → F (W) ⊗ X → Ω^{−1}E(W) ⊗ X.

The modules E(W) and F (W) are idempotent in the stable category, in the sense that there are isomorphisms E(W) ⊗ E(W) ≅ E(W) ⊕ (projective) and F (W) ⊗ F (W) ≅ F (W) ⊕ (projective).

A couple of properties of these modules will be particularly useful.

Lemma 8.1. Assume that W is as above.

(i) For any finitely generated module M with V_{G}(M) ∈ W we have that M ⊗ E(W) ≅ M ⊕ P for some projective module P.

(ii) E(W), and every object in C_{W} are filtered colimits of finitely generated modules whose varieties are elements of W.

Proof. The first statement is Lemma 5.14 of Rickard [18]. Part (ii) follows from the construction preceding Lemma 5.3 of that paper. Or viewed another way, Lemmas 5.3 and 5.17 of [18] assure us that the modules A_{i} of [18] are filtered colimits of objects in C_{W}, and E(W) is the colimit of the A_{i}’s.

If V is a closed homogeneous subvariety of V_{G}(k), then taking for W the set of all closed homogeneous subvarieties contained in V, we define E(V) = E(W) and F (V) = F (W), and we get a triangle

E(V) → k → F (V) → Ω^{−1}E(V).  (8.1)

Similarly, taking for W′ the set of all closed homogeneous subvarieties of V not containing any irreducible component of V, we define E′(V) = E(W′) and F′(V) = F (W′), and we

20 D. J. BENSON, JON F. CARLSON, AND J. RICKARD

get a triangle

E′(V) → k → F′(V) → Ω^{−1}E′(V).  (8.2)

Finally, we write κ(V) for the module E(V) ⊗ F′(V).

Lemma 8.2. If i : H → G is the inclusion of a subgroup H of G, and W is a collection of closed homogeneous subvarieties of V_{G}(k) closed under specialization, then the triangle

E(W) → k → F (W) → Ω^{−1}E(W)

restricts to

E(i_{∗}^{−1}(W)) → k → F (i_{∗}^{−1}(W)) → Ω^{−1}E(i_{∗}^{−1}(W)).

The module κ(V) restricts to κ(W), where W is the union of the irreducible components of i_{∗}^{−1}(V) whose images under i_{∗} are components of V.

Proof. Let C be the subcategory of stmod(kG) consisting of modules whose variety is in W, and let C′ be the subcategory of stmod(kH) consisting of modules whose variety is in i_{∗}^{−1}W. Then E(W) is in C^{⊕}, and so E(W)↓_{H} is in (C′)^{⊕}. If M is a kH-module in C′, then

Hom_{StMod(kH)}(M, F (W)↓_{H}) ≅ Hom_{StMod(kG)}(M ↑^{G}, F (W)) = 0,

so that F (W) ↓_{H} is local with respect to C′. The result now follows from the characterization of the triangle for i_{∗}^{−1}(W).

The statement about κ(V) follows by examining i_{∗}^{−1}(W) and i_{∗}^{−1}(W′) for the collections W and W′ used to define the modules E(V) and F′(V).

We end this section with some further properties of the idempotent modules which will be useful in our discussion of the cohomological varieties.

Lemma 8.3. Suppose that V and W are closed homogeneous subvarieties of V_{G}(k). (i) There is a canonical stable isomorphism

E(V) ⊗ E(W) → E(V ∩ W).

COMPLEXITY AND VARIETIES FOR INFINITELY GENERATED MODULES, II. 21

(ii) Provided that V does not contain any irreducible component of W, there is a canonical stable isomorphism E(V) ⊗ E′(W) → E(V ∩ W).

(iii) If V ⊆ W then E(V) ⊗ F (W) is projective.

(iv) If V ⊆ W and V does not contain any irreducible component of W then E(V) ⊗ F′(W) is projective.

Proof. Statement (i) is part (a) of Proposition 6.2 of [18]. The proof of statement (ii) is essentially the same. To prove statement (iii), look at the triangle

E(V) ⊗ E(W) → E(V) ⊗ k → E(V) ⊗ F (W) → E(V) ⊗ Ω^{−1}E(W)

and notice that the first map is the canonical stable isomorphism described in (i). The proof of statement (iv) is similar, using (ii).

Proposition 8.4. If V and W have the property that no irreducible component of one is contained in any irreducible component of the other (i.e., the intersection contains no irreducible component of either V or W), then we have the following.

(i) There is a canonical stable isomorphism F′(V ∪ W) → F′(V) ⊗ F′(W).

(ii) The module κ(V) ⊗ E′(W) is projective.

(iii) There is a canonical stable isomorphism E(V) ⊗ F′(V ∪ W) → κ(V).

(iv) There is a canonical stable isomorphism κ(V ∪ W) → κ(V) ⊕ κ(W).

Proof. Statement (i) is proved in the same way as Theorem 7.5 of [18]. To prove statement (ii), we use part (ii) of the lemma to deduce that there is a stable isomorphism

κ(V) ⊗ E′(W) = E(V) ⊗ F′(V) ⊗ E′(W) → E(V ∩ W) ⊗ F′(W),

and part (iv) of the lemma to see that this is projective.

To prove statement (iii), we use statement (i) to see that there is a canonical stable isomorphism

E(V) ⊗ F′(V ∪ W) → E(V) ⊗ F′(V) ⊗ F′(W) = κ(V) ⊗ F′(W).

Now apply statement (ii) to the triangle κ(V) ⊗ E′(W) → κ(V) ⊗ k → κ(V) ⊗ F′(W) → Ω^{−1}κ(V) ⊗ E′(W) to see that there is a canonical stable isomorphism κ(V) → κ(V) ⊗ F′(W).

To prove statement (iv), we consider the triangle

E(V ∩ W) ⊗ F′(V ∪ W) → (E(V) ⊕ E(W)) ⊗ F′(V ∪ W) → E(V ∪ W) ⊗ F′(V ∪ W) → Ω^{−1}E(V ∩ W) ⊗ F′(V ∪ W)

obtained by tensoring the Mayer–Vietoris triangle (Section 8 of [18]) with F′(V ∪ W). The left hand term is projective by part (iv) of the lemma, the middle term is canonically stably isomorphic to κ(V) ⊕ κ(W) by statement (iii), and the third term is κ(V ∪ W).

Theorem 8.5. Let V_{1}, …, V_{t} be the irreducible components of a closed homogeneous subvariety V ⊆ V_{G}(k). Then there is a canonical stable isomorphism between κ(V) and κ(V_{1}) ⊕ ⋯ ⊕ κ(V_{t}).

Proof. This follows by repeated application of part (iv) of the proposition.

9. Rank varieties for idempotent modules

An important and possibly confusing point regarding the last section is that the collections W used to define idempotent modules are collections of closed subvarieties of the cohomological variety V_{G}(k). For this section and the rest of the paper it is necessary to be careful about the distinction between it and the rank variety. For one thing, the rank variety is defined only for elementary abelian p-groups, though we shall see that restrictions to the elementary abelian p-subgroups determine all of the varieties.

Theorem 9.1. Let E be an elementary abelian p-group. Suppose that W is a collection of closed homogeneous subvarieties of V_{E}(k) and that W is closed under specialization and finite unions. Then the rank variety V_{E}^{r}(E(W)) is exactly the set of subvarieties β^{∗}(V) such that V ∈ W and V is irreducible.

In other words, if K is an algebraically closed extension of k of sufficiently large transcendence degree, then a point α ∈ V_{E}^{r}(K) is in V_{E}^{r}(K ⊗_{k} E(W)) if and only if α is a generic point of β^{∗}(V) for some V ∈ W.

COMPLEXITY AND VARIETIES FOR INFINITELY GENERATED MODULES, II. 23

Proof. The equivalence of the two statements is a consequence of Proposition 5.3. Suppose that V ∈ W is irreducible. Let M be a finitely generated module with V_{E}^{r}(M) = β^{∗}(V) (see [4], Corollary 5.9.2). Then V_{E}(M) = V and M ⊗ E(W) ≅ M ⊕ P with P a projective module, by Lemma 8.1. Now by the Tensor Product Theorem (Corollary 7.5), V ∈ V_{E}^{r}(E(W)).

For the converse suppose that V ∉ W. Choose α ∈ V_{G}(k) so that β^{∗}(α) is generic for β^{∗}(V). By Lemma 8.1 (ii), E(W) is a filtered colimit of some system {A_{i}}_{i∈I} with V_{E}(A_{i}) ∈ W and A_{i} finitely generated for all i. Let ⟨u_{α}⟩ be the cyclic shifted subgroup of KE determined by β^{∗}(α). Because V ∉ W we must have that β^{∗}(α) ∉ V_{E}^{r}(K ⊗_{k} A_{i}) = β^{∗}(V_{E}(K ⊗_{k} A_{i})). Hence A_{i} ↓_{⟨u_{α}⟩} is a projective K⟨u_{α}⟩-module by the definition of the rank variety. Because the restriction to ⟨u_{α}⟩ commutes with the colimit, we must have that E(W)↓_{⟨u_{α}⟩} is a free K⟨u_{α}⟩-module, and that β^{∗}(α) ∉ V_{E}^{r}(K ⊗_{k} E(W)). This implies that β^{∗}(V) ∉ V_{E}^{r}(E(W)) as desired.

Corollary 9.2. For E and W as in the Theorem, we have that

V_{E}^{r}(F (W)) = {β^{∗}(V) ∈ V_{E}^{r}(k) | V ∉ W}.

Also, if V ⊆ V_{E}(k) is a closed homogeneous subvariety then

V_{E}^{r}(κ(V)) = {β^{∗}(V_{1}), …, β^{∗}(V_{t})}

where V_{1}, …, V_{t} are the irreducible components of V.

Proof. The first part comes from the fact that E(W) ⊗ F (W) is projective. Hence V_{E}^{r}(E(W)) ∩ V_{E}^{r}(F (W)) = ∅. On the other hand if K is an algebraically closed extension of k of transcendence degree at least the rank of E then there is an exact sequence

0 → K ⊗_{k} E(W) → K ⊕ P → K ⊗_{k} F (W) → 0

where P is a projective module. Hence we must have that

V_{E}^{r}(K ⊕ P) = V_{E}^{r}(K) = V_{E}^{r}(K ⊗_{k} E(W)) ∪ V_{E}^{r}(K ⊗_{k} F (W)).

Now apply Propositions 5.3 and 3.1.

The calculation of the variety of κ(V) is a straightforward application of the Tensor Product Theorem (Corollary 7.5).

10. Cohomological varieties

In [5] we formulated a notion of a cohomological variety for an infinitely generated module. The notion had some defects which were pointed out in an appendix to the paper. Among other things, it did not satisfy a tensor product theorem, and it did not behave well on restriction to subgroups. In this section we introduce a different cohomological variety which repairs some of the defects of the old one. We use the notation V̂_{G}(M) to denote the collection which was referred to as the cohomological variety in [5]. The symbol V_{G}(M) will denote the new variety. We need to recall some basic facts about the complexity of modules.

For M a finitely generated kG-module the complexity of M is the dimension of V_{G}(M). Several equivalent extensions of the notion to infinitely generated modules were given in [5]. One of the most useful of these is that a kG-module M has complexity c if it is a filtered colimit of finitely generated modules of complexity at most c, but not a filtered colimit of finitely generated modules of complexity less than c. A couple of useful facts about complexity are contained in the following.

Lemma 10.1. (i) If M and N are kG-modules then the complexity of M ⊗ N is never greater than the complexity of either M or N.

(ii) Suppose that W is a collection of subvarieties of V_{G}(k) closed under specialization and finite unions. Then the complexity of E(W) is equal to the maximum of dim V for V ∈ W.

Proof. Part (i) follows directly from the definition, though it is also implicit in Proposition 8.2 of [5]. Part (ii) is a consequence of the fact discussed earlier, that E(W) is a filtered colimit of finitely generated modules whose varieties are in W. Hence the complexity of E(W) is at most the maximum value of dim V for V ∈ W. On the other hand, if N is a finitely generated module with V_{G}(N) ∈ W, then E(W) ⊗ N ≅ N ⊕ P

COMPLEXITY AND VARIETIES FOR INFINITELY GENERATED MODULES, II. 25

with P a projective kG-module. Hence the complexity of E(W) is at least dim V_{G}(N) by (i).

Definition 10.2. The cohomological variety V_{G}(M) of a kG-module M is the set of all closed homogeneous irreducible subvarieties V of V_{G}(k) with the property that M ⊗ E(V) has complexity equal to the dimension of V.

Notice here that V_{G}(k) is the set of all closed homogeneous irreducible subvarieties of V_{G}(k) by Lemma 10.1 (ii). Hence as a set, V_{G}(k) is the projective prime ideal spectrum Proj H^{∗}(G, k). For any kG-module M, V_{G}(M) is a subset of V_{G}(k). It can be checked that if M is finitely generated then V_{G}(M) is the set of all closed homogeneous irreducible subvarieties of V_{G}(M).

Lemma 10.3. Suppose that M is a kG-module and that V is a closed homogeneous irreducible subvariety of V_{G}(k). Then V ∈ V_{G}(M) if and only if κ(V) ⊗ M is not projective.

Proof. From the triangle (8.2) we see that if M ⊗ κ(V) = M ⊗ E(V) ⊗ F′(V) is projective then M ⊗ E(V) ⊗ E′(V) ≅ M ⊗ E(V) ⊕ (projective). But the complexity of M ⊗ E(V) ⊗ E′(V) is at most equal to the complexity of E′(V) which is dim V − 1. On the other hand if the complexity of M ⊗ E(V) is less than the dimension of V then M ⊗ E(V) is a filtered colimit of finitely generated modules whose varieties are proper closed subsets of V (tensor together filtered systems of modules of complexity at most dim V − 1 for M ⊗ E(V) and of modules whose variety is contained in V for E(V) to obtain a filtered system satisfying both conditions for M ⊗ E(V)). Any such module must have the property that its tensor product with F′(V) is projective. So M ⊗ κ(V) is a filtered colimit of projective modules and hence is projective.

Lemma 10.4. Let V be a closed homogeneous irreducible subvariety of V_{G}(k). Then the varieties of κ(V) and E(V) are given by V_{G}(κ(V)) = {V } and V_{G}(E(V)) = {W ∈ V_{G}(k) | W ⊆ V }.

Proof. The proof of both statements relies on Proposition 6.2 of Rickard [18], which says that E(V) ⊗ E(W) ≅ E(V ∩ W) ⊕ (projective). So κ(V) ⊗ κ(W) ≅ E(V ∩ W) ⊗

26 D. J. BENSON, JON F. CARLSON, AND J. RICKARD

F′(V) ⊗ F′(W) ⊕ (projective), which is projective if V ∩ W is a proper subvariety of either V or W. Since both V and W are irreducible, this says that W ∈ V_{G}(κ(V)) if and only if V = W. The proof of the second statement is similar.

Theorem 10.5. Suppose that E is an elementary abelian p-group and M is a kE-module. Then β^{∗}(V_{E}(M)) = V_{E}^{r}(M). Here, β^{∗}(V_{E}(M)) means the set {β^{∗}(V) | V ∈ V_{E}(M)}.

Proof. Notice that V ∈ V_{E}(M) if and only if κ(V) ⊗ M is not projective. But κ(V) ⊗ M is not projective if and only if V_{E}^{r}(κ(V)) ∩ V_{E}^{r}(M) ≠ ∅ by the Tensor Product Theorem (Corollary 7.5). Hence by Corollary 9.2 the last condition holds if and only if β^{∗}(V) ∈ V_{E}^{r}(M).

Let EA(G) denote the set of elementary abelian p-subgroups of G. For E ∈ EA(G) we have a restriction map res_{G,E} : H^{∗}(G, k) → H^{∗}(E, k), and a corresponding map res_{G,E}^{∗} : V_{E}(k) → V_{G}(k) on projective prime ideal spectra. One of the implications of Theorem 6.2 of Quillen [16] is that

V_{G}(k) = ⋃_{E∈EA(G)} res_{G,E}^{∗}(V_{E}(k)).

Our next task is to extend this theorem to modules. To do so we should first recall Chouinard’s Theorem [12], which says that a kG-module is projective if and only if its restriction to every elementary abelian p-subgroup of G is projective.

Theorem 10.6. Suppose that M is a kG-module. Then

V_{G}(M) = ⋃_{E∈EA(G)} res_{G,E}^{∗}(V_{E}(M ↓_{E})).

Proof. First suppose that V ∈ V_{G}(M). Then κ(V) ⊗ M is not projective, so by Chouinard’s Theorem, for some E ∈ EA(G), (κ(V) ⊗ M) ↓_{E} is not projective. Now by Lemma 8.2, κ(V) ↓_{E}≅ κ(W), where W = ⋃_{i=1}^{m} W_{i} is the union of the irreducible subvarieties W_{i} ∈ V_{E}(k) such that res_{G,E}^{∗}(W_{i}) = V. Now by Theorem 8.5 we know that κ(W) is stably isomorphic to κ(W_{1}) ⊕ ⋯ ⊕ κ(W_{m}). So for some W_{i}, (κ(W_{i}) ⊗ M) ↓_{E} is not projective and W_{i} ∈ V_{E}(M ↓_{E}). Hence V = res_{G,E}^{∗}(W_{i}) ∈ res_{G,E}^{∗}(V_{E}(M ↓_{E})).

COMPLEXITY AND VARIETIES FOR INFINITELY GENERATED MODULES, II. 27

Suppose conversely that V = res_{G,E}^{∗}(W) for some elementary abelian p-subgroup E and some W ∈ V_{E}(M ↓_{E}). Then (κ(W) ⊗ M)↓_{E} is not projective and is a direct summand of (κ(V) ⊗ M)↓_{E} as in the above argument. Hence V ∈ V_{G}(M) as desired.

We notice at this point that the theorem is not as strong as we would like. It says only that if V ∈ V_{G}(M) then there exists an elementary abelian p-subgroup E ∈ EA(G) such that V ∈ res_{G,E}^{∗}(V_{E}(M ↓_{E})). It does not tell us what the fiber of the restriction homomorphism res_{G,E}^{∗} is on the variety of M. For this we need the following. The next theorem is the analogue for infinitely generated modules of the Avrunin–Scott stratification theorem [3] for finitely generated modules.

Theorem 10.7. Suppose that V ∈ V_{G}(M) and that V = res_{G,E}^{∗}(W) for some E ∈ EA(G) and W ∈ V_{E}(k). Then W ∈ V_{E}(M ↓_{E}).

Proof. From the previous theorem, we get that there is an elementary abelian psubgroup E′ such that V = res_{G,E′}^{∗}(W′) for W′ ∈ V_{E′}(M ↓_{E′}). Now let E′′ ⊆ E′ be a minimal subgroup of E′ such that W′ is supported on E′′. That is, W′ = res_{E′,E′′}^{∗}(W′′) for some W′′ ∈ V_{E′′}(k), but W′ is not in the image of res^{∗} from any proper subgroup of E′′. Let K be an algebraically closed extension of large transcendence degree over k and let α ∈ V_{E′′}^{r}(K) be a generic point for β^{∗}(W′′). Then res_{E′,E′′}^{∗}(α) ∈ V_{E′}^{r}(K) is a generic point for β^{∗}(W′).

Let j : E′′ → E′ be the inclusion. Let u_{α} generate a cyclic shifted subgroup of KE′′ corresponding to α. Then j(u_{α}) generates a cyclic shifted subgroup of KE′ corresponding to res_{E′,E′′}^{∗}(α). In coordinates, and in the notation of Section 5, suppose that E′ = ⟨g_{1}, …, g_{r}⟩, E′′ = ⟨g_{1}, …, g_{s}⟩ for s ≤ r, and set X_{i} = g_{i} − 1. Then u_{α} = 1 + ∑_{i=1}^{s} λ_{i}X_{i} and j(u_{α}) = 1 + ∑_{i=1}^{r} λ_{i}X_{i}, where λ_{i} = 0 if s < i ≤ r. Since α ∈ V_{E′}^{r}(K ⊗_{k} M ↓_{E′})), the module (M ↓_{E′}) ↓_{⟨j(u_{α})⟩}= (M ↓_{E′′}) ↓_{⟨u_{α}⟩} is not projective. Hence β^{∗}(W′′) ∈ V_{E′′}^{r}(M ↓_{E′′}) and W′′ ∈ V_{E′′}^{r}(M ↓_{E′′}).

Now look at the stratification of the variety V_{G}(k) as given by Theorem 10.2 of Quillen [16] (see also Quillen [17]). The point is that there must exist g ∈ G such that gE′′g^{−1} ⊆ E and W = res_{E,gE′′g^{−1}}^{∗}(c_{g}(W′′)), where c_{g} denotes the action of conjugation by g on varieties.

Because M is a kG-module, c_{g}(W′′) ∈ V_{gE′′g^{−1}}(M ↓_{gE′′g^{−1}}) and by the previous theorem

W ∈ V_{E}(M ↓_{E}).

Theorem 10.8 (Tensor Product Theorem). Suppose that M and N are kG-modules. Then

V_{G}(M ⊗ N) = V_{G}(M) ∩ V_{G}(N).

Proof. First notice that the theorem is true if G is elementary abelian by Corollary 7.5

and Theorem 10.5.

Let V ∈ V_{G}(M ⊗ N). Then κ(V) ⊗ M ⊗ N is not projective, and by associativity of

tensor product κ(V) ⊗ M and κ(V) ⊗ N are also not projective. So V ∈ V_{G}(M) ∩ V_{G}(N).

Conversely, suppose that V ∈ V_{G}(M) ∩ V_{G}(N). Let E ∈ EA(G) be a subgroup such

that V = res_{G,E}^{∗}(W) for some W ∈ V_{E}(k). Then W is in V_{E}(M ↓_{E}) and V_{E}(N ↓_{E}) by

Theorem 10.7. Because the theorem is true for E we have that W ∈ V_{E}((M ⊗ N) ↓_{E}).

Therefore, by Theorem 10.6 we are finished.

References

[1] J. L. Alperin and L. Evens. Representations, resolutions, and Quillen’s dimension theorem. J. Pure Appl. Algebra 22 (1981), 1–9.

[2] M. Auslander. Representation theory of artin algebras I. Commun. in Alg. 1 (1974), 177–268.

[3] G. S. Avrunin and L. L. Scott. Quillen stratification for modules. Invent. Math. 66 (1982), 277–286.

[4] D. J. Benson. Representations and cohomology II: Cohomology of groups and modules. Cambridge studies in advanced mathematics, vol. 31, CUP, 1991.

[5] D. J. Benson, J. F. Carlson and J. Rickard. Complexity and varieties for infinitely generated modules. Math. Proc. Camb. Phil. Soc. 118 (1995), 223–243.

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

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

[8] J. F. Carlson. The decomposition of the trivial module in the complexity quotient category. J. Pure & Applied Algebra (to appear).

[9] J. F. Carlson, P. W. Donovan and W. W. Wheeler. Complexity and quotient categories for group algebras. J. Pure & Applied Algebra 93 (1994), 147–167.

[10] J. F. Carlson and C. Peng. Relative projectivity and ideals in cohomology rings. Preprint, 1995.

[11] J. F. Carlson and W. W. Wheeler. Varieties and localizations of module categories. J. Pure & Applied Algebra (to appear).

[12] L. Chouinard. Projectivity and relative projectivity over group rings. J. Pure & Applied Algebra 7 (1976), 287–302.

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

[14] L. Evens. The cohomology of groups. Oxford Science Publications, OUP, 1991.

[15] G. Köthe. Verallgemeinerte Abelsche Gruppen mit hyperkomplexen Operatorenring. Mat Z. 39 (1935), 31–44.

[16] D. G. Quillen. The spectrum of an equivariant cohomology ring, I, II. Ann. of Math. 94 (1971), 549–572, 573–602.

[17] D. G. Quillen. A cohomological criterion for p-nilpotence. J. Pure & Applied Algebra 1 (1971), 361–372.

[18] J. Rickard. Idempotent modules in the stable category. Preprint, 1994.

[19] O. Zariski and P. Samuel. Commutative Algebra, II. Graduate Texts in Mathematics, Vol. 29, Springer-Verlag, Berlin/New York, 1975.

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

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

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

   E-mail address: jfc@sloth.math.uga.edu

School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, England.

   E-mail address: J.Rickard@bristol.ac.uk