INFINITE DIMENSIONAL MODULES FOR FINITE GROUPS

D. J. BENSON

Abstract. In the first lecture, I give an introduction to some of the questions currently under investigation in the theory of finite dimensional modular representations of finite groups. In the second lecture, I give some idea of how the theory of infinite dimensional modules has recently had an impact on our understanding of the category of finite dimensional modules. In the third lecture, I present recent progress on understanding phantom maps. These may be viewed as the aspects of infinite dimensional module theory which are invisible from the point of view of the finite dimensional modules.

These notes form a reasonably faithful account of the three lectures I gave at the Euroconference on Infinite Length Modules in Bielefeld in September 1998. I hope they form a useful introduction to the role played by infinite dimensional modules in the representation theory of finite groups. I would like to thank Claus Ringel for his enthusiastic efforts in bringing this conference together and making it so worthwhile.

1. Blocks and Cohomology

This first lecture is intended as a gentle introduction to modular representation theory, with emphasis on describing some of the problems which have motivated recent developments in the subject. In recent years, progress has been made on a number of these problems, and the second lecture will describe some aspects of recent progress which have depended on developments in the theory of infinite dimensional modules. Rickard’s idempotent modules and functors have played a central role in these developments.

Let k be a field and G be a finite group. If the characteristic of k is zero or a prime not dividing the group order, then the representation theory is fairly well understood:

• Every kG-module can be written as a direct sum of simple (or irreducible) modules, so that simple and indecomposable are equivalent.

• Every kG-module is projective.

• H^{n}(G, M) = 0 for n > 0 for every kG-module M.

• The category mod(kG) of finitely generated kG-modules is easy to describe. Everything is determined by character theory: If k has characteristic zero, the character of a finitely generated kG-module M is the function on G sending the element g to the trace Tr(g, M). This is constant

Date: December 1998.

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

2 D. J. BENSON

on conjugacy classes, and modules are determined by their characters. If k has characteristic p, it is necessary to lift eigenvalues to characteristic zero before adding, to get a “Brauer trace”. If p does not divide the group order, then modules are again determined by their Brauer characters.

If p does divide the group order, the Brauer character exactly determines the set of composition factors of a finitely generated kG-module, but it does not indicate how these composition factors extend each other. The phrase “modular representation” refers to a representation in characteristic p with p dividing the group order.

As an example, let G be the group ℤ/2 × ℤ/2, of order four, generated by two commuting elements g_{1} and g_{2} of order two. Let k be a field of characteristic two. Then the representations

g_{1} ↦ (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}) g_{2} ↦ (\begin{matrix}1 & λ \\ 0 & 1 \end{matrix})

are indecomposable but not simple or projective. Different values of λ give pairwise nonisomorphic modules, so that if k is an infinite field then we get an infinite collection of isomorphism classes of two dimensional indecomposable modules this way.

There are also infinite dimensional indecomposable modules, and almost every conceivable pathology occurs in the infinite dimensional representation theory. For example, Krull–Schmidt fails for finite direct sums of indecomposables, and even cancellation fails. There are modules with no indecomposable direct summands, and there exists a module M which is isomorphic to M ⊕ M ⊕ M but not to M ⊕ M. Examples like this occur whenever there are infinitely many isomorphism classes of indecomposable finitely generated kG-modules, and that happens precisely when the Sylow p-subgroups of G are noncyclic, where p is the characteristic. This is related to the subject of Paul Eklof’s article in these proceedings.

Blocks

Decompose the group algebra kG as a Cartesian product of indecomposable two-sided ideals:

kG = B_{0} × B_{1} × ⋯ × B_{s}.

This corresponds to a decomposition of the identity

1 = e_{0} + e_{1} + ⋯ + e_{s}

into a sum of primitive orthogonal central idempotents. If M is any kG-module, we have a direct sum decomposition

M = e_{0}M ⊕ e_{1}M ⊕ ⋯ ⊕ e_{s}M.

If M is indecomposable, then there exists a unique index j such that M = e_{j}M and e_{i}M = 0 for i ≠ j. We then say that M is “in the block B_{j}.” We label so that the trivial module k (i.e., the

INFINITE DIMENSIONAL MODULES FOR FINITE GROUPS 3

field k with each group element acting as the identity) is in B_{0}, and we call B_{0} the “principal block.”

This way, we really regard a block as a receptacle for primitive central idempotents, two sided ideal direct summands of the group algebra, indecomposable modules, and so on. Another way to view it is that the category mod(kG), whose objects are the finitely generated kG-modules and whose morphisms are the module homomorphisms, decomposes as a sum of categories, one for each block.

Every block has associated to it a defect group D, which is a subgroup of G, minimal such that every module in B is a direct summand of a module induced from D. The defect groups of a block form a single conjugacy class of p-subgroups of G. For example, for B_{0}, the defect groups are the Sylow p-subgroups. The defect groups of a block B are trivial if and only if B is isomorphic to a matrix algebra over a division ring with k in its center. This is the case which most closely corresponds to the characteristic zero situation.

The first main theorem of Brauer says that there is a natural one-one correspondence between blocks of G with defect group D and blocks of N_{G}(D) with defect group D. This is called the Brauer correspondence.

Cohomology

We view Ext_{kG}^{n}(M, N) in terms of n-fold extensions, which are exact sequences of the form

0 → N → M_{1} → … → M_{n} → M → 0.

We regard two such exact sequences as equivalent if there is a diagram of the form

[figure: 0; N; M_{1}; ⋯; M_{n}; M; 0; 0; N; M′_{1}; ⋯; M′_{n}; M; 0]

This is not quite an equivalence relation, but we complete it to one, and the equivalence classes are the elements of Ext_{kG}^{n}(M, N). The zero element is represented by such an exact sequence which is a splice of split short exact sequences. We denote Ext_{kG}^{n}(k, M) by H^{n}(G, M).

There is a natural map from H^{∗}(G, k) to Ext_{kG}^{∗}(M, M) given by tensoring long exact sequences with M. Usually, when we talk of tensor products of modules, we are referring to the tensor product over the field, with the diagonal action of the group. This natural map is a ring homomorphism, and lands in the graded center of Ext_{kG}^{∗}(M, M). Via this map, we regard Ext_{kG}^{∗}(M, M) as a module over H^{∗}(G, k). Then its annihilator as a module is the same as the annihilator of the identity element, namely the kernel of this ring homomorphism.

Another way of viewing cohomology is as follows. We denote by Ω(M) the kernel of a surjection from a projective module to M, and Ω^{−1}(M) denotes the cokernel of an injective

4 D. J. BENSON

map from M to an injective module. These operations are not quite well defined in general, but Schanuel’s lemma shows that Ω is well defined up to adding and removing projective summands, and Ω^{−1} is well defined up to adding and removing injective summands. In the case we’re interested in, namely kG-modules, projective modules are the same as injective modules, and the operations Ω and Ω^{−1} are inverse, to the extent that they are well defined. With these definitions, we have

Ext_{kG}^{n}(M, N) ≅ \underline{Hom}_{kG}(Ω^{n}M, N) ≅ \underline{Hom}_{kG}(M, Ω^{−n}N)

where \underline{Hom} denotes homomorphisms modulo projective homomorphisms. A projective homomorphism is one which factors through some projective module.

If M and N are in different blocks, then Ext_{kG}^{n}(M, N) = 0, because we can decompose any extension of M by N, using block idempotents, as a direct sum of an extension of M by the zero module and an extension of the zero module by N. This can also be seen using the \underline{Hom} description given above. In particular, if M is not in the principal block B_{0} then H^{∗}(G, M) = 0.

Since kG is a symmetric algebra, simple modules M and N are in the same block if and only if for some n ≥ 0 there exists a chain of simple modules M_{1}, …, M_{n} such that Ext_{kG}^{1}(M, M_{1}) ≠ 0, Ext_{kG}^{1}(M_{1}, M_{2}) ≠ 0, …, Ext_{kG}^{1}(M_{n}, N) ≠ 0.

Some Problems

The following list of problems includes some which provide major impetus for current research, as well as some which are intended to motivate the discussion in the next lecture. In particular, I have included some which have recently been solved using the theory of infinite dimensional modules.

1. If S is a simple module in B_{0}, is H^{n}(G, S) necessarily nonzero for some n ≥ 0? Open

References: Linnell [36], Linnell and Stammbach [37, 38].

2. Under what circumstances is it true that for all finitely generated nonprojective modules M in B_{0} we have H^{n}(G, M) ≠ 0 for some n > 0? Solved

References: Benson, Carlson and Robinson [11], Benson [5].

3. Does there exist an integer n > 0 such that if G is a finite group, k is a field and M is a faithful absolutely irreducible kG-module then dim_{k} H^{1}(G, M) ≤ n? Can we take n = 2? Open

References: Guralnick [29, 30].

4. Does there exist a value of n > 0 such that if H^{i}(G, ℤ) = 0 for i ≤ n then G is the trivial group? Open

References: Loday originally conjectured that n can be taken to be 4. However, Milgram (1993)

INFINITE DIMENSIONAL MODULES FOR FINITE GROUPS 5

(see [1]) showed that H^{i}(M_{23}, ℤ) = 0 for i ≤ 5.

5. How do you tell whether a kG-module is projective? The reason why this is important is that a lot of effort goes into constructing projective resolutions for modules. For example, the construction of multiply periodic resolutions in [7] depends crucially on tests for projectivity. References: Much is known. Chouinard’s theorem [21], Dade’s lemma in the finitely generated [23] and infinitely generated [9] cases, varieties for modules [4, 19], etc.

6. If M is a finitely generated indecomposable periodic kG-module (i.e., there is a periodic projective resolution of M) does there exist a projective module P such that M ⊕ P is isomorphic to a module induced from a suitable p-local subgroup? Solved [A p-local subgroup is defined to be a normalizer of a non-trivial p-subgroup of the group] What about nonperiodic modules?

References: For the periodic case, see [5]. For the nonperiodic case, Carlson [20], Section 13, is probably about all that can be said.

7. Describe the thick subcategories of the stable category of finitely generated kG-modules. In other words, find all collections of modules closed under the following operations:

(i) finite direct sums and summands, (ii) Ω and Ω^{−1}, and (iii) extensions. Solved for p-groups but not in general

References: Benson, Carlson and Rickard [10].

8. (Broué’s Conjecture): If the defect groups D of a block B are abelian, then the derived category of bounded complexes of finitely generated modules in B is equivalent to the derived category for the block b of N_{G}(D) which corresponds to B under the Brauer correspondence:

D^{b}(mod(B)) ≃ D^{b}(mod(b)).

Solved for D cyclic (Rickard), and a few other cases. Open in general

References: Broué [15, 16, 17], Külshammer [34], Linckelmann [35], Rickard [42, 43]. [Jeremy Rickard’s second talk, in these proceedings, contains an account of some new progress on this problem.]

Two blocks B_{1} and B_{2} are said to be Morita equivalent if their categories of finitely generated modules are equivalent:

mod(B_{1}) ≃ mod(B_{2}).

6 D. J. BENSON

9. (Donovan’s conjecture) Given a field k of characteristic p and a p-group D, do there only exist a finite number of Morita equivalence classes of blocks of group algebras with defect group D? Known for D cyclic; known for G symmetric groups (Scopes); Open in general

References: Alperin [2], Scopes [48], Kessar [32].

10. (Alperin’s conjecture) Is the number of simple kG-modules equal to the sum over the conjugacy classes of p-subgroups Q of G, of the number of projective simple k(N_{G}(Q)/Q)- modules? Much studied, but still Open (There is also a block by block version of Alperin’s conjecture)

References: Alperin [3], Dade [24, 25], Knörr and Robinson [33], Külshammer [34], Robinson [45, 46], Robinson and Staszewski [47], Thévenaz [49, 50].

Rickard’s Idempotent Modules

In the next lecture, we shall describe how some recent progress on some of these problems has depended on Rickard’s infinite dimensional idempotent modules [44]. In preparation for this, we end this first lecture with a description of the easiest case of these idempotent modules,

Let

ζ ∈ H^{n}(G, k) ≅ Ext_{kG}^{n}(k, k)
≅ \underline{Hom}_{kG}(Ω^{n}k, k)
≅ \underline{Hom}_{kG}(k, Ω^{−n}k)
≅ \underline{Hom}_{kG}(Ω^{−n}k, Ω^{−2n}k)
≅ …

Then we can associate to ζ a filtered system of modules

k ⟶^{ζ̂} Ω^{−n}k ⟶^{ζ̂} Ω^{−2n}k → …

by choosing representative maps, and we define F (ζ) to be the colimit of this sequence. There is a choice involved, but different choices give colimits which are isomorphic up to adding and removing projective summands.

The module F (ζ) constructed in this manner comes with an obvious map k → F (ζ). We can make this surjective by adding a suitable projective module to k, and then the kernel is denoted E(ζ). These modules E(ζ) and F (ζ) are the Rickard idempotent modules associated

INFINITE DIMENSIONAL MODULES FOR FINITE GROUPS 7

to the element ζ in cohomology. Some of their properties are as follows.

E(ζ) ⊗ E(ζ) ≅ E(ζ) ⊕ (Projective)
F (ζ) ⊗ F (ζ) ≅ F (ζ) ⊕ (Projective)
E(ζ) ⊗ F (ζ) is projective

For a finitely generated kG-module M, the following are equivalent:

(i) Some power of ζ annihilates Ext_{kG}^{∗}(M, M). (ii) M ⊗ F (ζ) is projective (iii) M ⊗ E(ζ) ≅ M ⊕ (Projective)

To give an explicit example, let G = ℤ/2 × ℤ/2 with commuting generators g_{1} and g_{2} of order two, and k be a field of characteristic two. Then

H^{∗}(G, k) ≅ k[x, y],

a polynomial ring in two variables x and y in degree one. We represent Ω^{−1}k with the following diagram:

[figure: ∘; y ∘; ∘ x]

The vertices in this diagram represent basis elements as a k-vector space. Multiplication by g_{1} − 1 acts on these basis elements by going down and to the left along a diagonal line, or by zero if there is no such diagonal line. Multiplication by g_{2} − 1 acts similarly, but down and to the right.

The vertex marked x represents the image of the element

x ∈ H^{1}(G, k) ≅ Hom_{kG}(k, Ω^{−1}k),

and similarly for y.

With this notation, Ω^{−2}(k) is represented by the diagram

[figure: ∘; ∘; y^{2} ∘; ∘ xy; ∘ x^{2}]

The map represented by x from Ω^{−1}k to Ω^{−2}k embeds the first of the above diagrams as the right hand part of the second. Continuing this way and taking the colimit, we see that F (x) is the infinite dimensional module represented by the following diagram:

… [figure: ∘; ∘; ∘; ∘; ∘; ∘]

The map from k to F (x) has the rightmost vertex as image. The module E(x) is similar:

… [figure: ∘; ∘; ∘; ∘; ∘]

and the map from E(x) to k sends all except the rightmost vertex to zero.

The endomorphism rings of these modules are quite interesting:

\underline{End}_{kG}(F (x)) ≅ k[t] \underline{End}_{kG}(E(x)) ≅ k[[t]].

2. Varieties and the Nucleus

In this section, we introduce the machinery of varieties for modules, and indicate the relevance of Rickard’s idempotent modules to various questions about finite dimensional modules.

Varieties for Finitely Generated Modules

We denote by V_{G} the maximal ideal spectrum of the cohomology ring H^{∗}(G, k). This is not quite a commutative ring, but rather satisfies the graded commutativity relations xy = (−1)^{deg(x) deg(y)}yx. If p = 2 then it is strictly commutative. If p is odd then elements of odd degree square to zero, and are in the the nil radical. Modulo the ideal generated by the odd degree elements, the ring is commutative.

By a theorem of Evens [26], H^{∗}(G, k) is finitely generated as a k-algebra, and so by the above comments, its maximal ideal spectrum may be viewed as an affine variety over k. A choice of s generators for H^{∗}(G, k) (modulo its nil radical) gives an explicit embedding as a homogeneous subvariety of s-dimensional affine space over k, 𝔸^{s}(k).

As an example, if G ≅ (ℤ/p)^{r} is an elementary abelian p-group of rank r and k has characteristic p, then the cohomology ring is given as follows. If p = 2, it is a polynomial ring in r degree one generators. If p is odd, it is a tensor product of a polynomial ring on r degree two generators and an exterior algebra on r degree one independent generators. In the latter case, the degree one generators square to zero, and generate the nil radical. Modulo these, the ring is polynomial on the r degree two generators. So in either case, the variety V_{G} is isomorphic to 𝔸^{r}(k).

INFINITE DIMENSIONAL MODULES FOR FINITE GROUPS 9

An explicit description of V_{G} for a general finite group in terms of its elementary abelian p-subgroups was given by Quillen [40, 41]. For each conjugacy class of elementary abelian psubgroup E of the group, the corresponding affine space V_{E} is quotiented by the action of the normalizer of E, and then these quotients are glued together according to the maps given by the inclusions between elementary abelian subgroups.

If M is a finitely generated kG-module, then Ext_{kG}^{∗}(M, M) is finitely generated as a module over H^{∗}(G, k). Its annihilator I_{M} ⊆ H^{∗}(G, M) is a homogeneous ideal, so it defines a closed homogeneous subvariety of V_{G}, denoted V_{G}(M). This is what is referred to as the variety of M. These varieties for modules have the following properties:

(i) V_{G}(M ⊕ N) = V_{G}(M) ∪ V_{G}(N). (ii) V_{G}(M ⊗ N) = V_{G}(M) ∩ V_{G}(N). (iii) dim V_{G}(M) is equal to an invariant of M called the complexity, which measures the polynomial rate of growth of the minimal projective resolution of M. (iv) V_{G}(M) = {0} if and only if M is projective.

(v) If ζ̂ : Ω^{n}k → k represents ζ ∈ H^{n}(G, k) ≅ \underline{Hom}_{kG}(Ω^{n}k, k) then the module L_{ζ} defined by the short exact sequence

0 → L_{ζ} → Ω^{n}k ⟶^{ζ̂} k → 0

satisfies

V_{G}(L_{ζ}) = V_{G}⟨ζ⟩,

the hypersurface determined by ζ as an element of the coordinate ring of V_{G}. (vi) If

0 → M_{1} → M_{2} → M_{3} → 0

is a short exact sequence of kG-modules then for any permutation of the indices {i, j, k} = {1, 2, 3} we have

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

Properties (ii) and (v) show that every closed homogeneous subvariety is the variety of some module, namely an appropriate tensor product of modules of the form L_{ζ}.

It should be mentioned that it is possible to calculate the varieties V_{G}(M) without knowing anything about cohomology. The idea is that it is enough to calculate on the elementary abelian p-subgroups, and for an elementary abelian p-group the variety of the module is essentially the same as the rank variety introduced by Carlson [19]. This is a subvariety of affine space defined by looking at the restriction to certain cyclic subgroups of the group algebra, called cyclic shifted subgroups. The points of affine space index these cyclic shifted subgroups, and a point is in the

10 D. J. BENSON

variety of the module if and only if its restriction to the corresponding cyclic shifted subgroup is not projective. The rank variety is not just a calculational tool; it is used, for example, in the proof of the tensor product property (ii) above.

An important ingredient in setting up the theory of varieties for finitely generated modules is Dade’s Lemma [23], which says that a finitely generated kG-module is projective if and only if its restriction to every cyclic shifted subgroup is projective.

The Nucleus

A finite group G is said to be p-nilpotent if it has a normal subgroup N ⊴ G of order prime to p and index a power of p.

For any finite group G, the nucleus Y_{G} ⊆ V_{G} is defined to be the union of the images of V_{H} → V_{G} as H runs over the subgroups whose centralizers are not p-nilpotent. If there is no such subgroup, Y_{G} is defined to be {0}. In the definition of the nucleus, it suffices to let H run over the elementary abelian p-subgroups whose centralizers are not p-nilpotent. This definition comes from [5, 10, 11].

Theorem 2.1. The subvariety Y_{G} is equal to the union of the V_{G}(M) as M runs over the finitely generated kG-modules in the principal block B_{0} satisfying H^{n}(G, M) = 0 for all n > 0.

This is the main theorem of [5], and in the remainder of this lecture, we shall try to give some idea of how Rickard’s idempotent modules featured in the proof.

Corollary 2.2. The following conditions on a group G are equivalent:

(i) For every finitely generated nonprojective kG-module in B_{0}, H^{n}(G, M) ≠ 0 for all n > 0. (ii) Y_{G} = {0}. (iii) The centralizer of each element of order p in G is p-nilpotent.

This corollary answers Problem 2 of the first lecture.

Thick Subcategories

The stable category of finitely generated kG-modules stmod(kG) has the same objects as the module category mod(kG), but for the arrows, instead of Hom_{kG}(M, N) we use the quotient by the projective homomorphisms, \underline{Hom}_{kG}(M, N). This means that stmod(kG) is not an abelian category, because we can no longer tell whether a morphism is injective, or whether it is surjective. Instead, it carries a structure of triangulated category, with the distinguished triangles corresponding to the short exact sequences in mod(kG). The octahedral axiom for triangulated categories corresponds to the third isomorphism theorem in the module category.

INFINITE DIMENSIONAL MODULES FOR FINITE GROUPS 11

Similarly, if we do not restrict ourselves to finitely generated modules, we write Mod(kG) for the category of all kG-modules and module homomorphisms, and we write StMod(kG) for the corresponding stable category.

If V is a closed homogeneous subvariety of V_{G}, we define C(V) to be the subcategory of the stable module category stmod(kG) consisting of the finitely generated kG-modules M satisfying V_{G}(M) ⊆ V.

The category C(V) is a thick subcategory of stmod(kG), meaning that it is closed under Ω and Ω^{−1}, finite direct sums and direct summands, and extensions (i.e., completing an arrow to a triangle).

More generally, if V is a collection of closed homogeneous irreducible subvarieties of V_{G} which is closed under specialization (meaning that if W ⊆ V and V ∈ V then W ∈ V), then we have a thick subcategory C(V) of stmod(kG) consisting of the finitely generated kG-modules M such that V_{G}(M) is a finite union of elements of V.

Here are some more examples of thick subcategories of stmod(kG). If B is a block of kG then stmod(B) is a thick subcategory. The full subcategory of stmod(B_{0}) consisting of modules M satisfying H^{n}(G, M) = 0 for all n > 0 is a thick subcategory (the reason why it is closed under Ω and Ω^{−1} is that H^{n}(G, ΩM) ≅ H^{n−1}(G, M), and if H^{n}(G, M) ≠ 0 for some n > 0 then the same is true for infinitely many values of n > 0 [11]). A collection of finitely generated modules generates a thick subcategory, which is the smallest one which contains them. Further thick subcategories can be formed by taking intersections.

Idempotent Functors (Rickard [44])

If C is a thick subcategory of stmod(kG), then there are functors E_{C} and F_{C} on the stable category of all kG-modules StMod(kG) which fit into a triangle

E_{C}(M) → M → F_{C}(M).

If you are unfamiliar with triangulated categories, think of this as meaning that there is a short exact sequence

0 → E_{C}(M) → M ⊕ (projective) → F_{C}(M) → 0.

The properties of these functors are:

(i) E_{C}(M) is a filtered colimit of modules in C. (ii) F_{C}(M) is C-local in the sense that \underline{Hom}_{kG}(N, F_{C}(M)) = 0 for any module N in C.

The functors E_{C} and F_{C} are uniquely determined up to natural isomorphism by the above properties. This makes it easy in practice to check that candidates for these functors are what they are claimed to be.

12 D. J. BENSON

Now consider the case where C is a tensor ideal thick subcategory, in the sense that the tensor product of a module in C with any module in stmod(kG) lands in C. This is the case, for example, with the above defined categories C(V). In this case, the values of E_{C} and F_{C} are determined by the value on the trivial module. Namely, setting E_{C} = E_{C}(k) and F_{C} = F_{C}(k), in the stable category we have

E_{C}(M) ≅ E_{C} ⊗ M F_{C}(M) ≅ F_{C} ⊗ M.

As an example, if ζ ∈ H^{n}(G, k) and V is the set of closed homogeneous irreducible subvarieties of the hypersurface V ⟨ζ⟩ then we get the modules E(ζ) and F (ζ) described at the end of the first lecture. More generally, if C = C(V) we write E(V) and F (V) for E_{C} and F_{C}. If V consists of all the closed homogeneous irreducible subvarieties of a given closed homogeneous subvariety V then we write E(V) and F (V).

We are now in a position to be able to define the variety of an infinite dimensional module. Rather than a single variety, this is a collection of varieties, and is defined as follows. If M is a (not necessarily finitely generated) kG-module, then V_{G}(M) is defined to be the collection of nonzero closed homogeneous irreducible subvarieties V of V_{G} such that

M ⊗ E(V) ⊗ F ({all proper subvarieties of V })

is not projective.^{1}

Some of the properties of these varieties for not necessarily finitely generated modules are as follows:

(i) If M is finitely generated then V_{G}(M) is the set of closed homogeneous irreducible subvarieties of the previously defined cohomology variety V_{G}(M). (ii) M is projective if and only if V_{G}(M) = ∅. (iii) V_{G}(M ⊕ N) = V_{G}(M) ∪ V_{G}(N). (iv) V_{G}(M ⊗ N) = V_{G}(M) ∩ V_{G}(N).

(v) If V is a closed homogeneous irreducible subvariety of V_{G} and

M = E(V) ⊗ F ({all proper subvarieties of V })

then V_{G}(M) = {V }.

Combining properties (iii) and (v) shows that each collection of closed homogeneous irreducible subvarieties of V_{G} occurs as the variety of some kG-module.

1It should be noted that the original definition of cohomology variety given in [8] was superseded by the definition given in [9], which is equivalent to the definition given here. The original definition did not satisfy the crucial tensor product property (iv). Note that when we write “all proper subvarieties of V ”, we really mean “all proper closed homogeneous irreducible subvarieties of V ”.

INFINITE DIMENSIONAL MODULES FOR FINITE GROUPS 13

Just as in the finitely generated case, an important ingredient in the development of the varieties for infinitely generated modules is a version of Dade’s lemma in this context. As it stands, Dade’s lemma is false for infinitely generated modules. An easy example is given by the module F (U) where U is the set of all proper subvarieties of V_{G}, with G = ℤ/2 × ℤ/2 and where k has characteristic two. This is the module whose underlying k-vector space is k(t) ⊕ k(t) (t an indeterminate) with g_{1} acting as ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) and g_{2} acting as ( \begin{matrix}1 & t \\ 0 & 1 \end{matrix} ).

What is wrong in the above example is that despite the fact that k is algebraically closed, it is still not big enough. After forming a transcendental extension of k, we find that there are then cyclic shifted subgroups where the restriction is not projective. The infinite dimensional version of Dade’s lemma says that a kG-module is projective if and only if its restriction to every cyclic shifted subgroup defined over every extension field is projective. In the above example, we need to pass to a transcendental extension field of k before we can find a cyclic shifted subgroup such that the restriction is not projective. In some sense, the variety of the above module is “V_{G} with all the proper subvarieties defined over k removed”. The extension fields needed are the function fields of subvarieties, and then the cyclic shifted subgroups correspond to “generic points” for subvarieties. The details can be found in [9], and an informal account is given in [6].

We are now ready to sketch the proof of the thick subcategory theorem, which is a good illustration of how the infinitely generated idempotent modules can be used to get information about the category of finitely generated modules.

Theorem 2.3. If M is a finitely generated kG-module then the thick subcategory of stmod(kG) generated by all modules of the form M ⊗ X, as X ranges over finitely generated kG-modules, is equal to C(V_{G}(M)).

Proof. (Rickard) [10] Let ⟨⟨M⟩⟩ denote the thick subcategory generated by all such M ⊗ X. It is easy to see that ⟨⟨M⟩⟩ is contained in C(V) where V = V_{G}(M). The point is to prove the other inclusion.

For any finitely generated kG-modules N and X we have

0 = \underline{Hom}_{kG}(M ⊗ X, F_{⟨⟨M⟩⟩}E_{V} (N))
≅ \underline{Hom}_{kG}(X, M^{∗} ⊗ F_{⟨⟨M⟩⟩}E_{V} (N))

and so M^{∗} ⊗ F_{⟨⟨M⟩⟩}E_{V} (N) is projective. So we have

V_{G}(M) ∩ V_{G}(F_{⟨⟨M⟩⟩}E_{V} (N)) = ∅.

But the triangle

E_{⟨⟨M⟩⟩}E_{V} (N) → E_{V} (N) → F_{⟨⟨M⟩⟩}E_{V} (N)

14 D. J. BENSON

shows that

V_{G}(F_{⟨⟨M⟩⟩}E_{V} (N)) ⊆ V_{G}(M)

and so we have

V_{G}(F_{⟨⟨M⟩⟩}E_{V} (N)) = ∅.

It follows that F_{⟨⟨M⟩⟩}E_{V} (N) is projective. So the map

E_{⟨⟨M⟩⟩}(N) = E_{⟨⟨M⟩⟩}E_{V} (N) → E_{V} (N)

(where the first equality comes from the inclusion ⟨⟨M⟩⟩ ⊆ C(V)) is a stable isomorphism, and hence ⟨⟨M⟩⟩ is equal to C(V) as required.

The tensor ideal version of the thick subcategory theorem follows immediately.

Theorem 2.4. The tensor ideal thick subcategories of stmod(kG) are precisely the categories of the form C(V), as V, as V runs over collections of closed homogeneous irreducible subvarieties of V_{G}, closed under specialization.

For a p-group, the fact that every finitely generated module has a filtration with quotients isomorphic to the trivial module implies that every thick subcategory of stmod(kG) is a tensor ideal, and so we get a classification of thick subcategories in this case.

For a more general finite group, if the nucleus Y_{G} is equal to {0} then the classification of the thick subcategories of B_{0} goes the same way as in the above theorem.

If Y_{G} ≠ {0} then the same techniques can be used to give a classification of:

(i) the thick subcategories of stmod(B_{0}) avoiding Y_{G}, and (ii) the thick subcategories of the quotient category

stmod(B_{0})/stmod(B_{0}) ∩ C(Y_{G}).

It appears from examples that in the case where Y_{0} ≠ {0}, each line through the origin in Y_{G} supports an infinite number of thick subcategories of stmod(B_{0}).

This gives a partial answer to Problem 7 of the first lecture.

Induction and Restriction

It is about time to explain the origin of the role of the nucleus Y_{G} in the theory. This comes from a detailed examination of induction and restriction of idempotent modules. In this context, restriction behaves better than induction. Let i : H → G denote the inclusion of a subgroup H of G, and let i_{∗} : V_{H} → V_{G} be the corresponding map of cohomology varieties. If V is a collection of closed homogenous irreducible subvarieties of V_{G}, closed under specialization, then the triangle

E(V) → k → F (V)

in stmod(kG) restricts to

E(i_{∗}^{−1}(V)) → k → F (i_{∗}^{−1}(V))

in stmod(kH).

Induction is trickier, but we can get good information in a special case of interest. Let A ≤ G be an elementary abelian p-subgroup of G, and set C = C_{G}(A) and N = N_{G}(A). Let ℓ_{0} be a line through the origin in V_{A}, which is not in the image of V_{A′} → V_{A} for any proper subgroup A′ of A. Let D be the “diagonalizer” of ℓ_{0}; namely the subgroup of N consisting of the elements which fix ℓ_{0} setwise. The quotient D/C acts faithfully on ℓ_{0}, so it is a cyclic p′-group.

Denote by ℓ the image of ℓ_{0} in V_{D}, and by L the image of ℓ_{0} in V_{G}.

Theorem 2.5. E(ℓ)↑^{G}≅ E(L) ⊕ (projective).

Proof. This theorem is proved in [5].

Corollary 2.6. If V_{G}(M) = {L} then M ⊕ (projective) is induced from D.

Proof. Modulo projectives, we have

M ≅ M ⊗ E(L) ≅ M ⊗ E(ℓ)↑^{G}≅ (M ↓_{D} ⊗E(ℓ))↑^{G} .

Since every periodic finitely generated module has a variety of this form for a suitable choice of A and ℓ_{0}, this corollary gives an answer to Problem 6 of the first lecture.

Actually, since [5] came out, Carlson [18] (see also section 13 of [20]) found a proof of this corollary which does not depend on the idempotent module technology. But at the same time, he wrote down a generalization for certain varieties of larger dimension which has not been proved without this technology.

Further analysis at the level of D gives the following, which explains where the nucleus fits into the picture. Under the circumstances of the above corollary, if C is p-nilpotent and M is in B_{0}, then H^{∗}(G, M) ≠ 0. If C is not p-nilpotent, then it is possible to construct a finitely generated periodic module M with V_{G}(M) = L (so V_{G}(M) = {L}) such that H^{∗}(G, M) = 0.

We end this lecture with an open question. Consider the thick subcategory of stmod(B_{0}) consisting of the finitely generated modules M such that \underline{Hom}_{kG}(M, N) = 0 for all modules N satisfying H^{n}(G, N) = 0 for all n > 0. Is this the same as the thick subcategory generated by the trivial module?

Jon Carlson has made some recent progress on this question, and he will report on this elsewhere in these proceedings.

16 D. J. BENSON

3. Phantom Maps

In the previous two lectures, I have tried to give some indication of how certain infinite dimensional modules play a crucial role in understanding certain aspects of the category of finitely generated modules over a group algebra of a finite group. In this lecture, I want to switch attention to those properties of maps between infinitely generated modules that are invisible to the finite dimensional world.

The appropriate definition is borrowed from algebraic topology, so I shall begin by briefly reviewing the topological situation.

Phantoms in Topology

A map f : X → Y between CW complexes is said to be phantom if its restriction to each skeleton X^{n} is null homotopic. In other words, there exists a map ϕ : X^{n} × [0, 1] → Y such that ϕ(x, 0) = f(x) for all x ∈ X^{n}, and ϕ(x, 1) is independent of x ∈ X^{n}.

This is not the only possible definition of phantom in topology. For example, one might want to use finite subcomplexes instead of the skeleta, or one might want to work in the world of spectra (Christensen and Strickland [22]). A good survey of phantom maps in topology is given in McGibbon [39], where the following theorems are discussed.

Theorem 3.1. The set of homotopy classes of phantom maps from X to Y is in natural one-one correspondence with the elements of lim_{⟵}^{1}[X^{n}; ΩY].

Note that on the category of filtered systems of abelian groups, lim_{⟵} is left exact but not right exact, and lim_{⟵}^{i} is used to denote its ith right derived functor. In the context of the above theorem, however, the groups in question need not be abelian. In the context of filtered systems of groups, lim_{⟵} still has one well defined right derived functor lim_{⟵}^{1}. A discussion of the construction of this derived functor can be found in Chapter XI, §2 of Bousfield and Kan [14].

Theorem 3.2. The composite of any two phantom maps is null homotopic.

Theorem 3.3. There is a (weakly) universal phantom map

X → ⋁_{n=1}^{∞} ΣX^{n}.

In other words, every phantom map out of X factors through this one, but not uniquely.^{2}

In fact, the previous theorem follows from this one. Namely, if X → Y → Z is a composite of two phantom maps, then it factors as X → ⋁_{n=1}^{∞} ΣX^{n} → Y → Z, and any phantom map out of a wedge of finite dimensional complexes is null homotopic.

2Mike Hopkins tells me that I should call such a map “versal”: like universal but without the “uni”.

Example (Brayton Gray [28]). There are 2^{ℵ_{0}} homotopy classes of phantom maps from infinite dimensional complex projective space ℂP^{∞} to the three dimensional sphere S^{3}.

Theorem 3.4. If X and Y are nilpotent and of finite type, then a map X → Y is phantom if and only if its composite with the Sullivan profinite completion Y → Ŷ is null homotopic.

Phantoms in Representation Theory

In representation theory, we make the following definition. Let G be a finite group and k a field of characteristic p. A map M → N of kG-modules is said to be phantom if its restriction to every finitely generated submodule of M factors through a projective module.

A couple of years ago, I asked my graduate student Gilles Gnacadja to look at this definition and see whether there were any interesting examples, and to what extent the topological theorems carried over. The results of his inquiries were more interesting than I had hoped. The first example he came up with was the following.

Example (Gilles Gnacadja [27]). Let G be the group ℤ/2×ℤ/2 and let k be a field of characteristic two. We saw at the end of the first lecture that the cohomology ring is H^{∗}(G, k) = k[x, y], and we described the modules E(x) and F (x). It turns out that

dim_{k} \underline{Hom}_{kG}(F (x), F (y)) = 2^{ℵ_{0}}

and that all maps are phantom.

More generally, for G a finite group and k a field of characteristic p, if ζ, η is a regular sequence of length two consisting of homogeneous elements in H^{∗}(G, k) (actually, this places some restrictions on G), then

dim_{k} \underline{Hom}_{kG}(F (ζ), Ω^{−1}F (η)) = 2^{ℵ_{0}}

and all maps are phantom.

In the above example for ℤ/2 × ℤ/2, Ω^{−1}F (y) is (stably) isomorphic to F (y).

Write Ph(M, N) for the space of phantom maps from M to N, and \underline{Ph}(M, N) for the quotient of Ph(M, N) by the projective maps. Then in contrast to Gray’s example described above, we have the following.

Theorem 3.5 (Gnacadja [27]). If N is finite dimensional then \underline{Ph}(M, N) = 0.

Theorem 3.6 (Gnacadja [27]). Writing M as a filtered colimit of finitely generated kG-modules M = lim_{\overrightarrow{α}} M_{α}, we have

\underline{Ph}(M, N) ≅ lim_{\overleftarrow{α}}^{1}\underline{Hom}_{kG}(M_{α}, ΩN)

18 D. J. BENSON

Universal Phantoms

If M = lim_{\overrightarrow{α}} M_{α} as above, then writing I_{α} for the injective hull of M_{α}, we define M′ by the short exact sequence

0 → ⨁_{α} M_{α} → M ⊕ ⨁_{α} I_{α} → M′ → 0.

This is the same as completing ⨁_{α} M_{α} → M to a triangle in StMod(kG):

⨁_{α} M_{α} → M → M′.

Then it is shown in [27] that the resulting map M → M′ is a (weakly) universal phantom map out of M.

If either k is countable, or M is countably generated, or G has cyclic Sylow p-subgroups, then M′ is a direct sum of finitely generated modules, and the argument used in the topological situation shows that the composite of any two phantom maps out of M is projective. Otherwise, M′ is usually not a direct sum of finitely generated modules, and it seems likely that there are examples where the composite of two phantom maps is not projective, although this has not been proved.

As an explicit example, let G = ℤ/p × ℤ/p and k be an uncountable field of characteristic p. Let U be the collection of all proper subvarieties of V_{G}, and let M = F (U). Then the module M′ constructed above is definitely not a direct sum of finite dimensional modules, and there is probably a nonprojective composite of two phantom maps out of M.

Profinite Completion

Next, we look for the analog of Theorem 3.4 in the context of representation theory. To understand what should replace profinite completion, consider for a moment the situation for vector spaces. The collection of finite dimensional quotients of a vector space forms an inverse system, and we can form the inverse limit. This comes with a natural map from the original vector space. It turns out that this construction gives exactly the embedding of the vector space into its double dual. So the corresponding theorem is as follows.

Theorem 3.7. [12] A map of kG-modules M → N is phantom if and only if the composite M → N → N^{∗∗} is projective.

Purity

The key to understanding phantom maps turns out to be the concept of purity. The lectures of Birge Huisgen–Zimmermann in these proceedings give an excellent introduction to purity, which I shall not repeat. The relevant definitions are also recalled in [12], and a more extensive reference is the book of Jensen and Lenzing [31].

INFINITE DIMENSIONAL MODULES FOR FINITE GROUPS 19

We just mention that the functor Pext^{n} is constructed in the same way as Ext^{n}, except that instead of taking an exact sequence of projective modules resolving M, we take a pure exact sequence of pure projective modules resolving M. Then we take the cohomology of the complex formed by taking homomorphisms from this sequence into N. Just as in the case of Ext^{n}, this gives the same answer as if we had taken a pure exact sequence of pure injective modules resolving N, and then taken homomorphisms from M. The functor Pext^{1} parametrizes pure extensions.

Theorem 3.8. [12] \underline{Ph}(M, N) ≅ Pext_{kG}^{1}(M, ΩN).

In fact, a map of kG-modules M → N is phantom if and only if the corresponding extension

0 → ΩN → X → M → 0

is pure exact, where this extension corresponds to the original homomorphism via

\underline{Hom}_{kG}(M, N) ≅ Ext_{kG}^{1}(M, ΩN).

Following this correspondence through, one finds that the following hold.

Theorem 3.9. [12] For a kG-module M, the following are equivalent.

(i) M is pure projective. (ii) M is a direct sum of finitely generated modules. (iii) There are no phantom maps out of M.

Theorem 3.10. [12] For a kG-module M, the following are equivalent.

(i) M is pure injective. (ii) The natural inclusion M → M^{∗∗} splits. (iii) M is a direct summand of a direct product of finitely generated modules. (iv) M is a direct summand of the dual of some module.

(v) There are no phantom maps into M.

Composites of Phantoms

We define \underline{Ph}^{n}(M, N) to be the space of maps from M to N which factorize as a composite of n phantoms, where the intermediate modules are not specified.

Theorem 3.11. [12] \underline{Ph}^{n}(M, N) is isomorphic to the image of the natural map

Pext_{kG}^{n}(M, Ω^{n}N) → Ext_{kG}^{n}(M, Ω^{n}N)

via the natural isomorphism

Ext_{kG}^{n}(M, Ω^{n}N) ≅ \underline{Hom}_{kG}(M, N).

It is still not known whether the image of this map can ever be nonzero in degrees two and higher, although it seems likely that it can.

Cardinality

Theorem 3.12 (Gruson and Jensen). If R is a ring of cardinality ℵ_{t} then the pure global dimension of R is at most t + 1.

For kG, it turns out that if the representation type is wild^{3} then equality holds in the above theorem. So if |k| = ℵ_{t} then the pure global dimension of kG is exactly t + 1.

For ℤ/2 × ℤ/2 in characteristic two, the pure global dimension is one for k countable and two for k uncountable. For the other tame cases, the exact value of the pure global dimension is not known. For finite representation type, the pure global dimension is zero. Example For G = ℤ/2 × ℤ/4 in characteristic two, the pure global dimension is t + 1, where |k| = ℵ_{t}, but \underline{Ph}^{6}(M, N) is always zero in this case. To see this, we note that G has a subgroup H of index two and pure global dimension two. So the restriction to H of a composite of three phantom maps is projective. This implies that a composite of three phantom maps is always divisible by the degree one cohomology element x ∈ H^{1}(G, 𝔽_{2}) corresponding to H. So the composite of six phantom maps is divisible by x^{2}, which is zero. Example If G has p-rank two then the Rickard idempotent E-modules are pure injective, so there are no phantom maps into them. It is not known whether this holds in higher rank. Example Denote by U the collection of all proper subvarieties of V_{G}. Then H^{∗}(G, F (U)) is flat, and its projective dimension as a module over H^{∗}(G, k) is equal to the pure projective dimension of F (U) as a kG-module.

During the conference, Henning Krause and the author found a proof that F (U) is always pure injective. In fact, it is even endofinite, so that it can be thought of in some sense as the generic kG-module. It decomposes as a direct sum of indecomposables corresponding to the irreducible components of the variety V_{G}. In general, it is rare for Rickard’s idempotent F -modules to be pure injective. This can be deduced from the known properties of its endomorphism ring.

A Spectral Sequence

One of the tools for studying composites of phantom maps is the following spectral sequence [13]. We write M = lim_{\overrightarrow{α}} M_{α} as a filtered colimit of finitely generated kG-modules. Then we have a spectral sequence

lim_{\overleftarrow{α}}^{s}Ext_{kG}^{t}(M_{α}, ΩN) ⇒ Ext_{kG}^{s+t}(M, ΩN).

3The representation type of kG is finite if G has cyclic Sylow p-subgroups, tame if p = 2 and G has dihedral, semidihedral or generalized quaternion Sylow 2-subgroups, and wild otherwise.

The module in the second slot can be anything, but we have chosen to make it ΩN so that the indices come out right in the five term sequence, which is as follows:

0 → \underline{Ph}(M, N) → \underline{Hom}_{kG}(M, N) → lim_{\overleftarrow{α}} \underline{Hom}_{kG}(M_{α}, N)

→ Pext_{kG}^{2}(M, ΩN) → \underline{Ph}^{2}(M, Ω^{−1}N) → 0.

Usually, the five term sequence coming from a spectral sequence does not go to zero on the right hand end, but here we have replaced the usual version of the last term by the image of the last map.

It follows from this five term sequence that if the pure global dimension of kG is at least two, then either \underline{Ph}^{2} ≠ 0 or there exists an example of a filtered system in stmod(kG) which does not lift to a filtered system in mod(kG). We know of examples where the latter occurs, but we still do not know whether the former occurs.

Note that references [12, 13, 18] can be found at ftp://byrd.math.uga.edu/pub/html/archive.html

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

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