93g:20099 20J06 19D55 20C20 55P20

Benson, D. J. [Benson, David John] ^{(}4-OX^{)}

⋆Representations and cohomology. II. Cohomology of groups and modules. Cambridge Studies in Advanced Mathematics, 31. Cambridge University Press, Cambridge, 1991. x+278 pp. $59.95. ISBN 0-521-36135-4 Vol. I has been reviewed [MR 92m:20005].

Let me start by quoting from the author’s introduction: “We try to develop” cohomology of groups and modules “from both an algebraic and a topological viewpoint and demonstrate the connection between the two approaches. Having in mind the die-hard algebraist who refuses to have anything to do with topology, we have tried to make sure that if the reader omits all sections involving topology, the rest is still a coherent treatment of the subject.

“One of the most exciting developments in recent years in group cohomology is the theory of varieties for modules, expounded in Chapter 5. In a sense, this is the central chapter of the entire two volumes, since it shows how inextricably intertwined representation theory and cohomology really are.”

Let me briefly discuss the various chapters: In the first chapter the necessary topological background is presented—often by pointing out the essential idea of the proofs or giving exact references: Homotopy groups, loop spaces, CW -complexes and cellular homology, fibrations and fibre bundles, simplicial sets and the Milnor exact sequence of inclusions of simplicial sets.

In the second chapter the (co-)homology of G is approached via an Eilenberg-Mac Lane space K(G, 1) of G and it is shown that H_{n}(G, R) ≃ H_{n}(K(G, 1), R) and H^{n}(G, R) ≃ H^{n}(K(G, 1), R). It is a little bit confusing, that without mentioning it, on p. 36 R is the normal subgroup of relations and on p. 37 R is a commutative ring.Milnor’s construction of a universal principal G-bundle ξ_{G}: EG → BG is given and it is proved that H_{n}(G, R) ≃ H_{n}(BG, R) and H^{n}(G, R) ≃ H^{n}(BG, R). Next, topological K-theory of compact spaces is developed, and generalized cohomology theory is introduced. In this connection the Adams operations are introduced, the Bott periodicity theorem and Atiyah’s completion theorem, relating the Grothendieck ring of the finite-dimensional complex representations of G to the K-theory of the classifying space, are stated. As special cohomology classes associated to vector bundles, the Chern and Stiefel-Whitney classes are introduced. Transfer from H to G is considered as a stable map from BH to BG and G. Segal’s conjecture (proved in 1984 by G. Carlson) is discussed, which states essentially that stable maps between classifying spaces are generated by the transfers and the group homomorphisms. To this end stable cohomotopy of the classifying space of the finite group G is introduced, and is related to the Burnside ring of G. In the next section Quillen’s topological calculation using the Adams operations of the cohomology ring of the general linear groups over finite fields is presented. On the basis of this proof, Quillen’s “BGL^{+}”-construction is derived to define the higher algebraic K-groups of a ring. As an application the K-groups of finite fields are computed using the computations in connection with the general linear groups. The Hochschild homology of an algebra is defined and its additive structure is determined for group algebras. Then the Dennis trace map from algebraic Ktheory to Hochschild homology is defined. The author now turns to cyclic homology and proves—using spectral sequences these will be defined only in the next chapter—Connes’ long exact sequence linking Hochschild homology and cyclic homology. The Chern map from algebraic K-theory to cyclic homology composed with the above map is the Dennis trace map.

Chapter 3 is devoted to spectral sequences: The author starts by explaining spectral sequences as they arise in computing the cohomology of a filtered CW -complex and of a Serre fibration of CW -complexes. In detail he then deals with the following: (a) the spectral sequence of an exact couple, (b) the spectral sequence of a Serre fibration, (c) the spectral sequence of a double complex, (d) the Lyndon-Hochschild-Serre spectral sequence of a group extension,

(e) the Kunneth spectral sequence, (f) the Eilenberg-Moore spectral sequence associated to a pullback diagram of fibrations of CW - complexes and (g) the Atiyah spectral sequence, which invokes the Bott periodicity theorem, and goes from the cohomology of a finite group to the representation ring. Finally—using spectral sequences— the author presents Venkov’s topological proof of the finite generation of the cohomology ring of a finite (compact Lie) group in the setting of equivariant cohomology.

In Chapter 4, L. Evens’ algebraic proof that for an RG-module M, finitely generated over the Noetherian ring R, the H^{∗}(G, R)- module H^{∗}(G, M) is finitely generated, is presented using Evens’ norm map. Next the spectral sequence of the exact couple derived from the Bockstein homomorphism is explained, and its multiplicative properties are derived. It links integral and modular cohomology. Moreover, Serre’s theorem on the vanishing of certain products of Bockstein operators is proved. Finally the principal properties of the algebraically introduced Steenrod operations are derived. Most of the relevant material in Chapter 5 on the varieties of modules can also be found in a book by Evens [The cohomology of groups, Oxford Univ. Press, New York, 1991]. Let K be a field of characteristic p. Firstly it is proved that an element in Ext_{KG}^{∗}(M, M) is nilpotent if and only if its restriction to every elementary abelian p-subgroup is nilpotent (Quillen for M = K, Carlson in general). This is used to prove Chouinard’s theorem that a KG-module M is projective if and only if it is projective when restricted to every elementary abelian p-subgroup. As a corollary to the Alperin-Evens theorem, that the complexity c_{G}(M) of the KG-module M (i.e. the polynomial growth rate of a minimal projective resolution) is the maximum of the complexities c_{E}(M) for elementary abelian p-subgroups E of G, Quillen’s theorem is derived, which states that the Krull dimension of the cohomology ring (even-dimensional in case p is odd) H.(G, K) is equal to the p-rank r_{p}(G) of G. As an example the mod 2 cohomology ring of extra special 2-groups is computed.

After a crash course on algebraic geometry Quillen’s stratification of the maximal ideal spectrum V_{G} := max(H.(G, K)) as the disjoint union of locally closed subvarieties, which are closely related to V_{E}, one for each conjugacy class of elementary abelian p-subgroups of G, the cohomological variety V_{G}(M) of a KG-module M is defined to be the subvariety of V_{G} of the ideal in H.(G, K) of all elements annihilating Ext_{KG}^{∗}(M, M). It has dimension dim(V_{G}(M)) = c_{G}(M), the complexity of M. As in case M = K this variety is stratified by varieties related to the elementary abelian p-subgroups (Avrunin, Scott). It is then proved that Carlson’s rank variety of M, which is defined without cohomology, coincides with the cohomology variety of M (Avrunin, Scott), and that these varieties are connected provided M is indecomposable (Carlson). Finally Eisenbud’s theorem, that an indecomposable module of complexity one is periodic or projective, is generalized (Benson, Carlson) by showing that a module of complexity c possesses a c-fold periodic projective resolution, which may be expressed as a tensor product of periodic complexes. Two proofs are presented, one using varieties, the other by means of hypercohomology. The remaining sections deal with the theorem of Benson, Carlson and Robinson, that the Tate cohomology of a module vanishes, provided it vanishes for r consecutive integers for some r depending on G. Poincare duality and Cohen-Macaulay cohomology rings are discussed.

Chapter 6 deals with the G-simplical complexes, especially the complexes 𝒮_{p}(G) of nontrivial p-subgroups, 𝒜_{p}(G) of nontrivial elementary abelian p-subgroups and ℬ_{p}(G) of nontrivial p-subgroups P of G satisfying P = O_{p}(N_{G}(P)). It is proved that the inclusion 𝒜_{p}(G) → 𝒮_{p}(G) is a G-homotopy equivalence (Quillen, Thevenaz) as well as ℬ_{p}(G) → 𝒮_{p}(G) (Bouc, Thevenaz). Also K. Brown’s result that the Euler characteristic of 𝒮_{p}(G) is congruent to one modulo |G|_{p} is proved. Via the (reduced) Lefschetz module—the alternating sum of the chain groups—of a G-simplicial complex the generalized Steinberg module is introduced and shown to be a virtual projective module (Quillen, Webb). It is then proved that for a Chevalley group in characteristic p the generalized Steinberg module coincides up to sign with the classical Steinberg module defined as homology of the Tits building (Solomon, Tits). Finally Alperin’s conjecture is discussed.

The final chapter is devoted to the Ronan-Smith theory of sheaves on G-simplicial complexes. Smith’s theorem on simple modules for Chevalley groups in characteristic p is stated without proof.

This book is written on the level of specialists. It will surely give the research area of “group cohomology” new impetus. It contains most of the results known about cohomology of groups, much of which had previously only been available in research papers. Because of this and the author’s desire to develop everything, both from the algebraic and topological point of view, the book is by no means easy to read; in particular, proofs are often quite sketchy and sometimes the notation is not so clear. K. W. Roggenkamp (D-STGT-B)

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