Group Cohomology and Algebraic Cycles

For an arbitrary group G, the cohomology groups Hⁱ(G, –) of G are the right derived functors of the Hom_G (Z, –) functor, where Z is the additive group of integers considered as a trivial G-module. Thus, for an arbitrary G-module M, by definition the cohomology groups Hⁱ(G, M) are calculated using a projective resolution of the G-module Z or an injective resolution of the G-module M.

Many important properties of the group G, and the G-module M, are encoded in these cohomology groups. For example, the 0-th cohomology group H⁰(G, M)=M^G_, is the subgroup of elements of M fixed under the action of G. There are also interpretations of the first and second cohomology groups that keep track of several important properties of the given group G, its subgroups and quotients.

Historically, homological algebra had one its origins in the cohomology of groups. Parallel to these developments in group theory, (co)homology groups of topological spaces were also being developed and homological algebra can also trace its roots to algebraic topology. These two developments would intertwine nicely with the discovery of the classifying space BG of any group G: The cohomology groups of this topological space are isomorphic to the cohomology groups of G defined in the algebraic fashion recalled above. In particular, for any commutative ring R, the cohomology H*(BG, R) is a graded commutative ring that depends only on the group G.

Perhaps it is worthwhile to recall that the classifying space BG of a group G indeed classifies something: principal G-bundles over an arbitrary topological space X. By this token, the cohomology R-algebra H*(BG, R) gives useful information on the classification of principal G-bundles on X.

At this point, though with so little having been said in the previous paragraphs I may be asking for a suspension of disbelief, it may be clear that group cohomology is one of those wide bridges relating pure group theory and algebraic topology at several deep junctions.

There are many known deep properties distributed along this bridge. One particular and important fact is that the cohomology ring of a finite group is finitely generated. However, this fact is not enough to describe the whole cohomology ring of a finite group. For example, we do not know the degrees of the generators or the relations amongst them. P. Symonds made a recent advance towards these questions in his paper “On the Castelnuovo-Mumford Regularity of the Cohomology Ring of a Group”, J. Amer. Math. Soc. 23 (2010), 1159–1173: Given a finite group G, with a faithful ordinary representation of dimension at least 2, for any prime integer p, the cohomology ring H*(G, Z/p) has generators of degree at most p² . This is the first general bound on the order of the generators for the cohomology ring.

One of the aims of the monograph under review is to give a self-contained proof of this theorem in Chapter 4, in its natural setting of the regularity of group cohomology. Chapter 3 gives the necessary background on commutative algebra on depth and regularity.

There are several constructions of the classifying space of a group, each of them useful in the corresponding context. When the given group is an algebraic group, the author of the monograph under review (“The Chow Ring of a Classifying Space”, Proc. Symp. Pure Math. 67 AMS (1999), 249–281.), and independently Morel and Voevodsky (“A¹-homotopy theory of schemes” Pub. Math. IHES 90 (1999), 45–143), have shown that the classifying space BG can be seen as a limit of algebraic varieties in a natural way. This fundamental result brings along the heavy machinery of algebraic geometry, in particular several algebro-geometric analogs of the cohomology rings: the Chow rings of algebraic cycles of the involved varieties. One of the major aims of the monograph under review is the definition of the Chow ring of algebraic cycles on the classifying space of an algebraic group, and the establishment of several of their more important properties. In particular, since any finite group is an algebraic group, variants of Symonds’s theorem are also proved.

This important monograph collects in a systematic way, and in some cases improves on, most of the recent developments in this area. The first three chapters give the necessary background on group cohomology, commutative algebra and Chow groups, with some detail but mostly just giving a survey of the results that will be used later on. Starting in chapter four with the proof of Symonds’s theorem, the remaining chapters include generalizations and analogs of this theorem for the Chow ring in chapters 5 and 6.

Then the monograph goes back to the usual cohomology ring of a finite group, with generic examples (e.g., p-groups), explicit examples (e.g., groups of order p⁴), and several explicit calculations (e.g., Chow rings of groups of order 16). The whole book emphasizes the advantages provided by the algebro-geometric approach to group cohomology, and at the same time gives new perspectives to important open questions on algebraic geometry. Even taking into account the high level of the subject, the exposition is nicely systematic, making it accessible to graduate students and researchers in adjacent areas, and even including a chapter on open problems.

Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is fz@xanum.uam.mx.