Spectrum of group cohomology and support varieties

It is a great privilege to reflect on the results and influence of Daniel Quillen’s two papers in the Annals (1971) entitled “The spectrum of an equivariant cohomology ring, I, II” [26], [27]. As with other papers by Dan, these are very clearly written and reach their conclusions with elegance and efficiency. The object of study is the equivariant cohomology algebra of a compact Lie group G acting on a reasonable topological space. A case of particular interest is the action of a finite group on a point, in which case the ring in question is the cohomology algebra of the finite group. Dan writes: “It is the purpose of this series of papers to relate the invariants attached to such a ring by commutative algebra to the family of elementary abelian p-subgroups of G.” What follows is a brief introduction to Dan’s results and methods, followed by an idiosyncratic discussion of some subsequent developments. The specific goal of Dan’s first paper [26] is to give a proof of a conjecture by M. Atiyah (unpublished) and R. Swan [31] concerning the Krull dimension of the mod-p cohomology of a finite group. Those familiar with Dan’s style will not be surprised that in reaching his goal he lays out clearly and concisely the foundations for equivariant cohomology as introduced by A. Borel [7]. Although we do not address the many topological applications of equivariant cohomology or recent developments using equivariant theories in algebraic geometry, we would remiss if we did not point out that this paper establishes the definitions and techniques which a generation of mathematicians have actively pursued. The second paper [26] opens the way to many subsequent developments in the cohomology and representation theory of groups and related structures. Namely, Dan identifies up to (Zariski) homeomorphism the spectrum of the cohomology of a finite group in terms of its elementary abelian p-subgroups. More recent developments which are an outgrowth of Dan’s methods are the subject of the latter part of this survey. The lasting impact of this second paper is in large part due to its vision of investigating group theory using the commutative algebra of the cohomology of groups, leading to new roles for algebraic geometry and triangulated categories in the study of representation theory.