Spectrum of group cohomology and support varieties

Abstract

We give a brief introduction to two fundamental papers by Daniel Quillen appearing in the Annals, 1971. These papers established the foundations of equivariant cohomology and gave a qualitative description of the cohomology of an arbitrary finite group. We briefly describe some of the influence of these seminal papers in the study of cohomology and representations of finite groups, restricted Lie algebras, and related structures.

Varieties for Modules of Finite Dimensional Hopf Algebras

“…If A is a finite dimensional cocommutative Hopf algebra (equivalently, finite group scheme), the answers are also known due to work of many mathematicians building on work on finite groups, on restricted Lie algebras [25], and on infinitesimal group schemes [51]. See, e.g., [25,28,29,51], and the surveys [24,39]. In this case, one works with support varieties defined via Hopf algebra cohomology H * (A, k), which is known to satisfy conditions (fg1 ′ ) and (fg2 ′ ) [25,29], and with rank varieties, which are homeomorphic to the support varieties.…”

“…The theory has been adapted to many other settings, such as finite group schemes, algebraic groups, Lie superalgebras, quantum groups, and self-injective algebras. See, e.g., [1,2,15,18,19,24,26,34,37,38,51,47].…”

Support varieties for rational representations

“…[FP07]). The reader is referred to [Fri13] for a brief history of support varieties, beginning with the fundamental work of Quillen [Qui71a,Qui71b]. For brevity, we usually use 'rational G-module' to refer to a rational representation of G.…”

“…a structure V (G) M which incorporates the information of the support variety of the rational representation M of G when restricted to any Frobenius kernel G (r) ⊂ G. Our formulation is an extension of the approach of C. Bendel, A. Suslin, and the author [23]; we employ 1-parameter subgroups rather than traditional methods of cohomology (e.g., [1]) or the more recent methods of π-points (e.g., [6]). The reader is referred to [5] for a brief history of support varieties, beginning with the fundamental work of D. Quillen [17], [18]. For brevity, we usually use "rational G-module" to refer to a rational representation of G.…”