Representations and cohomology of finite group schemes

Abstract

This is a survey article covering developments in representation theory of finite group schemes over the last fifteen years. We start with the finite generation of cohomology of a finite group scheme and proceed to discuss various consequences and theories that ultimately grew out of that result. This includes the theory of one-parameter subgroups and rank varieties for infinitesimal group schemes; the π-points and Π-support spaces for finite group schemes, modules of constant rank and constant Jordan type, and construction of bundles on varieties closely related to Proj H • (G, k) for an infinitesimal group scheme G. The material is mostly complementary to the article of D. Benson on elementary abelian p-groups in the same volume; we concentrate on the aspects of the theory which either hold generally for any finite group scheme or are specific to finite group schemes which are not finite groups. In the last section we discuss varieties of elementary subalgebras of modular Lie algebras, generalizations of modules of constant Jordan type, and new constructions of bundles on projective varieties associated to a modular Lie algebra.

Cohomology for Drinfeld doubles of some infinitesimal group schemes

“…The schemes |kG (r) | have been extensively studied and, in conjunction with support varieties of G (r) -representations, provide one means of approaching modular representation theory. One can see the survey [34] for example, and the references therein. 6.1.…”

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.…”