Representation-theoretic support spaces for finite group schemes

Abstract

We introduce the space P ( G ) of abelian p -points of a finite group scheme over an algebraically closed field of characteristic p > 0. We construct a homeomorphism Ψ G : P ( G ) → Proj | G | from P ( G ) to the projectivization of the cohomology variety for any finite group G . For an elementary abelian p -group (respectively, an infinitesimal group scheme), P ( G ) can be identified with the projectivization of the variety of cyclic shifted subgroups (resp., variety of 1-parameter subgroups). For a finite dimensional G -module M , Ψ G restricts to a homeomorphism P ( G ) M → Proj | G | M , thereby giving a representation-theoretic interpretation of the cohomological support variety.