Representations and cohomology of a family of finite supergroup schemes

Abstract

We examine the cohomology and representation theory of a family of finite supergroup schemes of the form (Ga−×Ga−)⋊(Ga(r)×(Z/p)s). In particular, we show that a certain relation holds in the cohomology ring, and deduce that for finite supergroup schemes having this as a quotient, both cohomology mod nilpotents and projectivity of modules is detected on proper sub-supergroup schemes. This special case feeds into the proof of a more general detection theorem for unipotent finite supergroup schemes, in a separate work of the authors joint with Iyengar and Krause.We also completely determine the cohomology ring in the smallest cases, namely (Ga−×Ga−)⋊Ga(1) and (Ga−×Ga−)⋊Z/p. The computation uses the local cohomology spectral sequence for group cohomology, which we describe in the context of finite supergroup schemes.