\title[A family of finite supergroup schemes]{Representations and cohomology of a family of finite
  supergroup schemes}

\author[Benson and Pevtsova]{Dave Benson and Julia Pevtsova}

\address{Dave Benson \\ 
Institute of Mathematics\\ 
University of Aber\-deen\\ 
King's College\\ 
Aber\-deen AB24 3UE\\ 
Scotland U.K.}

\address{Julia Pevtsova\\ 
Department of Mathematics\\ 
University of Washington\\ 
Seattle, WA 98195\\ 
U.S.A.}


\begin{document}

\begin{abstract} 
We examine the cohomology and representation theory of a family of finite supergroup
schemes of the form 
$(\bbG_a^-\times \bbG_a^-)\rtimes (\bbG_{a(r)}\times (\bbZ/p)^s)$. 
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-super\-group 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 $(\bbG_a^- \times \bbG_a^-) \rtimes \bbG_{a(1)}$ and
$(\bbG_a^- \times \bbG_a^-) \rtimes \bbZ/p$. The computation uses the
local cohomology spectral sequence for group cohomology, which we
describe in the context of finite supergroup schemes.
\end{abstract}

\keywords{cohomology, finite supergroup scheme, invariant theory,
Steenrod operations, local cohomology spectral sequence}
\subjclass[2010]{16A61 (primary); 16A24, 20G10, 20J06 (secondary)}

\date{\today}

\thanks{This work was supported by the NSF grant DMS-1440140 while the
  authors were in residence at the MSRI. 
The second author was 
partially supported by the DMS-0500946 award and by the Simons foundation}