\title[Finite supergroup schemes]
{Detecting nilpotence and projectivity over finite unipotent supergroup schemes}

\author[BIK$\Pi$]{Dave Benson, Srikanth B. Iyengar,
  Henning Krause\\ and Julia Pevtsova}

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

\address{Srikanth B. Iyengar\\ 
Department of Mathematics\\
University of Utah\\ 
Salt Lake City, UT 84112\\ 
U.S.A.}

\address{Henning Krause\\ 
Fakult\"at f\"ur Mathematik\\ 
Universit\"at Bielefeld\\ 
33501 Bielefeld\\ 
Germany.}

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

\begin{document}

\begin{abstract}
This work concerns the representation theory and cohomology of a finite unipotent supergroup scheme $G$ over a perfect field $k$ of positive characteristic $p\ge 3$. It is proved that an element $x$ in the cohomology of $G$ is nilpotent if and only if for every extension field $K$ of $k$ and every elementary
sub-supergroup scheme $E\subseteq G_K$, the restriction of $x_K$ to $E$ is nilpotent.  It is also shown that a $kG$-module $M$ is projective if and only if for every extension field $K$ of $k$ and every elementary sub-supergroup scheme $E\subseteq G_K$, the restriction of $M_K$ to $E$ is projective. The statements are motivated by, and are analogues of, similar results for finite groups and finite group schemes, but the structure of elementary supergroups schemes necessary for detection is more complicated than in either of these cases. 
One application is a detection theorem for the nilpotence of cohomology, and projectivity of modules, over finite dimensional Hopf subalgebras of the Steenrod algebra.
\end{abstract}

\thanks{The work was supported by the NSF grant DMS-1440140 while DB, SBI and JP were in residence at the MSRI.  SBI was partly supported by NSF grant DMS-1700985 and JP was partly supported by NSF grants DMS-0953011 and DMS-1501146.}