  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.
