Representations and cohomology of a family of finite supergroup schemes

We examine the cohomology and representation theory of a family of finite supergroup schemes of the form $(\mathbb G_a^-\times \mathbb G_a^-)\rtimes (\mathbb G_{a®}\times (\mathbb Z/p)^s)$. 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-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 $({\mathbb{G}}_a^{-}\times {\mathbb{G}}_a^{-})⋊{\mathbb{G}}_{a(1)}$ and $({\mathbb{G}}_a^{-}\times {\mathbb{G}}_a^{-})⋊\mathbb{Z}\mathrm{/}p$. The computation uses the local cohomology spectral sequence for group cohomology, which we describe in the context of finite supergroup schemes.