Π-supports for modules for finite group schemes

Abstract

We introduce the space $\Pi (G)$ of equivalence classes of $\pi$-points of a finite group scheme $G$ and associate a subspace $\Pi (G{)}_M$ to any $G$-module $M$. Our results extend to arbitrary finite group schemes $G$ over arbitrary fields $k$ of positive characteristic and to arbitrarily large $G$-modules, the basic results about “cohomological support varieties” and their interpretation in terms of representation theory. In particular, we prove that the projectivity of any (possibly infinite-dimensional) $G$-module can be detected by its restriction along $\pi$-points of $G$. Unlike the cohomological support variety of a $G$-module $M$, the invariant $M\mapsto \Pi (G{)}_M$ satisfies good properties for all modules, thereby enabling us to determine the thick, tensor-ideal subcategories of the stable module category of finite-dimensional $G$-modules. Finally, using the stable module category of $G$, we provide $\Pi (G)$ with the structure of a ringed space which we show to be isomorphic to the scheme $\mathrm{Proj}{H}^{•}(G,k)$