Generalized Support Varieties for Finite Group Schemes

Abstract

We construct two families of refinements of the (projectivized) support variety of a finite dimensional module $M$ for a finite group scheme $G$. For an arbitrary finite group scheme, we associate a family of non-maximal rank varieties $\Gamma^j(G)_M$, $1\leq j \leq p-1$, to a $kG$-module $M$. For $G$ infinitesimal, we construct a finer family of locally closed subvarieties $V^{\underline a}(G)_M$ of the variety of one parameter subgroups of $G$ for any partition $\underline a$ of $\dim M$. For an arbitrary finite group scheme $G$, a $kG$-module $M$ of constant rank, and a cohomology class $\zeta$ in ${H}^1(G,M)$ we introduce the zero locus $Z(\zeta) \subset \Pi(G)$. We show that $Z(\zeta)$ is a closed subvariety, and relate it to the non-maximal rank varieties. We also extend the construction of $Z(\zeta)$ to an arbitrary extension class $\zeta \in {Ext}^n_G(M,N)$ whenever $M$ and $N$ are $kG$-modules of constant Jordan type.