Sheaves on support varieties and varieties of elementary subalgebras

Abstract

We present several results about two closely related types of objects: the projectivized scheme $\PG$ of one parameter subgroups of an infinitesimal group scheme $G$ and the variety $\bE(\fg)$ of maximal elementary subalgebras of a restricted Lie algebra $\fg$. We define and present background material for both objects. For $\PG$ we provide a partial answer to a question of Friedlander and Pevtsova on whether a certain sheaf $\Ho(M)$ constructed on $\PG$ from a representation $M$ is zero if and only if $M$ is projective. We also explicitly calculate the sheaves $\gker{M}$ for all indecomposable $\slt$-modules $M$ and we calculate $\F_i(V(\lambda))$ where $V(\lambda)$ is a Weyl module and $i \neq p$. This extends work of Friedlander and Pevtsova who calculated $\F_i(V(\lambda))$ when $\lambda \leq 2p - 2$. For $\bE(\fg)$ we explicitly calculate this variety when $\fg$ is the Lie algebra of a reductive algebraic group $G$ and $p$ is good and satisfies a separability condition with respect to $G$. This recovers work of Carlson, Friedlander, and Pevtsova who calculated $\bE(\gln)$, $\bE(\slt[n])$, and $\bE(\mathfrak{sp}_{2n})$.