On the Existence of Mock Injective Modules for Algebraic Groups

TL;DR

This paper explores the existence of mock injective modules in algebraic groups over fields of positive characteristic, providing conditions for their existence and examining cases where they have simple socles.

Contribution

It establishes necessary and sufficient conditions for the existence of non-injective mock injective modules and investigates their occurrence in different types of algebraic groups.

Findings

Mock injective modules can exist for reductive groups.

Such modules do not occur for Borel subgroups.

Conditions for existence depend on group structure.

Abstract

Let $G$ be an affine algebraic group scheme over an algebraically closed field $k$ of characteristic $p>0$, and let $G_r$ denote the $r$-th Frobenius kernel of $G$. Motivated by recent work of Friedlander, the authors investigate the class of mock injective $G$-modules, which are defined to be those rational $G$-modules that are injective on restriction to $G_r$ for all $r\geq 1$. In this paper the authors provide necessary and sufficient conditions for the existence of non-injective mock injective $G$-modules, thereby answering a question raised by Friedlander. Furthermore, the authors investigate the existence of non-injective mock injectives with simple socles. Interesting cases are discovered that show that this can occur for reductive groups, but will not occur for their Borel subgroups.