Modules of constant Jordan type, pullbacks of bundles and generic kernel filtrations

TL;DR

This paper studies the functors from modules of constant Jordan type over elementary abelian p-groups to vector bundles, revealing how restrictions relate to subgroup actions and analyzing kernel filtrations to compute these bundles.

Contribution

It introduces a method to compute vector bundles associated with modules via kernel filtrations, extending results to higher ranks of elementary abelian groups.

Findings

Restriction of sheaves corresponds to subgroup restrictions of modules.

Kernel filtrations help compute associated vector bundles.

Results generalized to higher ranks of elementary abelian groups.

Abstract

Let $kE$ denote the group algebra of an elementary abelian $p$-group of rank $r$ over an algebraically closed field of characteristic $p$. We investigate the functors $\mathcal{F}_i$ from $kE$-modules of constant Jordan type to vector bundles on $\mathbb{P}^{r-1}(k)$, constructed by Benson and Pevtsova. For a $kE$-module $M$ of constant Jordan type, we show that restricting the sheaf $\mathcal{F}_i(M)$ to a dimension $s-1$ linear subvariety of $\mathbb{P}^{r-1}(k)$ is equivalent to restricting $M$ along a corresponding rank $s$ shifted subgroup of $kE$ and then applying $\mathcal{F}_i$. In the case $r=2$, we examine the generic kernel filtration of $M$ in order to show that $\mathcal{F}_i(M)$ may be computed on certain subquotients of $M$ whose Loewy lengths are bounded in terms of $i$. More precise information is obtained by applying similar techniques to the $n$th power generic kernel filtration of $M$. The latter approach also allows us to generalise our results to higher ranks $r$.