GL-algebras in positive characteristic II: the polynomial ring

TL;DR

This paper extends key results on GL-equivariant modules over polynomial rings from characteristic zero to positive characteristic, revealing new structural properties and providing explicit generators for the derived category.

Contribution

It generalizes Snowden's linearization and shift theorems to positive characteristic, enabling the study of $S$-modules' structure and derived categories in this setting.

Findings

Finiteness of linear strands of higher slope in resolutions

Explicit generators for the derived category of $S$-modules

Extension of key theorems to positive characteristic setting

Abstract

We study GL-equivariant modules over the infinite variable polynomial ring $S = k[x_1, x_2, ..., x_n, ...]$ with $k$ an infinite field of characteristic $p > 0$. We extend many of Sam--Snowden's far-reaching results from characteristic zero to this setting. For example, while the Castelnuovo--Mumford regularity of a finitely generated GL-equivariant $S$-module need not be finite in positive characteristic, we show that the resolution still has finitely many "linear strands of higher slope". The crux of this paper is two technical results. The first is an extension to positive characteristic of Snowden's recent linearization of Draisma's embedding theorem which we use to study the generic category of $S$-modules. The second is a Nagpal-type "shift theorem" about torsion $S$-modules for which we introduce certain categorifications of the Hasse derivative. These two results together allow us to obtain explicit generators for the derived category. In a follow-up paper, we also use these results to prove finiteness results for local cohomology modules.