GL-algebras in positive characteristic II: the polynomial ring

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.

35 pages, 1 figure. introduction shortened; other routine updates

References in corpus (9)

• FI-modules and stability for representations of symmetric groups
• GL-equivariant modules over polynomial rings in infinitely many variables
• Syzygies of Segre embeddings and Delta-modules
• Big polynomial rings and Stillman's conjecture
• Noetherianity of some degree two twisted commutative algebras
• FI- and OI-modules with varying coefficients
• Castelnuovo-Mumford regularity up to symmetry
• Codimension and Projective Dimension up to Symmetry
• GL-algebras in positive characteristic I: the exterior algebra