GL-algebras in Positive Characteristic

Abstract

Over the last decade, multiple authors have proved interesting asymptotic and uniformity results in fields ranging from topology and number theory to algebraic statistics and commutative algebra. A unifying feature of these seemingly unrelated works is the prevalence of highly symmetric infinite-dimensional objects. The class of GL-algebras, i.e., commutative rings which admit an (appropriate) action of the infinite general linear group GL plays a central role in this endeavour. Sam–Snowden and other authors have extensively studied GL-algebras over a field of characteristic zero.

On the contrary, GL-algebras in positive characteristic have hardly been studied despite being intimately connected to other ubiquitous algebraic structures like twisted commutative algebras. We fill this gap in the literature and initiate the systematic study of GL-algebras in positive characteristic. In this thesis, we study the simplest non-trivial GL-algebra, namely the symmetric algebra of the standard representation of GL, which we identify as the infinite variable polynomial ring S.

We prove two technical results for GL-equivariant S-modules: we extend an embedding theorem of Snowden for torsion-free modules from characteristic zero to all characteristics, and we also prove a shift theorem for torsion modules inspired by Nagpal’s thesis on FI-modules. We then stitch together these two results to provide a comprehensive picture of the bounded derived category of GL-equivariant S-modules and prove novel finiteness results for their Betti tables.