GL-equivariant modules over polynomial rings in infinitely many variables

TL;DR

This paper investigates the algebraic and homological properties of finitely generated modules over an infinite-variable polynomial ring with a natural action of the infinite general linear group, establishing finiteness properties and connections to combinatorial categories.

Contribution

It introduces and proves finiteness properties for modules over an infinite polynomial ring with group action, linking them to a combinatorial quiver category and character polynomials.

Findings

Finiteness properties for Hilbert series, depth, and local cohomology are established.

The module category is equivalent to a simpler quiver category.

Explicit, computable invariants are introduced and illustrated.

Abstract

Consider the polynomial ring in countably infinitely many variables over a field of characteristic zero, together with its natural action of the infinite general linear group G. We study the algebraic and homological properties of finitely generated modules over this ring that are equipped with a compatible G-action. We define and prove finiteness properties for analogues of Hilbert series, systems of parameters, depth, local cohomology, Koszul duality, and regularity. We also show that this category is built out of a simpler, more combinatorial, quiver category which we describe explicitly. Our work is motivated by recent papers in the literature which study finiteness properties of infinite polynomial rings equipped with group actions. (For example, the paper by Church, Ellenberg and Farb on the category of FI-modules, which is equivalent to our category.) Along the way, we see several connections with the character polynomials from the representation theory of the symmetric groups. Several examples are given to illustrate that the invariants we introduce are explicit and computable.