# GL-equivariant modules over polynomial rings in infinitely many variables. II

## TL;DR

This paper extends the understanding of modules over twisted commutative algebras, specifically A_d, generalizing previous results for A_1 and linking to representation stability and algebraic geometry.

## Contribution

It establishes the structure of the category of A_d-modules for any d, generalizing earlier work on A_1-modules and connecting to syzygies and asymptotic algebra.

## Key findings

- Structure of A_d-modules characterized for all d
- Connections made to syzygies of Segre and Veronese embeddings
- Implications for asymptotic commutative algebra

## Abstract

Twisted commutative algebras (tca's) have played an important role in the nascent field of representation stability. Let A_d be the complex tca freely generated by d indeterminates of degree 1. In a previous paper, we determined the structure of the category of A_1-modules (which is equivalent to the category of FI-modules). In this paper, we establish analogous results for the category of A_d-modules, for any d. Modules over A_d are closely related to the structures used by the authors in previous works studying syzygies of Segre and Veronese embeddings, and we hope the results of this paper will eventually lead to improvements on those works. Our results also have implications in asymptotic commutative algebra.