Equivariant Free Resolutions of Sequences of Symmetric Module

TL;DR

This paper develops an algorithm to compute equivariant free resolutions of modules over polynomial rings with symmetric group actions, enabling analysis of infinite-variable ideals with symmetry.

Contribution

It introduces a novel algorithm for computing syzygies and equivariant differentials in FI-modules, advancing the understanding of symmetric invariants in polynomial rings.

Findings

Algorithm for syzygy computation in FI-modules

Application to truncations of equivariant resolutions

Finite generation of free modules up to symmetry

Abstract

Given a sequence of related modules $M_n$ over a sequence of related Noetherian polynomial rings, where each $M_n$ is a representation of the symmetric group on $n$ letters, one may ask how to simultaneously compute an equivariant free resolution of each $M_n$. In this article, we address this question. Working in the setting of FI-modules over a Noetherian polynomial FI-algebra, we provide an algorithm for computing syzygies and FI-equivariant differentials. As an application, we show how this result can be used to compute truncations of equivariant free resolutions of ideals in polynomial rings in infinitely many variables that are invariant under actions of the monoid of strictly increasing maps or of permutations. The free modules occurring in such a free resolution are finitely generated up to symmetry.