Equivariant Free Resolutions of Sequences of Symmetric Module

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.