Computing Gr\"obner Bases and Free Resolutions of OI-Modules

Given a sequence of related modules $M_n$ defined over a sequence of related polynomial rings, one may ask how to simultaneously compute a finite Gr\"obner basis for each $M_n$. Furthermore, one may ask how to simultaneously compute the module of syzygies of each $M_n$. In this paper we address both questions. Working in the setting of OI-modules over a Noetherian polynomial OI-algebra, we provide OI-analogues of Buchberger's Criterion, Buchberger's Algorithm for computing Gr\"obner bases, and Schreyer's Theorem for computing syzygies. We also establish a stabilization result for Gr\"obner bases.