Computing Gröbner 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 Gröbner 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 Gröbner bases, and Schreyer's Theorem for computing syzygies. We also establish a stabilization result for Gröbner bases.