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Given a sequence of related modules $M_n$ defined over a sequence of relatedpolynomial rings, one may ask how to simultaneously compute a finite Gr\"obnerbasis for each $M_n$. Furthermore, one may ask how to simultaneously computethe 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 forcomputing Gr\"obner bases, and Schreyer's Theorem for computing syzygies. Wealso establish a stabilization result for Gr\"obner bases.

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