  In this paper we introduce an inductive method to study $\mathrm{OI}$-modules
presented in finite degrees, where $\mathrm{OI}$ is a skeleton of the category
of finitely totally ordered sets and strictly increasing maps. As an
application, we obtain an explicit upper bound for the Castelnuovo-Mumford
regularity of $\mathrm{OI}$-modules.

Representation Theory;Commutative Algebra;Rings and Algebras