An inductive method for $\mathrm{OI}$-modules

TL;DR

This paper introduces an inductive approach to analyze $ ext{OI}$-modules, providing explicit bounds on their Castelnuovo-Mumford regularity, advancing understanding in algebraic and combinatorial structures.

Contribution

The paper presents a novel inductive method for studying finitely presented $ ext{OI}$-modules and derives explicit regularity bounds, a new result in the field.

Findings

Established an inductive framework for $ ext{OI}$-modules

Derived explicit upper bounds for Castelnuovo-Mumford regularity

Enhanced understanding of algebraic properties of $ ext{OI}$-modules

Abstract

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.