Gr\"obner coherent rings and modules

TL;DR

This paper introduces Gr"obner-coherent modules over graded rings, a class that combines coherence with effective computational tools via Gr"obner bases, expanding the algebraic framework for graded modules.

Contribution

It defines and studies Gr"obner-coherent modules, establishing their properties and computational advantages over traditional coherent modules.

Findings

Gr"obner-coherent modules form an abelian category closed under extension.

They admit an effective theory of Gr"obner bases.

These modules enable computational methods not available for coherent modules.

Abstract

Let $R$ be a graded ring. We introduce a class of graded $R$-modules called Gr\"obner-coherent modules. Roughly, these are graded $R$-modules that are coherent as ungraded modules because they admit an adequate theory of Gr\"obner bases. The class of Gr\"obner-coherent modules is formally similar to the class of coherent modules: for instance, it is an abelian category closed under extension. However, Gr\"obner-coherent modules come with tools for effective computation that are not present for coherent modules.