On minimal flat-injective presentations over local graded rings

TL;DR

This paper extends the concept of flat-injective presentations to local graded rings, providing a minimality criterion, an algorithmic reduction method, and a construction approach for finitely generated modules.

Contribution

It introduces a minimality criterion for flat-injective presentations over local graded rings and offers an algorithmic reduction procedure in the polynomial ring case.

Findings

Established a criterion for minimal flat-injective presentations

Developed an algorithmic reduction procedure for polynomial rings

Provided a construction method from scalar multiplication maps

Abstract

Flat-injective presentations were introduced by Miller (2020) to provide combinatorial descriptions of $\mathbb Z^n$-graded modules. We consider them in the setting of local graded rings $R$, with grading over an abelian group, and give a criterion for minimality of them. In the special case of the polynomial ring, this criterion reduces to a family of $k$-linear equations, and we are able to give an algorithmic procedure for reduction. Furthermore, we provide the description of a flat-injective presentation, which can be constructed from the scalar multiplication maps of a given finitely generated $R$-module. Thereby, we solve the construction problem for flat-injective presentations under strong finiteness assumptions.