On minimal flat-injective presentations over local graded rings
Fritz Grimpen, Anastasios Stefanou
TL;DR
The paper develops a comprehensive framework for flat-injective presentations over local graded rings, establishing a generator-minimality criterion and showing how Matlis duality yields cogenerator-minimality, thereby enabling a robust minimality theory (Theorem 2). It introduces an associated flat-injective presentation construction for modules when $R_0 \cong k$, producing a free-cofree presentation that integrates flat and injective structures. A practical reduction algorithm is then developed, leveraging monomial matrices to represent presentations and using truncations and solution spaces $L(A,b)$ to iteratively remove redundant generators (and dually cogenerators), culminating in a minimal presentation. In the polynomial ring setting, the method yields a monomial-matrix description and a finite, algorithmic reduction procedure, offering a direct path to minimal flat-injective presentations without first computing minimal free resolutions, with potential applications to topological data analysis and other areas requiring duality-friendly combinatorial descriptions of graded modules.
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.