On minimal flat-injective presentations over local graded rings

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.