Relative Gr\"obner bases of modules and applications in persistence theory

TL;DR

This paper develops a theory of relative Gröbner bases for modules over polynomial rings and applies it to solve problems in persistence theory, including computing free presentations and resolutions.

Contribution

It introduces a new framework of relative Gröbner bases for modules and demonstrates their use in persistence theory applications, such as chain complex presentations and module embeddings.

Findings

Computed free presentations of complexes of torsion-free modules.

Constructed algorithms for free resolutions from injective hull embeddings.

Showed minimization of free resolutions using standard reduction techniques.

Abstract

Finitely generated modules over the polynomial ring in $n$ indeterminates are isomorphic to quotients of finite rank free modules. We introduce a theory of relative Gr\"obner bases for those quotients of free modules and, equivalently, for pairs of submodules; we prove corresponding Buchberger- and Schreyer-type theorems. As applications of this theory, we consider three problems in persistence theory, which can be solved by relative Gr\"obner bases. First, we show that the relative Schreyer's theorem can be used to compute free presentations of complexes of finitely generated torsion-free modules. In contrast to previous approaches, this allows computation of free presentations for multicritical persistent homology directly at the chain module level without additional topological constructions. Second, any finitely generated Artinian module embeds in an Artinian injective hull, giving rise to a flat-injective presentation. We represent the embedding of the module in this injective hull by a quotient of a free module and apply the relative Schreyer's theorem to construct an algorithm for the computation of a free presentation from a flat-injective presentation. Third, we investigate how free presentations, and more generally free resolutions, obtained by the two preceding applications can be minimized by standard reduction techniques.