Relative Koszul coresolutions and relative Betti numbers

TL;DR

This paper introduces a new relative Koszul coresolution framework for modules over finite-dimensional algebras, enabling the computation of relative Betti numbers and applications to persistence modules.

Contribution

It defines a relative Koszul coresolution for modules and shows how to compute relative Betti numbers, aiding in the analysis of persistence modules.

Findings

Defined $ ext{I}$-relative Koszul coresolutions for modules.

Expressed relative Betti numbers via homology of Koszul complexes.

Applied framework to persistence modules for interval decomposability.

Abstract

Let $G$ be a finitely generated right $A$-module for a finite-dimensional algebra $A$ over a filed $\Bbbk$, and $\mathcal{I}$ the additive closure of $G$. We will define a $\mathcal{I}$-relative Koszul coresolution $\mathcal{K}^{\bullet}(V)$ of an indecomposable direct summand $V$ of $G$, and show that for a finitely generated $A$-module $M$, the $\mathcal{I}$-relative $i$-th Betti number for $M$ at $V$ is given as the $\Bbbk$-dimension of the $i$-th homology of the $\mathcal{I}$-relative Koszul complex $\mathcal{K}_V(M)_{\bullet}:=\operatorname{Hom}_A(\mathcal{K}^{\bullet}(V),M)$ of $M$ at $V$ for all $i \ge 0$. This is applied to investigate the minimal interval resolution/coresolution of a persistence module $M$, e.g., to check the interval decomposability of $M$, and to compute the interval approximation of $M$.