Relative Koszul coresolutions and relative Betti numbers

Abstract

Let $G$ be a finitely generated right $A$-module for a finite-dimensional algebra $A$ over a filed $\mathbb{k}$, 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 $\mathbb{k}$-dimension of the $i$-th homology of the $\mathcal{I}$-relative Koszul complex ${\mathcal{K}}_V(M{)}_{\bullet }:={\mathrm{Hom}}_A({\mathcal{K}}^{\bullet }(V),M)$ 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 $M$.

36 pages. Main Theorem (3.9) is improved, now without the generator/cogenerator assumption on the module G. In the introduction, an outline of the solution is written in more detail. Some examples were added that treat the case where the endomorphism algebra of G is isomorphic to a lower semi-lattice. Criterion for right/left interval approximation was improved in Subsection 4.1