Relative Koszul coresolutions and relative Betti numbers

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 $M$.