  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 $M$.
