Relative Koszul coresolutions and relative Betti numbers

Let G be a finitely generated right A-module for a finite-dimensional algebra A over a field $k$, and $I$ the additive closure of G. We will define an $I$-relative Koszul coresolution

of an indecomposable direct summand V of G, and show that for a finitely generated A-module M, the $I$-relative i-th Betti number for M at V is given as the $k$-dimension of the i-th homology of the $I$-relative Koszul complex

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.