An approach to Martsinkovsky's invariant via Auslander's approximation theory

Auslander developed a theory of the $\delta$-invariant for finitely generated modules over commutative Gorenstein local rings, and Martsinkovsky extended this theory to the $\xi$-invariant for finitely generated modules over general commutative noetherian local rings. In this paper, we approach Martsinkovsky$'$s $\xi$-invariant by considering a non-decreasing sequence of integers that converges to it. We investigate Auslander$'$s approximation theory and provide methods for computing this non-decreasing sequence using the approximation.