An approach to Martsinkovsky's invariant via Auslander's approximation theory
Yuya Otake
TL;DR
This work defines an approximated sequence $\{\xi_n(M)\}$ that converges to Martsinkovsky's $\xi$-invariant by exploiting Auslander–Bridger AB-approximation theory over two-sided noetherian local rings. It links each $\xi_n(M)$ to concrete invariants derived from $n$-AB approximations, $n$-origin extensions, and $n$-FPD hulls in the subcategories $\mathscr{A}_n(R)$, $\mathscr{E}_n(R)$, and $\mathscr{H}_n(R)$, providing computable formulas and structural insights. The main framework unifies Auslander’s $\delta$-invariant and Martsinkovsky’s $\xi$-invariant and yields elementary proofs that avoid heavier DG-structures. It also studies stabilization and behavior under regular elements, offering practical tools for invariants in Cohen–Macaulay representation theory.
Abstract
Auslander developed a theory of the $δ$-invariant for finitely generated modules over commutative Gorenstein local rings, and Martsinkovsky extended this theory to the $ξ$-invariant for finitely generated modules over general commutative noetherian local rings. In this paper, we approach Martsinkovsky$'$s $ξ$-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.