Extension closedness of syzygies and local Gorensteinness of commutative rings

TL;DR

This paper refines a key theorem on the extension closedness of syzygies and uses it to connect Serre's condition with local Gorensteinness in commutative rings, extending previous results.

Contribution

It provides a converse to Evans and Griffith's theorem linking Serre's condition and Gorensteinness, generalizing recent work to all local rings.

Findings

Refined Auslander-Reiten theorem on n-th syzygies

Established converse of Evans-Griffith theorem

Extended results to arbitrary local rings

Abstract

We refine a well-known theorem of Auslander and Reiten about the extension closedness of n-th syzygies over noether algebras. Applying it, we obtain the converse of a celebrated theorem of Evans and Griffith on Serre's condition (S_n) and the local Gorensteiness of a commutative ring in height less than n. This especially extends a recent result of Araya and Iima concerning a Cohen-Macaulay local ring with canonical module to an arbitrary local ring.