When are syzygies of the residue field self-dual?

TL;DR

This paper investigates when syzygies of the residue field are self-dual in local rings, revealing that for rings with depth at least 2, such self-duality typically occurs only in hypersurface or regular local rings.

Contribution

It characterizes the conditions under which syzygies of the residue field are self-dual, especially in rings of depth at least 2, extending previous work on reflexive modules.

Findings

Self-duality of syzygies occurs mainly in hypersurface or regular local rings for depth ≥ 2.

The study connects self-duality with the structure of the ring, particularly hypersurface and regular cases.

Provides a partial characterization of rings where all syzygies of the residue field are self-dual.

Abstract

Finitely generated reflexive modules over commutative Noetherian rings form a key component of Auslander and Bridger's stable module theory and are likewise essential in the study of Cohen--Macaulay representations. Recently, H. Dao characterized Arf local rings as exactly those one-dimensional Cohen--Macaulay local rings over which every finitely generated reflexive module is self-dual, and raised the general question of characterizing rings over which every finitely generated reflexive module is self-dual. Motivated by this, in this article, we study the question of self-duality of syzygies of the residue field of a local ring when they are known to be reflexive. We show that for local rings of depth at least 2, the answer is given by hypersurface or regular local rings in most cases.