Some criteria for regular and Gorenstein local rings via syzygy modules

TL;DR

This paper establishes new criteria for identifying Gorenstein and regular Cohen-Macaulay local rings using properties of syzygy modules, including conditions involving semidualizing summands and Ext/Tors vanishing.

Contribution

It proves that syzygy modules of maximal Cohen-Macaulay modules lack semidualizing summands and characterizes Gorenstein rings via syzygies of canonical modules, providing new criteria based on Ext and Tors.

Findings

Syzygy modules of MCM modules have no semidualizing summands for all n ≥ 1.

A Cohen-Macaulay ring is Gorenstein iff some syzygy of a canonical module has a free summand.

Criteria involving Ext and Tors vanishings characterize regular and Gorenstein rings.

Abstract

Let $ R $ be a Cohen-Macaulay local ring. We prove that the $ n $th syzygy module of a maximal Cohen-Macaulay $ R $-module cannot have a semidualizing direct summand for every $ n \ge 1 $. In particular, it follows that $ R $ is Gorenstein if and only if some syzygy of a canonical module of $ R $ has a non-zero free direct summand. We also give a number of necessary and sufficient conditions for a Cohen-Macaulay local ring of minimal multiplicity to be regular or Gorenstein. These criteria are based on vanishing of certain Exts or Tors involving syzygy modules of the residue field.