Characterizations of regular local rings via syzygy modules of the residue field

TL;DR

This paper characterizes regular local rings by examining properties of syzygy modules of the residue field, establishing conditions under which the ring's regularity can be deduced from module-theoretic properties.

Contribution

It provides new criteria for regularity of local rings based on the structure of syzygy modules of the residue field, linking module decompositions to ring regularity.

Findings

Syzygy modules surjecting onto semidualizing modules imply regularity.

Existence of a syzygy module with a non-zero summand of finite injective dimension characterizes regularity.

Conditions on direct sums of syzygy modules determine the regularity of the ring.

Abstract

Let $R$ be a commutative Noetherian local ring with residue field $k$. We show that if a finite direct sum of syzygy modules of $k$ surjects onto `a semidualizing module' or `a non-zero maximal Cohen-Macaulay module of finite injective dimension', then $R$ is regular. We also prove that $R$ is regular if and only if some syzygy module of $k$ has a non-zero direct summand of finite injective dimension.