Cohen-Macaulay approximations and the $\text{SC}_r$-condition

TL;DR

This paper explores the relationship between Cohen-Macaulay approximations and the $ ext{SC}_r$-condition in local rings, generalizing previous results and providing new criteria for modules to satisfy these conditions.

Contribution

It generalizes Kato's result linking the $ ext{SC}_2$-condition to UFDs and introduces criteria for modules to satisfy the $ ext{SC}_r$-condition based on syzygies.

Findings

Established a criterion for $ ext{SC}_r$-condition for MCM modules.

Generalized Kato's theorem to higher $ ext{SC}_r$-conditions.

Connected $ ext{SC}_r$-conditions with module syzygies.

Abstract

We study the relation between MCM approximations and FID hulls of modules over a Cohen-Macaulay local ring $R$ with canonical module, specifically when $R$ is generically Gorenstein. We then generalize a result of Kato, who proved that a Gorenstein complete local ring $R$ satisfies the $\text{SC}_{2}$-condition if and only if $R$ is a UFD. For $r \geq 3$, we prove a criterion for when an MCM $R$-module $M$ satisfies the $\text{SC}_{r}$-condition, assuming that its first syzygy $\Omega_{R}^{1}(M)$ satisfies the $\text{SC}_{r-1}$-condition.