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

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 $r\ge 3$, we prove a criterion for when an MCM $R$-module $M$ satisfies the ${\text{SC}}_r$-condition, assuming that its first syzygy $\hat{I}{\text{c◯}}_R^1(M)$ satisfies the ${\text{SC}}_{r-1}$-condition.

23 pages. Corrected statement of Proposition 2.4 (c). Revised Propositions 2.6, 2.8, and 2.9