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.
Categories: math.AC