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arXiv · 2507.14424

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

Richard F. Bartels
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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.

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Richard F. Bartels. 2025-07-19. Cohen-Macaulay approximations and the $\text{SC}_r$-condition. https://arxiv.org/abs/2507.14424

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