Cohen-Macaulay approximations and the $\text{SC}_r$-condition
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915730555928576 |
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| author | Bartels, Richard F. |
| author_facet | Bartels, Richard F. |
| contents | 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 $Ω_{R}^{1}(M)$ satisfies the $\text{SC}_{r-1}$-condition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_14424 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cohen-Macaulay approximations and the $\text{SC}_r$-condition Bartels, Richard F. Commutative Algebra 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 $Ω_{R}^{1}(M)$ satisfies the $\text{SC}_{r-1}$-condition. |
| title | Cohen-Macaulay approximations and the $\text{SC}_r$-condition |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2507.14424 |