Cohen-Macaulayness of formal fibers and dimension of local cohomology modules
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910203534901248 |
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| author | Chau, Tran Do Minh |
| author_facet | Chau, Tran Do Minh |
| contents | Let $(R, \mathfrak{m} )$ be a Noetherian local ring, $M$ a finitely generated $R$-module of dimension $d$. Set $\mathfrak{a}(M):=\mathfrak{a}_0(M)\cdots \mathfrak{a}_{d-1}(M)$, where $\mathfrak{a}_i(M):={\rm Ann}_RH^i_{\mathfrak{m}}(M)$ for $i\geq 0$. In this paper, we study the Cohen-Macaulayness of formal fibers of $R$ in the relation with the dimension ${\rm dim} (R/\mathfrak{a}(M)).$ We prove that ${\rm dim} (R/\mathfrak{a}(M))<d$ if and only if $R/\mathfrak{p}$ is unmixed and the generic formal fiber of $R/\mathfrak{p}$ is Cohen-Macaulay for all $\mathfrak{p}\in{\rm Supp}_R(M)$ with ${\rm dim} (R/\mathfrak{p})=d.$ In general, $R/\mathfrak{p}$ is unmixed and the generic formal fiber of $R/\mathfrak{p}$ is Cohen-Macaulay for all $\mathfrak{p}\in{\rm Supp}_R(M)$ with ${\rm dim} (R/\mathfrak{p})>{\rm dim} (R/\mathfrak{a}(M)).$ As applications, we explore the structure of local rings and the dimension, the closedness of non Cohen-Macaulay locus of finitely generated modules. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_08579 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Cohen-Macaulayness of formal fibers and dimension of local cohomology modules Chau, Tran Do Minh Commutative Algebra 13D45, 13H10 Let $(R, \mathfrak{m} )$ be a Noetherian local ring, $M$ a finitely generated $R$-module of dimension $d$. Set $\mathfrak{a}(M):=\mathfrak{a}_0(M)\cdots \mathfrak{a}_{d-1}(M)$, where $\mathfrak{a}_i(M):={\rm Ann}_RH^i_{\mathfrak{m}}(M)$ for $i\geq 0$. In this paper, we study the Cohen-Macaulayness of formal fibers of $R$ in the relation with the dimension ${\rm dim} (R/\mathfrak{a}(M)).$ We prove that ${\rm dim} (R/\mathfrak{a}(M))<d$ if and only if $R/\mathfrak{p}$ is unmixed and the generic formal fiber of $R/\mathfrak{p}$ is Cohen-Macaulay for all $\mathfrak{p}\in{\rm Supp}_R(M)$ with ${\rm dim} (R/\mathfrak{p})=d.$ In general, $R/\mathfrak{p}$ is unmixed and the generic formal fiber of $R/\mathfrak{p}$ is Cohen-Macaulay for all $\mathfrak{p}\in{\rm Supp}_R(M)$ with ${\rm dim} (R/\mathfrak{p})>{\rm dim} (R/\mathfrak{a}(M)).$ As applications, we explore the structure of local rings and the dimension, the closedness of non Cohen-Macaulay locus of finitely generated modules. |
| title | Cohen-Macaulayness of formal fibers and dimension of local cohomology modules |
| topic | Commutative Algebra 13D45, 13H10 |
| url | https://arxiv.org/abs/2605.08579 |