Normality of Ideals and Modules
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866915749584437248 |
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| author | Endo, Naoki Goto, Shiro Hong, Jooyoun Ulrich, Bernd |
| author_facet | Endo, Naoki Goto, Shiro Hong, Jooyoun Ulrich, Bernd |
| contents | We investigate when the Rees algebra of an integrally closed $\mathfrak{m}$-primary ideal in a regular local ring is a Cohen-Macaulay normal domain. While this property always holds in dimension two, it fails in general in higher dimensions, prompting a search for sufficient conditions on the ideal. We show that if an integrally closed ideal contains a part of regular system of parameters of length $d-2$, where $d$ is the dimension of the regular local ring, then its Rees algebra is Cohen-Macaulay and normal. We also extend results of Goto and Ciupercă by proving the same conclusion when the minimal number of generators of an ideal is at most $d+2$. Furthermore, we treat the case of integrally closed zero-dimensional ideals generated by $d+3$ homogeneous polynomials. Finally, using generic Bourbaki ideals, we generalize these results to integrally closed torsionfree modules of finite colength. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_16339 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Normality of Ideals and Modules Endo, Naoki Goto, Shiro Hong, Jooyoun Ulrich, Bernd Commutative Algebra 13H10, 13B22, 13B25, 13A30 We investigate when the Rees algebra of an integrally closed $\mathfrak{m}$-primary ideal in a regular local ring is a Cohen-Macaulay normal domain. While this property always holds in dimension two, it fails in general in higher dimensions, prompting a search for sufficient conditions on the ideal. We show that if an integrally closed ideal contains a part of regular system of parameters of length $d-2$, where $d$ is the dimension of the regular local ring, then its Rees algebra is Cohen-Macaulay and normal. We also extend results of Goto and Ciupercă by proving the same conclusion when the minimal number of generators of an ideal is at most $d+2$. Furthermore, we treat the case of integrally closed zero-dimensional ideals generated by $d+3$ homogeneous polynomials. Finally, using generic Bourbaki ideals, we generalize these results to integrally closed torsionfree modules of finite colength. |
| title | Normality of Ideals and Modules |
| topic | Commutative Algebra 13H10, 13B22, 13B25, 13A30 |
| url | https://arxiv.org/abs/2601.16339 |