On the factorial case of Huneke's conjecture for local cohomology modules
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917662874927104 |
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| author | Dosea, André Miranda-Neto, Cleto B. |
| author_facet | Dosea, André Miranda-Neto, Cleto B. |
| contents | A conjecture raised in 1990 by C. Huneke predicts that, for a $d$-dimensional Noetherian local ring $R$, local cohomology modules of finitely generated $R$-modules have finitely many associated primes. Although counterexamples do exist, the conjecture has been confirmed in several cases, for instance if $d\leq 3$, and witnessed some progress in special cases for higher $d$. In this paper, assuming that $R$ is a factorial domain, we establish the case $d=4$, and under different additional conditions (in a couple of results) also the case $d=5$. Finally, when $R$ is regular and contains a field, we apply the Hartshorne-Lichtenbaum vanishing theorem as a tool to deal with the case $d=6$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_06862 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the factorial case of Huneke's conjecture for local cohomology modules Dosea, André Miranda-Neto, Cleto B. Commutative Algebra A conjecture raised in 1990 by C. Huneke predicts that, for a $d$-dimensional Noetherian local ring $R$, local cohomology modules of finitely generated $R$-modules have finitely many associated primes. Although counterexamples do exist, the conjecture has been confirmed in several cases, for instance if $d\leq 3$, and witnessed some progress in special cases for higher $d$. In this paper, assuming that $R$ is a factorial domain, we establish the case $d=4$, and under different additional conditions (in a couple of results) also the case $d=5$. Finally, when $R$ is regular and contains a field, we apply the Hartshorne-Lichtenbaum vanishing theorem as a tool to deal with the case $d=6$. |
| title | On the factorial case of Huneke's conjecture for local cohomology modules |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2405.06862 |