Annihilators of (co)homology and their influence on the trace Ideal
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914941303259136 |
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| author | Lyle, Justin Maitra, Sarasij |
| author_facet | Lyle, Justin Maitra, Sarasij |
| contents | Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring, and suppose $R$ is Cohen-Macaulay with canonical module $ω_R$. We develop new tools for analyzing questions involving annihilators of several homologically defined objects. Using these, we study a generalization introduced by Dao-Kobayashi-Takahashi of the famous Tachikawa conjecture, asking in particular whether the vanishing of $\mathfrak{m} \operatorname{Ext}_R^i(ω_R,R)$ should force the trace ideal of $ω_R$ to contain $\mathfrak{m}$, i.e., for $R$ to be nearly Gorenstein. We show this question has an affirmative answer for numerical semigroup rings of minimal multiplicity, but that the answer is negative in general. Our proofs involve a technical analysis of homogeneous ideals in a numerical semigroup ring, and exploit the behavior of Ulrich modules in this setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_04686 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Annihilators of (co)homology and their influence on the trace Ideal Lyle, Justin Maitra, Sarasij Commutative Algebra 13D07, 13C13, 13D02 Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring, and suppose $R$ is Cohen-Macaulay with canonical module $ω_R$. We develop new tools for analyzing questions involving annihilators of several homologically defined objects. Using these, we study a generalization introduced by Dao-Kobayashi-Takahashi of the famous Tachikawa conjecture, asking in particular whether the vanishing of $\mathfrak{m} \operatorname{Ext}_R^i(ω_R,R)$ should force the trace ideal of $ω_R$ to contain $\mathfrak{m}$, i.e., for $R$ to be nearly Gorenstein. We show this question has an affirmative answer for numerical semigroup rings of minimal multiplicity, but that the answer is negative in general. Our proofs involve a technical analysis of homogeneous ideals in a numerical semigroup ring, and exploit the behavior of Ulrich modules in this setting. |
| title | Annihilators of (co)homology and their influence on the trace Ideal |
| topic | Commutative Algebra 13D07, 13C13, 13D02 |
| url | https://arxiv.org/abs/2409.04686 |