Some Homological Conjectures Over Idealization Rings
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913372908290048 |
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| author | Nascimento, Igor Jorge-Pérez, Victor Freitas, Thiago |
| author_facet | Nascimento, Igor Jorge-Pérez, Victor Freitas, Thiago |
| contents | Let $(R,\mathfrak{m},k)$ be a Noetherian local ring and let $M$ be a finitely generated $R$-module. The main focus of this paper is to give positive answers for some long-standing homological conjectures over the idealization ring $R\ltimes M$. First, if $N$ is a $R\ltimes k$-module, we show that the vanishing of $\operatorname{Ext}_{R\ltimes k}^{i}(N,N\oplus (R\ltimes k))$ for $i=1,2,3$ gives that $N$ is free, and this provides a sharpened version of the Auslander-Reiten conjecture over $R\ltimes k$. Also, we give a characterization of the Betti numbers of an $R$-module over the idealization ring $R\ltimes M$ and, as a biproduct, we derive that the Jorgensen-Leuschke conjecture holds true for $R\ltimes M$. Further, we show that the true of Buchsbaum-Eisenbud-Horrocks and Total Rank conjectures over $R$ implies the true over $R\ltimes M$. This establishes particular answers for both conjectures for modules with infinite projective dimension, especially when $R$ is regular or a complete intersection ring. As applications of the idealization ring theory, we show that the
Zariski-Lipman conjecture holds for any ring $R$ provided the Betti numbers of the $R$-derivation module $\operatorname{Der}_k(R)$, seen as $R\ltimes k$-module, satisfy the inequality $β_{n}^{R\ltimes k}(\operatorname{Der}_k(R))\leqβ_{n-1}^{R\ltimes k}(\operatorname{Der}_k(R))$ for some $n>0$. Some implications regarding the Herzog-Vasconcelos conjecture are also provided. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_06745 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some Homological Conjectures Over Idealization Rings Nascimento, Igor Jorge-Pérez, Victor Freitas, Thiago Commutative Algebra 13D22, 13D07, 13H05, 13H10, 13C10 Let $(R,\mathfrak{m},k)$ be a Noetherian local ring and let $M$ be a finitely generated $R$-module. The main focus of this paper is to give positive answers for some long-standing homological conjectures over the idealization ring $R\ltimes M$. First, if $N$ is a $R\ltimes k$-module, we show that the vanishing of $\operatorname{Ext}_{R\ltimes k}^{i}(N,N\oplus (R\ltimes k))$ for $i=1,2,3$ gives that $N$ is free, and this provides a sharpened version of the Auslander-Reiten conjecture over $R\ltimes k$. Also, we give a characterization of the Betti numbers of an $R$-module over the idealization ring $R\ltimes M$ and, as a biproduct, we derive that the Jorgensen-Leuschke conjecture holds true for $R\ltimes M$. Further, we show that the true of Buchsbaum-Eisenbud-Horrocks and Total Rank conjectures over $R$ implies the true over $R\ltimes M$. This establishes particular answers for both conjectures for modules with infinite projective dimension, especially when $R$ is regular or a complete intersection ring. As applications of the idealization ring theory, we show that the Zariski-Lipman conjecture holds for any ring $R$ provided the Betti numbers of the $R$-derivation module $\operatorname{Der}_k(R)$, seen as $R\ltimes k$-module, satisfy the inequality $β_{n}^{R\ltimes k}(\operatorname{Der}_k(R))\leqβ_{n-1}^{R\ltimes k}(\operatorname{Der}_k(R))$ for some $n>0$. Some implications regarding the Herzog-Vasconcelos conjecture are also provided. |
| title | Some Homological Conjectures Over Idealization Rings |
| topic | Commutative Algebra 13D22, 13D07, 13H05, 13H10, 13C10 |
| url | https://arxiv.org/abs/2405.06745 |