On some Rings of differentiable type
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914829967556608 |
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| author | Islam, Sayed Sadiqul Puthenpurakal, Tony J. |
| author_facet | Islam, Sayed Sadiqul Puthenpurakal, Tony J. |
| contents | Let $K$ be a field of characteristic 0 and $S=K[x_1,\ldots,x_m]/I$ be an affine domain. Consider $R=S_P$ where $P\in Spec(S)$ such that $R$ is regular. In this paper we construct a field $F$ which is contained in $R$ such that
(1) The residue field of $R$ is a finite extension of $F$.
(2) $D_F(R)$, the ring of $F$-linear differential operators on $R$ is left and right Noetherian with finite global dimension.
(3) The Bernstein class of $D_F(R)$ is closed under localization at one element of $R$.
We also prove a similar result for $R^h$, the Henselization of $R$. As an application we prove that $\frac{D_F(R)}{D_F(R)P}\cong E(κ(P))$ where $E(κ(P))$ is the injective hull of the residue field of $R$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_05390 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On some Rings of differentiable type Islam, Sayed Sadiqul Puthenpurakal, Tony J. Commutative Algebra Primary 13N10, Secondary 13N15, 13D45 Let $K$ be a field of characteristic 0 and $S=K[x_1,\ldots,x_m]/I$ be an affine domain. Consider $R=S_P$ where $P\in Spec(S)$ such that $R$ is regular. In this paper we construct a field $F$ which is contained in $R$ such that (1) The residue field of $R$ is a finite extension of $F$. (2) $D_F(R)$, the ring of $F$-linear differential operators on $R$ is left and right Noetherian with finite global dimension. (3) The Bernstein class of $D_F(R)$ is closed under localization at one element of $R$. We also prove a similar result for $R^h$, the Henselization of $R$. As an application we prove that $\frac{D_F(R)}{D_F(R)P}\cong E(κ(P))$ where $E(κ(P))$ is the injective hull of the residue field of $R$. |
| title | On some Rings of differentiable type |
| topic | Commutative Algebra Primary 13N10, Secondary 13N15, 13D45 |
| url | https://arxiv.org/abs/2406.05390 |