A note on Deligne's formula
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911933637066752 |
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| author | Schenzel, Peter |
| author_facet | Schenzel, Peter |
| contents | Let $R$ denote a Noetherian ring and an ideal $J \subset R$ with $U = \operatorname{Spec R} \setminus V(J)$. For an $R$-module $M$ there is an isomorphism $Γ(U, \tilde{M}) \cong \varinjlim \operatorname{Hom}_R(J^n,M)$ known as Deligne's formula (see [R. Hartshorne: Algebraic Geometry, Springer, 1983] and Deligne's Appendix in [R. Hartshorne: Residues and Duality, Lecture Notes in Math. 20, Springer,1966] ). We extend the isomorphism for any $R$-module $M$ in the non-Noetherian case of $R$ and $J = (x_1,\ldots,x_k)$ a certain finitely generated ideal. Moreover, we recall a corresponding sheaf construction. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_18185 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A note on Deligne's formula Schenzel, Peter Commutative Algebra 13D45, 14B15, 14F06, 13C11 Let $R$ denote a Noetherian ring and an ideal $J \subset R$ with $U = \operatorname{Spec R} \setminus V(J)$. For an $R$-module $M$ there is an isomorphism $Γ(U, \tilde{M}) \cong \varinjlim \operatorname{Hom}_R(J^n,M)$ known as Deligne's formula (see [R. Hartshorne: Algebraic Geometry, Springer, 1983] and Deligne's Appendix in [R. Hartshorne: Residues and Duality, Lecture Notes in Math. 20, Springer,1966] ). We extend the isomorphism for any $R$-module $M$ in the non-Noetherian case of $R$ and $J = (x_1,\ldots,x_k)$ a certain finitely generated ideal. Moreover, we recall a corresponding sheaf construction. |
| title | A note on Deligne's formula |
| topic | Commutative Algebra 13D45, 14B15, 14F06, 13C11 |
| url | https://arxiv.org/abs/2406.18185 |