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A note on Deligne's formula

Peter Schenzel

Abstract

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.

A note on Deligne's formula

Abstract

Let denote a Noetherian ring and an ideal with . For an -module there is an isomorphism 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 -module in the non-Noetherian case of and a certain finitely generated ideal. Moreover, we recall a corresponding sheaf construction.

Paper Structure

This paper contains 5 sections, 8 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Recalls about Sequences
  3. The Homomorphisms
  4. Proof of the Main Results
  5. The associated sheaf

Key Result

Theorem 1.1

Let $J = (x_1,\ldots,x_k)R$ denote a finitely generated ideal in a commutative ring $R$. For an $R$-module $M$ there is a commutative diagram \xymatrix{ & \mathcal{D}_J(M) \ar[dl]_{\theta_M} \ar[dr]^{\rho_M} & \\ \varprojlim_{x \in J}M_x \ar[rr]^-{\sigma_M} & & \hole \check{D}^0_{\underli

Theorems & Definitions (17)

  • Theorem 1.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • ...and 7 more