A note on Deligne's formula

TL;DR

This paper extends Deligne's formula, originally valid for Noetherian rings, to non-Noetherian rings with finitely generated ideals, and discusses related sheaf constructions.

Contribution

It generalizes Deligne's isomorphism to non-Noetherian rings and explores associated sheaf constructions.

Findings

Extended Deligne's formula to non-Noetherian rings.

Provided sheaf construction for the extended isomorphism.

Clarified conditions under which the isomorphism holds.

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 $\Gamma(U, \tilde{M}) \cong \varinjlim \operatorname{Hom}_R(J^n,M)$ 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.