A note on Deligne's formula

Abstract

Let R denote a Noetherian ring and an ideal $J\subset R$ with $U=\mathrm{Spec}R\setminus V\left(J\right)$. For an R-module M there is an isomorphism $\Gamma (U,\tilde{M})\cong \underrightarrow{\lim }{\mathrm{Hom}}_R(J^n,M)$ known as Deligne's formula (see [8, p. 217] and Deligne's Appendix in [7]). We extend the isomorphism for any R-module M in the non-Noetherian case of R and $J=(x_1,\dots ,x_k)$ a certain finitely generated ideal. Moreover, we recall a corresponding sheaf construction.