Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $I$ be an ideal of a commutative ring $R$. $M$ be an $R$-module. In Local cohomology: an algebraic introduction with geometric applications of Brodmann M. P., Sharp R. Y we have $$D_I(M)=\mathop {\lim }\limits_{\begin{subarray}{c} \longrightarrow \\ \end{subarray}} Hom_R(I^{n},M) $$ called ideal transform of $M$ respect to $I$ or $I-transform$ of $M$. I heard someone talked about Deligne's formula but i can not find it. Can anyone help me to find it? I think that, it is $$D_I(M)=\mathop {\lim }\limits_{\begin{subarray}{c} \longrightarrow \\ \end{subarray}} M_a $$ where $M_a$ is the localization of $M$ with respect to multiplicative systems of powers of a single $M$-regular element $a$ in $I$.

share|cite|improve this question

1 Answer 1

up vote 3 down vote accepted

I've seen the following called Deligne's formula (it is in Hartshorne, Algebraic Geometry, Chapter III, Exercise 3.7), and I think essentially answers your question.

It says that if $Z = V(I)$ (the closed subset of $X = \text{Spec} R$) and $U = X \setminus Z$, then $$ \Gamma(U, \widetilde{M}) = \lim_{\to} \text{ Hom}_R(I^n, M) $$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.