Lemma 47.10.2 (0956)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 47: Dualizing Complexes
Section 47.10: Local cohomology for Noetherian rings
Lemma 47.10.2 (cite)

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Lemma 47.10.2. Let $A$ be a Noetherian ring and let $I = (f_1, \ldots , f_ r)$ be an ideal of $A$ . Set $Z = V(I) \subset \mathop{\mathrm{Spec}}(A)$ . There are canonical isomorphisms

$R\Gamma _ I(A) \to (A \to \prod \nolimits _{i_0} A_{f_{i_0}} \to \prod \nolimits _{i_0 < i_1} A_{f_{i_0}f_{i_1}} \to \ldots \to A_{f_1\ldots f_ r}) \to R\Gamma _ Z(A)$

in $D(A)$ . If $M$ is an $A$ -module, then we have similarly

$R\Gamma _ I(M) \cong (M \to \prod \nolimits _{i_0} M_{f_{i_0}} \to \prod \nolimits _{i_0 < i_1} M_{f_{i_0}f_{i_1}} \to \ldots \to M_{f_1\ldots f_ r}) \cong R\Gamma _ Z(M)$

in $D(A)$ .

Proof. This follows from Lemma 47.10.1 and the computation of the functor $R\Gamma _ Z$ in Lemma 47.9.1. $\square$

Comments (3)

Comment #3596 by Kestutis Cesnavicius on September 28, 2018 at 04:28

One should say in the statement of the lemma what $Z$ is. It would also be good to mention that $I$ is an ideal of $A$ .

Comment #3598 by Kestutis Cesnavicius on September 28, 2018 at 04:39

Also, it would be nice to state this with coefficients, that is, for $R\Gamma_I(A, M)$ , etc. (Sorry for a double comment.)

Comment #3713 by Johan on October 22, 2018 at 19:26

Thanks and changes are here.

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