Lemma 47.9.3 (0ALZ)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 47: Dualizing Complexes
Section 47.9: Local cohomology
Lemma 47.9.3 (cite)

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Lemma 47.9.3. Let $A \to B$ be a ring homomorphism and let $I \subset A$ be a finitely generated ideal. Set $J = IB$ . Let $Z = V(I)$ and $Y = V(J)$ . Then

$R\Gamma _ Z(K) \otimes _ A^\mathbf {L} B = R\Gamma _ Y(K \otimes _ A^\mathbf {L} B)$

functorially in $K \in D(A)$ .

Proof. Write $I = (f_1, \ldots , f_ r)$ . Then $J$ is generated by the images $g_1, \ldots , g_ r \in B$ of $f_1, \ldots , f_ r$ . Then we have

$(A \to \prod A_{f_{i_0}} \to \ldots \to A_{f_1\ldots f_ r}) \otimes _ A B = (B \to \prod B_{g_{i_0}} \to \ldots \to B_{g_1\ldots g_ r})$

as complexes of $B$ -modules. Represent $K$ by a K-flat complex $K^\bullet$ of $A$ -modules. Since the total complexes associated to

$K^\bullet \otimes _ A (A \to \prod A_{f_{i_0}} \to \ldots \to A_{f_1\ldots f_ r}) \otimes _ A B$

and

$K^\bullet \otimes _ A B \otimes _ B (B \to \prod B_{g_{i_0}} \to \ldots \to B_{g_1\ldots g_ r})$

represent the left and right hand side of the displayed formula of the lemma (see Lemma 47.9.1) we conclude. $\square$

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