Lemma 36.6.13 (0G7T)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 36: Derived Categories of Schemes
Section 36.6: Cohomology with support in a closed subset
Lemma 36.6.13 (cite)

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Lemma 36.6.13. Let $X$ be a scheme. Let $Z \to X$ be a closed immersion of finite presentation whose conormal sheaf $\mathcal{C}_{Z/X}$ is locally free of rank $c$ . Then there is a canonical map

$c : \wedge ^ c(\mathcal{C}_{Z/X})^\vee \otimes _{\mathcal{O}_ Z} i^*\mathcal{F} \longrightarrow \mathcal{H}_ Z^ c(\mathcal{F})$

functorial in the quasi-coherent module $\mathcal{F}$ .

Proof. Follows from the construction in Remark 36.6.10 and the independence of the choice of generators of the ideal sheaf shown in Lemma 36.6.12. Some details omitted. $\square$

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