Lemma 20.42.8 (08I1)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 20: Cohomology of Sheaves
Section 20.42: Internal hom in the derived category
Lemma 20.42.8 (cite)

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Lemma 20.42.8. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $L$ be an object of $D(\mathcal{O}_ X)$ . Set $L^\vee = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, \mathcal{O}_ X)$ . For $M$ in $D(\mathcal{O}_ X)$ there is a canonical map

20.42.8.1

$\begin{equation} \label{cohomology-equation-eval} M \otimes ^\mathbf {L}_{\mathcal{O}_ X} L^\vee \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, M) \end{equation}$

which induces a canonical map

$H^0(X, M \otimes ^\mathbf {L}_{\mathcal{O}_ X} L^\vee ) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(L, M)$

functorial in $M$ in $D(\mathcal{O}_ X)$ .

Proof. The map (20.42.8.1) is a special case of Lemma 20.42.5 using the identification $M = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ X, M)$ . $\square$

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