The Stacks project

Lemma 21.34.9. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $L$ be an object of $D(\mathcal{O})$. Set $L^\vee = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, \mathcal{O})$. For $M$ in $D(\mathcal{O})$ there is a canonical map

21.34.9.1
\begin{equation} \label{sites-cohomology-equation-eval} L^\vee \otimes ^\mathbf {L}_\mathcal {O} M \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, M) \end{equation}

which induces a canonical map

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

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

Proof. The map (21.34.9.1) is a special case of Lemma 21.34.6 using the identification $M = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}, M)$. $\square$


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