Lemma 20.46.9. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{E}^\bullet $, $\mathcal{F}^\bullet $ be complexes of $\mathcal{O}_ X$-modules with $\mathcal{E}^\bullet $ strictly perfect. Then the internal hom $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{E}^\bullet , \mathcal{F}^\bullet )$ is represented by the complex $\mathcal{H}^\bullet $ with terms
\[ \mathcal{H}^ n = \bigoplus \nolimits _{n = p + q} \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{E}^{-q}, \mathcal{F}^ p) \]
and differential as described in Section 20.41.
Proof.
Choose a quasi-isomorphism $\mathcal{F}^\bullet \to \mathcal{I}^\bullet $ into a K-injective complex. Let $(\mathcal{H}')^\bullet $ be the complex with terms
\[ (\mathcal{H}')^ n = \prod \nolimits _{n = p + q} \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{E}^{-q}, \mathcal{I}^ p) \]
which represents $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{E}^\bullet , \mathcal{F}^\bullet )$ by the construction in Section 20.42. It suffices to show that the map
\[ \mathcal{H}^\bullet \longrightarrow (\mathcal{H}')^\bullet \]
is a quasi-isomorphism. Given an open $U \subset X$ we have by inspection
\[ H^0(\mathcal{H}^\bullet (U)) = \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O}_ U)}(\mathcal{E}^\bullet |_ U, \mathcal{K}^\bullet |_ U) \to H^0((\mathcal{H}')^\bullet (U)) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}(\mathcal{E}^\bullet |_ U, \mathcal{K}^\bullet |_ U) \]
By Lemma 20.46.8 the sheafification of $U \mapsto H^0(\mathcal{H}^\bullet (U))$ is equal to the sheafification of $U \mapsto H^0((\mathcal{H}')^\bullet (U))$. A similar argument can be given for the other cohomology sheaves. Thus $\mathcal{H}^\bullet $ is quasi-isomorphic to $(\mathcal{H}')^\bullet $ which proves the lemma.
$\square$
Comments (1)
Comment #8625 by nkym on