Lemma 22.35.4. Let $R$ be a ring. Let $\mathcal{C}$ be a site. Let $\mathcal{O}$ be a sheaf of commutative $R$-algebras. Let $K^\bullet $ be a complex of $\mathcal{O}$-modules. The functor of Lemma 22.35.3 has the following property: For every $M$, $N$ in $D(E, \text{d})$ there is a canonical map
\[ R\mathop{\mathrm{Hom}}\nolimits (M, N) \longrightarrow R\mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(M \otimes _ E^\mathbf {L} K^\bullet , N \otimes _ E^\mathbf {L} K^\bullet ) \]
in $D(R)$ which on cohomology modules gives the maps
\[ \mathop{\mathrm{Ext}}\nolimits ^ n_{D(E, \text{d})}(M, N) \to \mathop{\mathrm{Ext}}\nolimits ^ n_{D(\mathcal{O})} (M \otimes _ E^\mathbf {L} K^\bullet , N \otimes _ E^\mathbf {L} K^\bullet ) \]
induced by the functor $- \otimes _ E^\mathbf {L} K^\bullet $.
Proof.
The right hand side of the arrow is the global derived hom introduced in Cohomology on Sites, Section 21.36 which has the correct cohomology modules. For the left hand side we think of $M$ as a $(R, A)$-bimodule and we have the derived $\mathop{\mathrm{Hom}}\nolimits $ introduced in Section 22.31 which also has the correct cohomology modules. To prove the lemma we may assume $M$ and $N$ are differential graded $E$-modules with property (P); this does not change the left hand side of the arrow by Lemma 22.31.3. By Lemma 22.31.5 this means that the left hand side of the arrow becomes $\mathop{\mathrm{Hom}}\nolimits _{\text{Mod}^{dg}_{(B, \text{d})}}(M, N)$. In Lemmas 22.35.1, 22.35.2, and 22.35.3 we have constructed a functor
\[ - \otimes _ E K^\bullet : \text{Mod}^{dg}_{(E, \text{d})} \longrightarrow \text{Comp}^{dg}(\mathcal{O}) \]
of differential graded categories and we have shown that $- \otimes _ E^\mathbf {L} K^\bullet $ is computed by evaluating this functor on differential graded $E$-modules with property (P). Hence we obtain a map of complexes of $R$-modules
\[ \mathop{\mathrm{Hom}}\nolimits _{\text{Mod}^{dg}_{(B, \text{d})}}(M, N) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{\text{Comp}^{dg}(\mathcal{O})} (M \otimes _ E K^\bullet , N \otimes _ E K^\bullet ) \]
For any complexes of $\mathcal{O}$-modules $\mathcal{F}^\bullet $, $\mathcal{G}^\bullet $ there is a canonical map
\[ \mathop{\mathrm{Hom}}\nolimits _{\text{Comp}^{dg}(\mathcal{O})} (\mathcal{F}^\bullet , \mathcal{G}^\bullet ) = \Gamma (\mathcal{C}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{F}^\bullet , \mathcal{G}^\bullet )) \longrightarrow R\mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(\mathcal{F}^\bullet , \mathcal{G}^\bullet ). \]
Combining these maps we obtain the desired map of the lemma.
$\square$
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