Lemma 24.29.9. Let (\mathcal{C}, \mathcal{O}) be a ringed site. Let \mathcal{A} be a differential graded \mathcal{O}-algebra. Let \mathcal{M} be a differential graded \mathcal{A}-module. Let n \in \mathbf{Z}. We have
H^ n(\mathcal{C}, \mathcal{M}) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{A}, \text{d})}(\mathcal{A}, \mathcal{M}[n])
where on the left hand side we have the cohomology of \mathcal{M} viewed as a complex of \mathcal{O}-modules.
Proof.
To prove the formula, observe that
R\Gamma (\mathcal{C}, \mathcal{M}) = \Gamma (\mathcal{C}, \mathcal{I})
where \mathcal{M} \to \mathcal{I} is a quasi-isomorphism to a graded injective and K-injective differential graded \mathcal{A}-module \mathcal{I} (combine Lemmas 24.29.1 and 24.29.8). By Lemma 24.26.7 we have
\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{A}, \text{d})}(\mathcal{A}, \mathcal{M}[n]) = \mathop{\mathrm{Hom}}\nolimits _{K(\textit{Mod}(\mathcal{A}, \text{d}))}(\mathcal{M}, \mathcal{I}[n]) = H^0(\Gamma (\mathcal{C}, \mathcal{I}[n])) = H^ n(\Gamma (\mathcal{C}, \mathcal{I}))
Combining these two results we obtain our equality.
\square
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