The Stacks project

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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