The Stacks project

Lemma 24.26.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. The functor $H^0 : \textit{Mod}(\mathcal{A}, \text{d}) \to \textit{Mod}(\mathcal{O})$ of Section 24.13 factors through a functor

\[ H^0 : K(\textit{Mod}(\mathcal{A}, \text{d})) \to \textit{Mod}(\mathcal{O}) \]

which is homological in the sense of Derived Categories, Definition 13.3.5.

Proof. It follows immediately from the definitions that there is a commutative diagram

\[ \xymatrix{ \textit{Mod}(\mathcal{A}, \text{d}) \ar[r] \ar[d] & K(\textit{Mod}(\mathcal{A}, \text{d})) \ar[d] \\ \text{Comp}(\mathcal{O}) \ar[r] & K(\textit{Mod}(\mathcal{O})) } \]

Since $H^0(\mathcal{M})$ is defined as the zeroth cohomology sheaf of the underlying complex of $\mathcal{O}$-modules of $\mathcal{M}$ the lemma follows from the case of complexes of $\mathcal{O}$-modules which is a special case of Derived Categories, Lemma 13.11.1. $\square$

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