The Stacks project

Lemma 24.26.7. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $\mathcal{M}$ and $\mathcal{N}$ be differential graded $\mathcal{A}$-modules. Let $\mathcal{N} \to \mathcal{I}$ be a quasi-isomorphism with $\mathcal{I}$ a graded injective and K-injective differential graded $\mathcal{A}$-module. Then

\[ \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{A}, \text{d})}(\mathcal{M}, \mathcal{N}) = \mathop{\mathrm{Hom}}\nolimits _{K(\textit{Mod}(\mathcal{A}, \text{d}))}(\mathcal{M}, \mathcal{I}) \]

Proof. Since $\mathcal{N} \to \mathcal{I}$ is a quasi-isomorphism we see that

\[ \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{A}, \text{d})}(\mathcal{M}, \mathcal{N}) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{A}, \text{d})}(\mathcal{M}, \mathcal{I}) \]

In the discussion preceding Definition 24.26.4 we found, using Lemma 24.25.10, that any morphism $\mathcal{M} \to \mathcal{I}$ in $D(\mathcal{A}, \text{d})$ can be represented by a morphism $f : \mathcal{M} \to \mathcal{I}$ in $K(\textit{Mod}(\mathcal{A}, \text{d}))$. Now, if $f, f' : \mathcal{M} \to \mathcal{I}$ are two morphism in $K(\textit{Mod}(\mathcal{A}, \text{d}))$, then they define the same morphism in $D(\mathcal{A}, \text{d})$ if and only if there exists a quasi-isomorphism $g : \mathcal{I} \to \mathcal{K}$ in $K(\textit{Mod}(\mathcal{A}, \text{d}))$ such that $g \circ f = g \circ f'$, see Categories, Lemma 4.27.6. However, by Lemma 24.25.10 there exists a map $h : \mathcal{K} \to \mathcal{I}$ such that $h \circ g = \text{id}_\mathcal {I}$ in in $K(\textit{Mod}(\mathcal{A}, \text{d}))$. Thus $g \circ f = g \circ f'$ implies $f = f'$ and the proof is complete. $\square$

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