Lemma 24.23.2. et (\mathcal{C}, \mathcal{O}) be a ringed site. Let \mathcal{A} be a sheaf of differential graded algebras on (\mathcal{C}, \mathcal{O}). Let 0 \to \mathcal{P} \to \mathcal{P}' \to \mathcal{P}'' \to 0 be an admissible short exact sequence of differential graded \mathcal{A}-modules. If two-out-of-three of these modules are good, so is the third.
Proof. For condition (1) this is immediate as the sequence is a direct sum at the graded level. For condition (2) note that for any left differential graded \mathcal{A}-module, the sequence
0 \to \mathcal{P} \otimes _\mathcal {A} \mathcal{N} \to \mathcal{P}' \otimes _\mathcal {A} \mathcal{N} \to \mathcal{P}'' \otimes _\mathcal {A} \mathcal{N} \to 0
is an admissible short exact sequence of differential graded \mathcal{O}-modules (since forgetting the differential the tensor product is just taken in the category of graded modules). Hence if two out of three are exact as complexes of \mathcal{O}-modules, so is the third. Finally, the same argument shows that given a morphism (f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}') \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) of ringed topoi, a differential graded \mathcal{O}'-algebra \mathcal{A}', and a map \varphi : f^{-1}\mathcal{A} \to \mathcal{A}' of differential graded f^{-1}\mathcal{O}-algebras we have that
0 \to f^*\mathcal{P} \to f^*\mathcal{P}' \to f^*\mathcal{P}'' \to 0
is an admissible short exact sequence of differential graded \mathcal{A}'-modules and the same argument as above applies here. \square
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