Lemma 24.23.6. Let (\mathcal{C}, \mathcal{O}) be a ringed site. Let \mathcal{A} be a differential graded \mathcal{A}-algebra. Let \mathcal{S} be a sheaf of graded sets on \mathcal{C}. Then the free graded module \mathcal{A}[\mathcal{S}] on \mathcal{S} endowed with differential as in Remark 24.23.5 is a good differential graded \mathcal{A}-module.
Proof. Let \mathcal{N} be a left graded \mathcal{A}-module. Then we have
\mathcal{A}[\mathcal{S}] \otimes _\mathcal {A} \mathcal{N} = \mathcal{O}[\mathcal{S}] \otimes _\mathcal {O} \mathcal{N} = \mathcal{N}[\mathcal{S}]
where \mathcal{N}[\mathcal{S} is the graded \mathcal{O}-module whose degree n part is the sheaf associated to the presheaf
U \longmapsto \bigoplus \nolimits _{s \in \mathcal{S}(U)} s \cdot \mathcal{N}^{n - \deg (s)}(U)
It is clear that \mathcal{N} \to \mathcal{N}[\mathcal{S}] is an exact functor, hence \mathcal{A}[\mathcal{S} is flat as a graded \mathcal{A}-module. Next, suppose that \mathcal{N} is a differential graded left \mathcal{A}-module. Then we have
H^*(\mathcal{A}[\mathcal{S}] \otimes _\mathcal {A} \mathcal{N}) = H^*(\mathcal{O}[\mathcal{S}] \otimes _\mathcal {O} \mathcal{N})
as graded sheaves of \mathcal{O}-modules, which by the flatness (over \mathcal{O}) is equal to
H^*(\mathcal{N})[\mathcal{S}]
as a graded \mathcal{O}-module. Hence if \mathcal{N} is acyclic, then \mathcal{A}[\mathcal{S}] \otimes _\mathcal {A} \mathcal{N} is acyclic.
Finally, consider 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. Then it is straightforward to see that
f^*\mathcal{A}[\mathcal{S}] = \mathcal{A}'[f^{-1}\mathcal{S}]
which finishes the proof that our module is good. \square
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