The Stacks project

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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