Lemma 24.23.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Then $j_!\mathcal{A}_ U$ is a good differential graded $\mathcal{A}$-module.
Proof. Let $\mathcal{N}$ be a left graded $\mathcal{A}$-module. By Lemma 24.10.2 we have
\[ j_!\mathcal{A}_ U \otimes _\mathcal {A} \mathcal{N} = j_!(\mathcal{A}_ U \otimes _{\mathcal{A}_ U} \mathcal{N}|_ U) = j_!(\mathcal{N}_ U) \]
as graded modules. Since both restriction to $U$ and $j_!$ are exact this proves condition (1). The same argument works for (2) using Lemma 24.19.2.
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. We have to show that
\[ f^*j_!\mathcal{A}_ U = f^{-1}j_!\mathcal{A}_ U \otimes _{f^{-1}\mathcal{A}} \mathcal{A}' \]
satisfies (1) and (2) for the ringed topos $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}')$ endowed with the sheaf of differential graded $\mathcal{O}'$-algebras $\mathcal{A}'$. To prove this we may replace $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ and $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}')$ by equivalent ringed topoi. Thus by Modules on Sites, Lemma 18.7.2 we may assume that $f$ comes from a morphism of sites $f : \mathcal{C} \to \mathcal{C}'$ given by the continuous functor $u : \mathcal{C} \to \mathcal{C}'$. In this case, set $U' = u(U)$ and denote $j' : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}'/U') \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}')$ the corresponding localization morphism. We obtain a commutative square of morphisms of ringed topoi
\[ \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'/U'), \mathcal{O}'_{U'}) \ar[rr]_{(j', (j')^\sharp )} \ar[d]_{(f', (f')^\sharp )} & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}') \ar[d]^{(f, f^\sharp )} \\ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U), \mathcal{O}_ U) \ar[rr]^{(j, j^\sharp )} & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}). } \]
and we have $f'_*(j')^{-1} = j^{-1}f_*$. See Modules on Sites, Lemma 18.20.1. By uniqueness of adjoints we obtain $f^{-1}j_! = j'_!(f')^{-1}$. Thus we obtain
\begin{align*} f^*j_!\mathcal{A}_ U & = f^{-1}j_!\mathcal{A}_ U \otimes _{f^{-1}\mathcal{A}} \mathcal{A}' \\ & = j'_!(f')^{-1}\mathcal{A}_ U \otimes _{f^{-1}\mathcal{A}} \mathcal{A}' \\ & = j'_!\left( (f')^{-1}\mathcal{A}_ U \otimes _{f^{-1}\mathcal{A}|_{U'}} \mathcal{A}'|_{U'}\right) \\ & = j'_!\mathcal{A}'_{U'} \end{align*}
The first equation is the definition of the pullback of $j_!\mathcal{A}_ U$ to a differential graded module over $\mathcal{A}'$. The second equation because $f^{-1}j_! = j'_!(f')^{-1}$. The third equation by Lemma 24.19.2 applied to the ringed site $(\mathcal{C}', f^{-1}\mathcal{O})$ with sheaf of differential graded algebras $f^{-1}\mathcal{A}$ and with differential graded modules $(f')^{-1}\mathcal{A}_ U$ on $\mathcal{C}'/U'$ and $\mathcal{A}'$ on $\mathcal{C}'$. The fourth equation holds because of course we have $(f')^{-1}\mathcal{A}_ U = f^{-1}\mathcal{A}|_{U'}$. Hence we see that the pullback is another module of the same kind and we've proven conditions (1) and (2) for it above. $\square$
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