Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and denote
the corresponding localization morphism (Modules on Sites, Section 18.19). Below we will use the following fact: for $\mathcal{O}_ U$-modules $\mathcal{M}_ i$, $i = 1, 2$ and a $\mathcal{O}$-module $\mathcal{A}$ there is a canonical map
Namely, we have $j_!(\mathcal{M}_1 \otimes _{\mathcal{O}_ U} \mathcal{A}|_ U) = j_!\mathcal{M}_1 \otimes _\mathcal {O} \mathcal{A}$ by Modules on Sites, Lemma 18.27.9.
Let $\mathcal{A}$ be a differential graded $\mathcal{O}$-algebra. We will denote $\mathcal{A}_ U$ the restriction of $\mathcal{A}$ to $\mathcal{C}/U$, in other words, we have $\mathcal{A}_ U = j^*\mathcal{A} = j^{-1}\mathcal{A}$. In Section 24.18 we have constructed adjoint functors
with $j^*$ left adjoint to $j_*$. We claim there is in addition an exact functor
right adjoint to $j_*$. Namely, given a differential graded $\mathcal{A}_ U$-module $\mathcal{M}$ we define $j_!\mathcal{M}$ to be the graded $\mathcal{A}$-module constructed in Section 24.10 with differentials $j_!\text{d} : j_!\mathcal{M}^ n \to j_!\mathcal{M}^{n + 1}$. Given a homogeneous map $f : \mathcal{M} \to \mathcal{M}'$ of degree $n$ of differential graded $\mathcal{A}_ U$-modules, we obtain a homogeneous map $j_!f : j_!\mathcal{M} \to j_!\mathcal{M}'$ of degree $n$ of differential graded $\mathcal{A}$-modules. We omit the straightforward verification that this construction is compatible with differentials. Thus we obtain our functor.
Lemma 24.19.1. In the situation above we have
\[ \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{dg}(\mathcal{A}, \text{d})}( j_!\mathcal{M}, \mathcal{N}) = \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{dg}(\mathcal{A}_ U, \text{d})}( \mathcal{M}, j^*\mathcal{N}) \]
Proof. Omitted. Hint: We have seen in Lemma 24.10.1 that the lemma is true on graded level. Thus all that needs to be checked is that the resulting isomorphism is compatible with differentials. $\square$
Lemma 24.19.2. In the situation above, let $\mathcal{M}$ be a right differential graded $\mathcal{A}_ U$-module and let $\mathcal{N}$ be a left differential graded $\mathcal{A}$-module. Then as complexes of $\mathcal{O}$-modules functorially in $\mathcal{M}$ and $\mathcal{N}$.
\[ j_!\mathcal{M} \otimes _\mathcal {A} \mathcal{N} = j_!(\mathcal{M} \otimes _{\mathcal{A}_ U} \mathcal{N}|_ U) \]
Proof. As graded modules, this follows from Lemma 24.10.2. We omit the verification that this isomorphism is compatible with differentials. $\square$
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