## 24.18 Pull and push for sheaves of differential graded modules

Let $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. Let $\mathcal{A}$ be a differential graded $\mathcal{O}_\mathcal {C}$-algebra. Let $\mathcal{B}$ be a differential graded $\mathcal{O}_\mathcal {D}$-algebra. Suppose we are given a map

$\varphi : f^{-1}\mathcal{B} \to \mathcal{A}$

of differential graded $f^{-1}\mathcal{O}_\mathcal {D}$-algebras. By the adjunction of restriction and extension of scalars, this is the same thing as a map $\varphi : f^*\mathcal{B} \to \mathcal{A}$ of differential graded $\mathcal{O}_\mathcal {C}$-algebras or equivalently $\varphi$ can be viewed as a map

$\varphi : \mathcal{B} \to f_*\mathcal{A}$

of differential graded $\mathcal{O}_\mathcal {D}$-algebras. See Remark 24.12.2.

Let us define a functor

$f_* : \textit{Mod}(\mathcal{A}, \text{d}) \longrightarrow \textit{Mod}(\mathcal{B}, \text{d})$

Given a differential graded $\mathcal{A}$-module $\mathcal{M}$ we define $f_*\mathcal{M}$ to be the graded $\mathcal{B}$-module constructed in Section 24.9 with differential given by the maps $f_*d : f_*\mathcal{M}^ n \to f_*\mathcal{M}^{n + 1}$. The construction is clearly functorial in $\mathcal{M}$ and we obtain our functor.

Let us define a functor

$f^* : \textit{Mod}(\mathcal{B}, \text{d}) \longrightarrow \textit{Mod}(\mathcal{A}, \text{d})$

Given a differential graded $\mathcal{B}$-module $\mathcal{N}$ we define $f^*\mathcal{N}$ to be the graded $\mathcal{A}$-module constructed in Section 24.9. Recall that

$f^*\mathcal{N} = f^{-1}\mathcal{N} \otimes _{f^{-1}\mathcal{B}} \mathcal{A}$

Since $f^{-1}\mathcal{N}$ comes with the differentials $f^{-1}\text{d} : f^{-1}\mathcal{N}^ n \to f^{-1}\mathcal{N}^{n + 1}$ we can view this tensor product as an example of the tensor product discussed in Section 24.17 which provides us with a differential. The construction is clearly functorial in $\mathcal{N}$ and we obtain our functor $f^*$.

The functors $f_*$ and $f^*$ are readily enhanced to give functors of differential graded categories

$f_* : \textit{Mod}^{dg}(\mathcal{A}, \text{d}) \longrightarrow \textit{Mod}^{dg}(\mathcal{B}, \text{d}) \quad \text{and}\quad f^* : \textit{Mod}^{dg}(\mathcal{B}, \text{d}) \longrightarrow \textit{Mod}^{dg}(\mathcal{A}, \text{d})$

which do the same thing on underlying objects and are defined by functoriality of the constructions on homogenous morphisms of degree $n$.

Lemma 24.18.1. In the situation above we have

$\mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{dg}(\mathcal{B}, \text{d})}( \mathcal{N}, f_*\mathcal{M}) = \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{dg}(\mathcal{A}, \text{d})}( f^*\mathcal{N}, \mathcal{M})$

Proof. Omitted. Hints: This is true for the underlying graded categories by Lemma 24.9.1. A calculation shows that these isomorphisms are compatible with differentials. $\square$

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