Lemma 24.28.1. In the situation above, the functor (24.28.0.1) composed with the localization functor $K(\textit{Mod}(\mathcal{A}', \text{d})) \to D(\mathcal{A}', \text{d})$ has a left derived extension $D(\mathcal{B}, \text{d}) \to D(\mathcal{A}', \text{d})$ whose value on a good right differential graded $\mathcal{B}$-module $\mathcal{P}$ is $f^*\mathcal{P} \otimes _\mathcal {A} \mathcal{N}$.
24.28 Derived pullback
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.
In addition to the above, let $\mathcal{A}'$ be a second differential graded $\mathcal{O}_\mathcal {C}$-algebra and let $\mathcal{N}$ be a differential graded $(\mathcal{A}, \mathcal{A}')$-bimodule. In this setting we can consider the functor
\[ \textit{Mod}(\mathcal{B}, \text{d}) \longrightarrow \textit{Mod}(\mathcal{A}', \text{d}),\quad \mathcal{M} \longmapsto f^*\mathcal{M} \otimes _{\mathcal{A}} \mathcal{N} \]
Observe that this extends to a functor
\[ \textit{Mod}^{dg}(\mathcal{B}, \text{d}) \longrightarrow \textit{Mod}^{dg}(\mathcal{A}', \text{d}),\quad \mathcal{M} \longmapsto f^*\mathcal{M} \otimes _{\mathcal{A}} \mathcal{N} \]
of differential graded categories by the discussion in Sections 24.18 and 24.17. It follows formally that we also obtain an exact functor
24.28.0.1
\begin{equation} \label{sdga-equation-pullback} K(\textit{Mod}(\mathcal{B}, \text{d})) \longrightarrow K(\textit{Mod}(\mathcal{A}', \text{d})),\quad \mathcal{M} \longmapsto f^*\mathcal{M} \otimes _{\mathcal{A}} \mathcal{N} \end{equation}
of triangulated categories.
Proof. Recall that for any (right) differential graded $\mathcal{B}$-module $\mathcal{M}$ there exists a quasi-isomorphism $\mathcal{P} \to \mathcal{M}$ with $\mathcal{P}$ a good differential graded $\mathcal{B}$-module. See Lemma 24.23.7. Hence by Derived Categories, Lemma 13.14.15 it suffices to show that given a quasi-isomorphism $\mathcal{P} \to \mathcal{P}'$ of good differential graded $\mathcal{B}$-modules the induced map
\[ f^*\mathcal{P} \otimes _\mathcal {A} \mathcal{N} \longrightarrow f^*\mathcal{P}' \otimes _\mathcal {A} \mathcal{N} \]
is a quasi-isomorphism. The cone $\mathcal{P}''$ on $\mathcal{P} \to \mathcal{P}'$ is a good differential graded $\mathcal{A}$-module by Lemma 24.23.2. Since we have a distinguished triangle
\[ \mathcal{P} \to \mathcal{P}' \to \mathcal{P}'' \to \mathcal{P}[1] \]
in $K(\textit{Mod}(\mathcal{B}, \text{d}))$ we obtain a distinguished triangle
\[ f^*\mathcal{P} \otimes _\mathcal {A} \mathcal{N} \to f^*\mathcal{P}' \otimes _\mathcal {A} \mathcal{N} \to f^*\mathcal{P}'' \otimes _\mathcal {A} \mathcal{N} \to f^*\mathcal{P}[1] \otimes _\mathcal {A} \mathcal{N} \]
in $K(\textit{Mod}(\mathcal{A}', \text{d}))$. By Lemma 24.23.8 the differential graded module $f^*\mathcal{P}'' \otimes _\mathcal {A} \mathcal{N}$ is acyclic and the proof is complete. $\square$
Definition 24.28.2. Derived tensor product and derived pullback.
Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$, $\mathcal{B}$ be differential graded $\mathcal{O}$-algebras. Let $\mathcal{N}$ be a differential graded $(\mathcal{A}, \mathcal{B})$-bimodule. The functor $D(\mathcal{A}, \text{d}) \to D(\mathcal{B}, \text{d})$ constructed in Lemma 24.28.1 is called the derived tensor product and denoted $- \otimes _\mathcal {A}^\mathbf {L} \mathcal{N}$.
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. Let $\varphi : \mathcal{B} \to f_*\mathcal{A}$ be a homomorphism of differential graded $\mathcal{O}_\mathcal {D}$-algebras. The functor $D(\mathcal{B}, \text{d}) \to D(\mathcal{A}, \text{d})$ constructed in Lemma 24.28.1 is called derived pullback and denote $Lf^*$.
With this language in place we can express some obvious compatibilities.
Lemma 24.28.3. In Lemma 24.28.1 the functor $D(\mathcal{B}, \text{d}) \to D(\mathcal{A}', \text{d})$ is equal to $\mathcal{M} \mapsto Lf^*\mathcal{M} \otimes _\mathcal {A}^\mathbf {L} \mathcal{N}$.
Proof. Immediate from the fact that we can compute these functors by representing objects by good differential graded modules and because $f^*\mathcal{P}$ is a good differential graded $\mathcal{A}$-module if $\mathcal{P}$ is a good differential graded $\mathcal{B}$-module. $\square$
Lemma 24.28.4. Let $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}')$ and $(g, g^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}') \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}''), \mathcal{O}'')$ be morphisms of ringed topoi. Let $\mathcal{A}$, $\mathcal{A}'$, and $\mathcal{A}''$ be a differential graded $\mathcal{O}$-algebra, $\mathcal{O}'$-algebra, and $\mathcal{O}''$-algebra. Let $\varphi : \mathcal{A}' \to f_*\mathcal{A}$ and $\varphi ' : \mathcal{A}'' \to g_*\mathcal{A}'$ be a homomorphism of differential graded $\mathcal{O}'$-algebras and $\mathcal{O}''$-algebras. Then we have $L(g \circ f)^* = Lf^* \circ Lg^* : D(\mathcal{A}'', \text{d}) \to D(\mathcal{A}, \text{d})$.
Proof. Immediate from the fact that we can compute these functors by representing objects by good differential graded modules and because $f^*\mathcal{P}$ is a good differential graded $\mathcal{A}'$-module of $\mathcal{P}$ is a good differential graded $\mathcal{A}$-module. $\square$
Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$, $\mathcal{B}$ be differential graded $\mathcal{O}$-algebras. Let $\mathcal{N} \to \mathcal{N}'$ be a homomorphism of differential graded $(\mathcal{A}, \mathcal{B})$-bimodules. Then we obtain canonical maps
\[ t : \mathcal{M} \otimes _\mathcal {A}^\mathbf {L} \mathcal{N} \longrightarrow \mathcal{M} \otimes _\mathcal {A}^\mathbf {L} \mathcal{N}' \]
functorial in $\mathcal{M}$ in $D(\mathcal{A}, \text{d})$ which define a natural transformation between exact functors $D(\mathcal{A}, \text{d}) \to D(\mathcal{B}, \text{d})$ of triangulated categories. The value of $t$ on a good differential graded $\mathcal{A}$-module $\mathcal{P}$ is the obvious map
\[ \mathcal{P} \otimes _\mathcal {A}^\mathbf {L} \mathcal{N} = \mathcal{P} \otimes _\mathcal {A} \mathcal{N} \longrightarrow \mathcal{P} \otimes _\mathcal {A} \mathcal{N}' = \mathcal{P} \otimes _\mathcal {A}^\mathbf {L} \mathcal{N}' \]
Lemma 24.28.5. In the situation above, if $\mathcal{N} \to \mathcal{N}'$ is an isomorphism on cohomology sheaves, then $t$ is an isomorphism of functors $(- \otimes _\mathcal {A}^\mathbf {L} \mathcal{N}) \to (- \otimes _\mathcal {A}^\mathbf {L} \mathcal{N}')$.
Proof. It is enough to show that $\mathcal{P} \otimes _\mathcal {A} \mathcal{N} \to \mathcal{P} \otimes _\mathcal {A} \mathcal{N}'$ is an isomorphism on cohomology sheaves for any good differential graded $\mathcal{A}$-module $\mathcal{P}$. To do this, let $\mathcal{N}''$ be the cone on the map $\mathcal{N} \to \mathcal{N}'$ as a left differential graded $\mathcal{A}$-module, see Definition 24.22.2. (To be sure, $\mathcal{N}''$ is a bimodule too but we don't need this.) By functoriality of the tensor construction (it is a functor of differential graded categories) we see that $\mathcal{P} \otimes _\mathcal {A} \mathcal{N}''$ is the cone (as a complex of $\mathcal{O}$-modules) on the map $\mathcal{P} \otimes _\mathcal {A} \mathcal{N} \to \mathcal{P} \otimes _\mathcal {A} \mathcal{N}'$. Hence it suffices to show that $\mathcal{P} \otimes _\mathcal {A} \mathcal{N}''$ is acyclic. This follows from the fact that $\mathcal{P}$ is good and the fact that $\mathcal{N}''$ is acyclic as a cone on a quasi-isomorphism. $\square$
Lemma 24.28.6. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$, $\mathcal{B}$ be differential graded $\mathcal{O}$-algebras. Let $\mathcal{N}$ be a differential graded $(\mathcal{A}, \mathcal{B})$-bimodule. If $\mathcal{N}$ is good as a left differential graded $\mathcal{A}$-module, then we have $\mathcal{M} \otimes _\mathcal {A}^\mathbf {L} \mathcal{N} = \mathcal{M} \otimes _\mathcal {A} \mathcal{N}$ for all differential graded $\mathcal{A}$-modules $\mathcal{M}$.
Proof. Let $\mathcal{P} \to \mathcal{M}$ be a quasi-isomorphism where $\mathcal{P}$ is a good (right) differential graded $\mathcal{A}$-module. To prove the lemma we have to show that $\mathcal{P} \otimes _\mathcal {A} \mathcal{N} \to \mathcal{M} \otimes _\mathcal {A} \mathcal{N}$ is a quasi-isomorphism. The cone $C$ on the map $\mathcal{P} \to \mathcal{M}$ is an acyclic right differential graded $\mathcal{A}$-module. Hence $C \otimes _\mathcal {A} \mathcal{N}$ is acyclic as $\mathcal{N}$ is assumed good as a left differential graded $\mathcal{A}$-module. Since $C \otimes _\mathcal {A} \mathcal{N}$ is the cone on the maps $\mathcal{P} \otimes _\mathcal {A} \mathcal{N} \to \mathcal{M} \otimes _\mathcal {A} \mathcal{N}$ as a complex of $\mathcal{O}$-modules we conclude. $\square$
Lemma 24.28.7. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$, $\mathcal{A}'$, $\mathcal{A}''$ be differential graded $\mathcal{O}$-algebras. Let $\mathcal{N}$ and $\mathcal{N}'$ be a differential graded $(\mathcal{A}, \mathcal{A}')$-bimodule and $(\mathcal{A}', \mathcal{A}'')$-bimodule. Assume that the canonical map in $D(\mathcal{A}'', \text{d})$ is a quasi-isomorphism. Then we have as functors $D(\mathcal{A}, \text{d}) \to D(\mathcal{A}'', \text{d})$.
\[ \mathcal{N} \otimes _{\mathcal{A}'}^\mathbf {L} \mathcal{N}' \longrightarrow \mathcal{N} \otimes _{\mathcal{A}'} \mathcal{N}' \]
\[ (\mathcal{M} \otimes _\mathcal {A}^\mathbf {L} \mathcal{N}) \otimes _{\mathcal{A}'}^\mathbf {L} \mathcal{N}' = \mathcal{M} \otimes _\mathcal {A}^\mathbf {L} (\mathcal{N} \otimes _{\mathcal{A}'} \mathcal{N}') \]
Proof. Choose a good differential graded $\mathcal{A}$-module $\mathcal{P}$ and a quasi-isomorphism $\mathcal{P} \to \mathcal{M}$, see Lemma 24.23.7. Then
\[ \mathcal{M} \otimes _\mathcal {A}^\mathbf {L} (\mathcal{N} \otimes _{\mathcal{A}'} \mathcal{N}') = \mathcal{P} \otimes _\mathcal {A} \mathcal{N} \otimes _{\mathcal{A}'} \mathcal{N}' \]
and we have
\[ (\mathcal{M} \otimes _\mathcal {A}^\mathbf {L} \mathcal{N}) \otimes _{\mathcal{A}'}^\mathbf {L} \mathcal{N}' = (\mathcal{P} \otimes _\mathcal {A} \mathcal{N}) \otimes _{\mathcal{A}'}^\mathbf {L} \mathcal{N}' \]
Thus we have to show the canonical map
\[ (\mathcal{P} \otimes _\mathcal {A} \mathcal{N}) \otimes _{\mathcal{A}'}^\mathbf {L} \mathcal{N}' \longrightarrow \mathcal{P} \otimes _\mathcal {A} \mathcal{N} \otimes _{\mathcal{A}'} \mathcal{N}' \]
is a quasi-isomorphism. Choose a quasi-isomorphism $\mathcal{Q} \to \mathcal{N}'$ where $\mathcal{Q}$ is a good left differential graded $\mathcal{A}'$-module (Lemma 24.23.7). By Lemma 24.28.6 the map above as a map in the derived category of $\mathcal{O}$-modules is the map
\[ \mathcal{P} \otimes _\mathcal {A} \mathcal{N} \otimes _{\mathcal{A}'} \mathcal{Q} \longrightarrow \mathcal{P} \otimes _\mathcal {A} \mathcal{N} \otimes _{\mathcal{A}'} \mathcal{N}' \]
Since $\mathcal{N} \otimes _{\mathcal{A}'} \mathcal{Q} \to \mathcal{N} \otimes _{\mathcal{A}'} \mathcal{N}'$ is a quasi-isomorphism by assumption and $\mathcal{P}$ is a good differential graded $\mathcal{A}$-module this map is an quasi-isomorphism by Lemma 24.28.5 (the left and right hand side compute $\mathcal{P} \otimes _\mathcal {A}^\mathbf {L} (\mathcal{N} \otimes _{\mathcal{A}'} \mathcal{Q})$ and $\mathcal{P} \otimes _\mathcal {A}^\mathbf {L} (\mathcal{N} \otimes _{\mathcal{A}'} \mathcal{N}')$ or you can just repeat the argument in the proof of the lemma). $\square$
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