The Stacks project

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21.19 Derived pullback

Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}')$ be a morphism of ringed topoi. We can use K-flat resolutions to define a derived pullback functor

\[ Lf^* : D(\mathcal{O}') \to D(\mathcal{O}) \]

Lemma 21.19.1. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C})$ be a ringed topos. For any complex of $\mathcal{O}_\mathcal {C}$-modules $\mathcal{G}^\bullet $ there exists a quasi-isomorphism $\mathcal{K}^\bullet \to \mathcal{G}^\bullet $ such that $f^*\mathcal{K}^\bullet $ is a K-flat complex of $\mathcal{O}_\mathcal {D}$-modules for any morphism $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C})$ of ringed topoi.

Proof. In the proof of Lemma 21.18.10 we find a quasi-isomorphism $\mathcal{K}^\bullet = \mathop{\mathrm{colim}}\nolimits _ i \mathcal{K}_ i^\bullet \to \mathcal{G}^\bullet $ where each $\mathcal{K}_ i^\bullet $ is a bounded above complex of flat $\mathcal{O}_\mathcal {C}$-modules. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C})$ be a morphism of ringed topoi. By Modules on Sites, Lemma 18.38.1 we see that $f^*\mathcal{F}_ i^\bullet $ is a bounded above complex of flat $\mathcal{O}_\mathcal {D}$-modules. Hence $f^*\mathcal{K}^\bullet = \mathop{\mathrm{colim}}\nolimits _ i f^*\mathcal{K}_ i^\bullet $ is K-flat by Lemmas 21.18.7 and 21.18.8. $\square$

Lemma 21.19.2. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}')$ be a morphism of ringed topoi. There exists an exact functor

\[ Lf^* : D(\mathcal{O}') \longrightarrow D(\mathcal{O}) \]

of triangulated categories so that $Lf^*\mathcal{K}^\bullet = f^*\mathcal{K}^\bullet $ for any complex as in Lemma 21.19.1 and in particular for any bounded above complex of flat $\mathcal{O}'$-modules.

Proof. To see this we use the general theory developed in Derived Categories, Section 13.15. Set $\mathcal{D} = K(\mathcal{O}')$ and $\mathcal{D}' = D(\mathcal{O})$. Let us write $F : \mathcal{D} \to \mathcal{D}'$ the exact functor of triangulated categories defined by the rule $F(\mathcal{G}^\bullet ) = f^*\mathcal{G}^\bullet $. We let $S$ be the set of quasi-isomorphisms in $\mathcal{D} = K(\mathcal{O}')$. This gives a situation as in Derived Categories, Situation 13.15.1 so that Derived Categories, Definition 13.15.2 applies. We claim that $LF$ is everywhere defined. This follows from Derived Categories, Lemma 13.15.15 with $\mathcal{P} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$ the collection of complexes $\mathcal{K}^\bullet $ as in Lemma 21.19.1. Namely, (1) follows from Lemma 21.19.1 and to see (2) we have to show that for a quasi-isomorphism $\mathcal{K}_1^\bullet \to \mathcal{K}_2^\bullet $ between elements of $\mathcal{P}$ the map $f^*\mathcal{K}_1^\bullet \to f^*\mathcal{K}_2^\bullet $ is a quasi-isomorphism. To see this write this as

\[ f^{-1}\mathcal{K}_1^\bullet \otimes _{f^{-1}\mathcal{O}'} \mathcal{O} \longrightarrow f^{-1}\mathcal{K}_2^\bullet \otimes _{f^{-1}\mathcal{O}'} \mathcal{O} \]

The functor $f^{-1}$ is exact, hence the map $f^{-1}\mathcal{K}_1^\bullet \to f^{-1}\mathcal{K}_2^\bullet $ is a quasi-isomorphism. The complexes $f^{-1}\mathcal{K}_1^\bullet $ and $f^{-1}\mathcal{K}_2^\bullet $ are K-flat complexes of $f^{-1}\mathcal{O}'$-modules by our choice of $\mathcal{P}$ because we can consider the morphism of ringed topoi $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), f^{-1}\mathcal{O}') \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}')$. Hence Lemma 21.18.11 guarantees that the displayed map is a quasi-isomorphism. Thus we obtain a derived functor

\[ LF : D(\mathcal{O}') = S^{-1}\mathcal{D} \longrightarrow \mathcal{D}' = D(\mathcal{O}) \]

see Derived Categories, Equation ( Finally, Derived Categories, Lemma 13.15.15 also guarantees that $LF(\mathcal{K}^\bullet ) = F(\mathcal{K}^\bullet ) = f^*\mathcal{K}^\bullet $ when $\mathcal{K}^\bullet $ is in $\mathcal{P}$. Since the proof of Lemma 21.19.1 shows that bounded above complexes of flat modules are in $\mathcal{P}$ we win. $\square$

Lemma 21.19.3. Consider morphisms of ringed topoi $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ and $g : (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{E}), \mathcal{O}_\mathcal {E})$. Then $Lf^* \circ Lg^* = L(g \circ f)^*$ as functors $D(\mathcal{O}_\mathcal {E}) \to D(\mathcal{O}_\mathcal {C})$.

Proof. Let $E$ be an object of $D(\mathcal{O}_\mathcal {E})$. By construction $Lg^*E$ is computed by choosing a complex $\mathcal{K}^\bullet $ as in Lemma 21.19.1 representing $E$ and setting $Lg^*E = g^*\mathcal{K}^\bullet $. By transitivity of pullback functors the complex $g^*\mathcal{K}^\bullet $ pulled back by any morphism of ringed topoi $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}') \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ is K-flat. Hence $g^*\mathcal{K}^\bullet $ is a complex as in Lemma 21.19.1 representing $Lg^*E$. We conclude $Lf^*Lg^*E$ is given by $f^*g^*\mathcal{K}^\bullet = (g \circ f)^*\mathcal{K}^\bullet $ which also represents $L(g \circ f)^*E$. $\square$

Lemma 21.19.4. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ be a morphism of ringed topoi. There is a canonical bifunctorial isomorphism

\[ Lf^*( \mathcal{F}^\bullet \otimes _{\mathcal{O}'}^{\mathbf{L}} \mathcal{G}^\bullet ) = Lf^*\mathcal{F}^\bullet \otimes _{\mathcal{O}}^{\mathbf{L}} Lf^*\mathcal{G}^\bullet \]

for $\mathcal{F}^\bullet , \mathcal{G}^\bullet \in \mathop{\mathrm{Ob}}\nolimits (D(\mathcal{O}'))$.

Proof. By Lemma 21.19.1 we may assume that $\mathcal{F}^\bullet $ and $\mathcal{G}^\bullet $ are K-flat complexes of $\mathcal{O}'$-modules such that $f^*\mathcal{F}^\bullet $ and $f^*\mathcal{G}^\bullet $ are K-flat complexes of $\mathcal{O}$-modules. In this case $\mathcal{F}^\bullet \otimes _{\mathcal{O}'}^{\mathbf{L}} \mathcal{G}^\bullet $ is just the total complex associated to the double complex $\mathcal{F}^\bullet \otimes _{\mathcal{O}'} \mathcal{G}^\bullet $. By Lemma 21.18.5 $\text{Tot}(\mathcal{F}^\bullet \otimes _{\mathcal{O}'} \mathcal{G}^\bullet )$ is K-flat also. Hence the isomorphism of the lemma comes from the isomorphism

\[ \text{Tot}(f^*\mathcal{F}^\bullet \otimes _{\mathcal{O}} f^*\mathcal{G}^\bullet ) \longrightarrow f^*\text{Tot}(\mathcal{F}^\bullet \otimes _{\mathcal{O}'} \mathcal{G}^\bullet ) \]

whose constituents are the isomorphisms $f^*\mathcal{F}^ p \otimes _{\mathcal{O}} f^*\mathcal{G}^ q \to f^*(\mathcal{F}^ p \otimes _{\mathcal{O}'} \mathcal{G}^ q)$ of Modules on Sites, Lemma 18.26.1. $\square$

Lemma 21.19.5. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}')$ be a morphism of ringed topoi. There is a canonical bifunctorial isomorphism

\[ \mathcal{F}^\bullet \otimes _\mathcal {O}^{\mathbf{L}} Lf^*\mathcal{G}^\bullet = \mathcal{F}^\bullet \otimes _{f^{-1}\mathcal{O}_ Y}^{\mathbf{L}} f^{-1}\mathcal{G}^\bullet \]

for $\mathcal{F}^\bullet $ in $D(\mathcal{O})$ and $\mathcal{G}^\bullet $ in $D(\mathcal{O}')$.

Proof. Let $\mathcal{F}$ be an $\mathcal{O}$-module and let $\mathcal{G}$ be an $\mathcal{O}'$-module. Then $\mathcal{F} \otimes _{\mathcal{O}} f^*\mathcal{G} = \mathcal{F} \otimes _{f^{-1}\mathcal{O}'} f^{-1}\mathcal{G}$ because $f^*\mathcal{G} = \mathcal{O} \otimes _{f^{-1}\mathcal{O}'} f^{-1}\mathcal{G}$. The lemma follows from this and the definitions. $\square$

Lemma 21.19.6. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{K}^\bullet $ be a complex of $\mathcal{O}$-modules.

  1. If $\mathcal{K}^\bullet $ is K-flat, then for every point $p$ of the site $\mathcal{C}$ the complex of $\mathcal{O}_ p$-modules $\mathcal{K}_ p^\bullet $ is K-flat in the sense of More on Algebra, Definition 15.57.3

  2. If $\mathcal{C}$ has enough points, then the converse is true.

Proof. Proof of (2). If $\mathcal{C}$ has enough points and $\mathcal{K}_ p^\bullet $ is K-flat for all points $p$ of $\mathcal{C}$ then we see that $\mathcal{K}^\bullet $ is K-flat because $\otimes $ and direct sums commute with taking stalks and because we can check exactness at stalks, see Modules on Sites, Lemma 18.14.4.

Proof of (1). Assume $\mathcal{K}^\bullet $ is K-flat. Choose a quasi-isomorphism $a : \mathcal{L}^\bullet \to \mathcal{K}^\bullet $ such that $\mathcal{L}^\bullet $ is K-flat and such that any pullback of $\mathcal{L}^\bullet $ is K-flat, see Lemma 21.19.1. In particular the stalk $\mathcal{L}_ p^\bullet $ is a K-flat complex of $\mathcal{O}_ p$-modules. Thus the cone $C(a)$ on $a$ is a K-flat (Lemma 21.18.6) acyclic complex of $\mathcal{O}$-modules and it suffuces to show the stalk of $C(a)$ is K-flat (by More on Algebra, Lemma 15.57.7). Thus we may assume that $\mathcal{K}^\bullet $ is K-flat and acyclic.

Assume $\mathcal{K}^\bullet $ is acyclic and K-flat. Before continuing we replace the site $\mathcal{C}$ by another one as in Sites, Lemma 7.29.5 to insure that $\mathcal{C}$ has all finite limits. This implies the category of neighbourhoods of $p$ is filtered (Sites, Lemma 7.33.1) and the colimit defining the stalk of a sheaf is filtered. Let $M$ be a finitely presented $\mathcal{O}_ p$-module. It suffices to show that $\mathcal{K}^\bullet \otimes _{\mathcal{O}_ p} M$ is acyclic, see More on Algebra, Lemma 15.57.11. Since $\mathcal{O}_ p$ is the filtered colimit of $\mathcal{O}(U)$ where $U$ runs over the neighbourhoods of $p$, we can find a neighbourhood $(U, x)$ of $p$ and a finitely presented $\mathcal{O}(U)$-module $M'$ whose base change to $\mathcal{O}_ p$ is $M$, see Algebra, Lemma 10.126.6. By Lemma 21.18.4 we may replace $\mathcal{C}, \mathcal{O}, \mathcal{K}^\bullet $ by $\mathcal{C}/U, \mathcal{O}_ U, \mathcal{K}^\bullet |_ U$. We conclude that we may assume there exists an $\mathcal{O}$-module $\mathcal{F}$ such that $M \cong \mathcal{F}_ p$. Since $\mathcal{K}^\bullet $ is K-flat and acyclic, we see that $\mathcal{K}^\bullet \otimes _\mathcal {O} \mathcal{F}$ is acyclic (as it computes the derived tensor product by definition). Taking stalks is an exact functor, hence we get that $\mathcal{K}^\bullet \otimes _{\mathcal{O}_ p} M$ is acyclic as desired. $\square$

Lemma 21.19.7. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}')$ be a morphism of ringed topoi. If $\mathcal{C}$ has enough points, then the pullback of a K-flat complex of $\mathcal{O}'$-modules is a K-flat complex of $\mathcal{O}$-modules.

Proof. This follows from Lemma 21.19.6, Modules on Sites, Lemma 18.35.4, and More on Algebra, Lemma 15.57.5. $\square$

Remark 21.19.8. The pullback of a K-flat complex is K-flat for a morphism of ringed topoi with enough points, see Lemma 21.19.7. This slightly improves the result of Lemma 21.19.1. However, in applications it seems rather that the explicit form of the K-flat complexes constructed in Lemma 21.18.10 is what is useful and not the plain fact that they are K-flat. Note for example that the terms of the complex constructed are each direct sums of modules of the form $j_{U!}\mathcal{O}_ U$, see Lemma 21.18.9.

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