The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

21.18 Flat resolutions

In this section we redo the arguments of Cohomology, Section 20.27 in the setting of ringed sites and ringed topoi.

Lemma 21.18.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{G}^\bullet $ be a complex of $\mathcal{O}$-modules. The functor

\[ K(\textit{Mod}(\mathcal{O})) \longrightarrow K(\textit{Mod}(\mathcal{O})), \quad \mathcal{F}^\bullet \longmapsto \text{Tot}(\mathcal{F}^\bullet \otimes _\mathcal {O} \mathcal{G}^\bullet ) \]

is an exact functor of triangulated categories.

Proof. Omitted. Hint: See More on Algebra, Lemmas 15.57.1 and 15.57.2. $\square$

Definition 21.18.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. A complex $\mathcal{K}^\bullet $ of $\mathcal{O}$-modules is called K-flat if for every acyclic complex $\mathcal{F}^\bullet $ of $\mathcal{O}$-modules the complex

\[ \text{Tot}(\mathcal{F}^\bullet \otimes _\mathcal {O} \mathcal{K}^\bullet ) \]

is acyclic.

Lemma 21.18.3. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{K}^\bullet $ be a K-flat complex. Then the functor

\[ K(\textit{Mod}(\mathcal{O})) \longrightarrow K(\textit{Mod}(\mathcal{O})), \quad \mathcal{F}^\bullet \longmapsto \text{Tot}(\mathcal{F}^\bullet \otimes _\mathcal {O} \mathcal{K}^\bullet ) \]

transforms quasi-isomorphisms into quasi-isomorphisms.

Proof. Follows from Lemma 21.18.1 and the fact that quasi-isomorphisms are characterized by having acyclic cones. $\square$

Lemma 21.18.4. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U$ be an object of $\mathcal{C}$. If $\mathcal{K}^\bullet $ is a K-flat complex of $\mathcal{O}$-modules, then $\mathcal{K}^\bullet |_ U$ is a K-flat complex of $\mathcal{O}_ U$-modules.

Proof. Let $\mathcal{G}^\bullet $ be an exact complex of $\mathcal{O}_ U$-modules. Since $j_{U!}$ is exact (Modules on Sites, Lemma 18.19.3) and $\mathcal{K}^\bullet $ is a K-flat complex of $\mathcal{O}$-modules we see that the complex

\[ j_{U!}(\text{Tot}(\mathcal{G}^\bullet \otimes _{\mathcal{O}_ U} \mathcal{K}^\bullet |_ U)) = \text{Tot}(j_{U!}\mathcal{G}^\bullet \otimes _\mathcal {O} \mathcal{K}^\bullet ) \]

is exact. Here the equality comes from Modules on Sites, Lemma 18.27.7 and the fact that $j_{U!}$ commutes with direct sums (as a left adjoint). We conclude because $j_{U!}$ reflects exactness by Modules on Sites, Lemma 18.19.4. $\square$

Lemma 21.18.5. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. If $\mathcal{K}^\bullet $, $\mathcal{L}^\bullet $ are K-flat complexes of $\mathcal{O}$-modules, then $\text{Tot}(\mathcal{K}^\bullet \otimes _\mathcal {O} \mathcal{L}^\bullet )$ is a K-flat complex of $\mathcal{O}$-modules.

Proof. Follows from the isomorphism

\[ \text{Tot}(\mathcal{M}^\bullet \otimes _\mathcal {O} \text{Tot}(\mathcal{K}^\bullet \otimes _\mathcal {O} \mathcal{L}^\bullet )) = \text{Tot}(\text{Tot}(\mathcal{M}^\bullet \otimes _\mathcal {O} \mathcal{K}^\bullet ) \otimes _\mathcal {O} \mathcal{L}^\bullet ) \]

and the definition. $\square$

Lemma 21.18.6. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{K}_1^\bullet , \mathcal{K}_2^\bullet , \mathcal{K}_3^\bullet )$ be a distinguished triangle in $K(\textit{Mod}(\mathcal{O}))$. If two out of three of $\mathcal{K}_ i^\bullet $ are K-flat, so is the third.

Proof. Follows from Lemma 21.18.1 and the fact that in a distinguished triangle in $K(\textit{Mod}(\mathcal{O}))$ if two out of three are acyclic, so is the third. $\square$

Lemma 21.18.7. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. A bounded above complex of flat $\mathcal{O}$-modules is K-flat.

Proof. Let $\mathcal{K}^\bullet $ be a bounded above complex of flat $\mathcal{O}$-modules. Let $\mathcal{L}^\bullet $ be an acyclic complex of $\mathcal{O}$-modules. Note that $\mathcal{L}^\bullet = \mathop{\mathrm{colim}}\nolimits _ m \tau _{\leq m}\mathcal{L}^\bullet $ where we take termwise colimits. Hence also

\[ \text{Tot}(\mathcal{K}^\bullet \otimes _\mathcal {O} \mathcal{L}^\bullet ) = \mathop{\mathrm{colim}}\nolimits _ m \text{Tot}( \mathcal{K}^\bullet \otimes _\mathcal {O} \tau _{\leq m}\mathcal{L}^\bullet ) \]

termwise. Hence to prove the complex on the left is acyclic it suffices to show each of the complexes on the right is acyclic. Since $\tau _{\leq m}\mathcal{L}^\bullet $ is acyclic this reduces us to the case where $\mathcal{L}^\bullet $ is bounded above. In this case the spectral sequence of Homology, Lemma 12.22.6 has

\[ {}'E_1^{p, q} = H^ p(\mathcal{L}^\bullet \otimes _ R \mathcal{K}^ q) \]

which is zero as $\mathcal{K}^ q$ is flat and $\mathcal{L}^\bullet $ acyclic. Hence we win. $\square$

Lemma 21.18.8. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{K}_1^\bullet \to \mathcal{K}_2^\bullet \to \ldots $ be a system of K-flat complexes. Then $\mathop{\mathrm{colim}}\nolimits _ i \mathcal{K}_ i^\bullet $ is K-flat.

Proof. Because we are taking termwise colimits it is clear that

\[ \mathop{\mathrm{colim}}\nolimits _ i \text{Tot}( \mathcal{F}^\bullet \otimes _\mathcal {O} \mathcal{K}_ i^\bullet ) = \text{Tot}(\mathcal{F}^\bullet \otimes _\mathcal {O} \mathop{\mathrm{colim}}\nolimits _ i \mathcal{K}_ i^\bullet ) \]

Hence the lemma follows from the fact that filtered colimits are exact. $\square$

Lemma 21.18.9. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. For any complex $\mathcal{G}^\bullet $ of $\mathcal{O}$-modules there exists a commutative diagram of complexes of $\mathcal{O}$-modules

\[ \xymatrix{ \mathcal{K}_1^\bullet \ar[d] \ar[r] & \mathcal{K}_2^\bullet \ar[d] \ar[r] & \ldots \\ \tau _{\leq 1}\mathcal{G}^\bullet \ar[r] & \tau _{\leq 2}\mathcal{G}^\bullet \ar[r] & \ldots } \]

with the following properties: (1) the vertical arrows are quasi-isomorphisms, (2) each $\mathcal{K}_ n^\bullet $ is a bounded above complex whose terms are direct sums of $\mathcal{O}$-modules of the form $j_{U!}\mathcal{O}_ U$, and (3) the maps $\mathcal{K}_ n^\bullet \to \mathcal{K}_{n + 1}^\bullet $ are termwise split injections whose cokernels are direct sums of $\mathcal{O}$-modules of the form $j_{U!}\mathcal{O}_ U$. Moreover, the map $\mathop{\mathrm{colim}}\nolimits \mathcal{K}_ n^\bullet \to \mathcal{G}^\bullet $ is a quasi-isomorphism.

Proof. The existence of the diagram and properties (1), (2), (3) follows immediately from Modules on Sites, Lemma 18.28.7 and Derived Categories, Lemma 13.28.1. The induced map $\mathop{\mathrm{colim}}\nolimits \mathcal{K}_ n^\bullet \to \mathcal{G}^\bullet $ is a quasi-isomorphism because filtered colimits are exact. $\square$

Lemma 21.18.10. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. For any complex $\mathcal{G}^\bullet $ of $\mathcal{O}$-modules there exists a $K$-flat complex $\mathcal{K}^\bullet $ and a quasi-isomorphism $\mathcal{K}^\bullet \to \mathcal{G}^\bullet $. Moreover, each $\mathcal{K}^ n$ is a flat $\mathcal{O}$-module.

Proof. Choose a diagram as in Lemma 21.18.9. Each complex $\mathcal{K}_ n^\bullet $ is a bounded above complex of flat modules, see Modules on Sites, Lemma 18.28.6. Hence $\mathcal{K}_ n^\bullet $ is K-flat by Lemma 21.18.7. The induced map $\mathop{\mathrm{colim}}\nolimits \mathcal{K}_ n^\bullet \to \mathcal{G}^\bullet $ is a quasi-isomorphism by construction. Since $\mathop{\mathrm{colim}}\nolimits \mathcal{K}_ n^\bullet $ is K-flat by Lemma 21.18.8 we win. $\square$

Lemma 21.18.11. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\alpha : \mathcal{P}^\bullet \to \mathcal{Q}^\bullet $ be a quasi-isomorphism of K-flat complexes of $\mathcal{O}$-modules. For every complex $\mathcal{F}^\bullet $ of $\mathcal{O}$-modules the induced map

\[ \text{Tot}(\text{id}_{\mathcal{F}^\bullet } \otimes \alpha ) : \text{Tot}(\mathcal{F}^\bullet \otimes _\mathcal {O} \mathcal{P}^\bullet ) \longrightarrow \text{Tot}(\mathcal{F}^\bullet \otimes _\mathcal {O} \mathcal{Q}^\bullet ) \]

is a quasi-isomorphism.

Proof. Choose a quasi-isomorphism $\mathcal{K}^\bullet \to \mathcal{F}^\bullet $ with $\mathcal{K}^\bullet $ a K-flat complex, see Lemma 21.18.10. Consider the commutative diagram

\[ \xymatrix{ \text{Tot}(\mathcal{K}^\bullet \otimes _\mathcal {O} \mathcal{P}^\bullet ) \ar[r] \ar[d] & \text{Tot}(\mathcal{K}^\bullet \otimes _\mathcal {O} \mathcal{Q}^\bullet ) \ar[d] \\ \text{Tot}(\mathcal{F}^\bullet \otimes _\mathcal {O} \mathcal{P}^\bullet ) \ar[r] & \text{Tot}(\mathcal{F}^\bullet \otimes _\mathcal {O} \mathcal{Q}^\bullet ) } \]

The result follows as by Lemma 21.18.3 the vertical arrows and the top horizontal arrow are quasi-isomorphisms. $\square$

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{F}^\bullet $ be an object of $D(\mathcal{O})$. Choose a K-flat resolution $\mathcal{K}^\bullet \to \mathcal{F}^\bullet $, see Lemma 21.18.10. By Lemma 21.18.1 we obtain an exact functor of triangulated categories

\[ K(\mathcal{O}) \longrightarrow K(\mathcal{O}), \quad \mathcal{G}^\bullet \longmapsto \text{Tot}(\mathcal{G}^\bullet \otimes _\mathcal {O} \mathcal{K}^\bullet ) \]

By Lemma 21.18.3 this functor induces a functor $D(\mathcal{O}) \to D(\mathcal{O})$ simply because $D(\mathcal{O})$ is the localization of $K(\mathcal{O})$ at quasi-isomorphisms. By Lemma 21.18.11 the resulting functor (up to isomorphism) does not depend on the choice of the K-flat resolution.

Definition 21.18.12. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{F}^\bullet $ be an object of $D(\mathcal{O})$. The derived tensor product

\[ - \otimes _\mathcal {O}^{\mathbf{L}} \mathcal{F}^\bullet : D(\mathcal{O}) \longrightarrow D(\mathcal{O}) \]

is the exact functor of triangulated categories described above.

It is clear from our explicit constructions that there is a canonical isomorphism

\[ \mathcal{F}^\bullet \otimes _\mathcal {O}^{\mathbf{L}} \mathcal{G}^\bullet \cong \mathcal{G}^\bullet \otimes _\mathcal {O}^{\mathbf{L}} \mathcal{F}^\bullet \]

for $\mathcal{G}^\bullet $ and $\mathcal{F}^\bullet $ in $D(\mathcal{O})$. Hence when we write $\mathcal{F}^\bullet \otimes _\mathcal {O}^{\mathbf{L}} \mathcal{G}^\bullet $ we will usually be agnostic about which variable we are using to define the derived tensor product with.

Definition 21.18.13. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{F}$, $\mathcal{G}$ be $\mathcal{O}$-modules. The Tor's of $\mathcal{F}$ and $\mathcal{G}$ are defined by the formula

\[ \text{Tor}_ p^\mathcal {O}(\mathcal{F}, \mathcal{G}) = H^{-p}(\mathcal{F} \otimes _\mathcal {O}^\mathbf {L} \mathcal{G}) \]

with derived tensor product as defined above.

This definition implies that for every short exact sequence of $\mathcal{O}$-modules $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ we have a long exact cohomology sequence

\[ \xymatrix{ \mathcal{F}_1 \otimes _\mathcal {O} \mathcal{G} \ar[r] & \mathcal{F}_2 \otimes _\mathcal {O} \mathcal{G} \ar[r] & \mathcal{F}_3 \otimes _\mathcal {O} \mathcal{G} \ar[r] & 0 \\ \text{Tor}_1^\mathcal {O}(\mathcal{F}_1, \mathcal{G}) \ar[r] & \text{Tor}_1^\mathcal {O}(\mathcal{F}_2, \mathcal{G}) \ar[r] & \text{Tor}_1^\mathcal {O}(\mathcal{F}_3, \mathcal{G}) \ar[ull] } \]

for every $\mathcal{O}$-module $\mathcal{G}$. This will be called the long exact sequence of $\text{Tor}$ associated to the situation.

Lemma 21.18.14. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{F}$ be an $\mathcal{O}$-module. The following are equivalent

  1. $\mathcal{F}$ is a flat $\mathcal{O}$-module, and

  2. $\text{Tor}_1^\mathcal {O}(\mathcal{F}, \mathcal{G}) = 0$ for every $\mathcal{O}$-module $\mathcal{G}$.

Proof. If $\mathcal{F}$ is flat, then $\mathcal{F} \otimes _\mathcal {O} -$ is an exact functor and the satellites vanish. Conversely assume (2) holds. Then if $\mathcal{G} \to \mathcal{H}$ is injective with cokernel $\mathcal{Q}$, the long exact sequence of $\text{Tor}$ shows that the kernel of $\mathcal{F} \otimes _\mathcal {O} \mathcal{G} \to \mathcal{F} \otimes _\mathcal {O} \mathcal{H}$ is a quotient of $\text{Tor}_1^\mathcal {O}(\mathcal{F}, \mathcal{Q})$ which is zero by assumption. Hence $\mathcal{F}$ is flat. $\square$


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