The Stacks project

103.3 The lisse-étale and the flat-fppf sites

The section is the analogue of Cohomology of Stacks, Section 102.14 for derived categories.

Lemma 103.3.1. Let $\mathcal{X}$ be an algebraic stack. Notation as in Cohomology of Stacks, Lemmas 102.14.2 and 102.14.4.

  1. The functor $g_! : \textit{Ab}(\mathcal{X}_{lisse,{\acute{e}tale}}) \to \textit{Ab}(\mathcal{X}_{\acute{e}tale})$ has a left derived functor

    \[ Lg_! : D(\mathcal{X}_{lisse,{\acute{e}tale}}) \longrightarrow D(\mathcal{X}_{\acute{e}tale}) \]

    which is left adjoint to $g^{-1}$ and such that $g^{-1}Lg_! = \text{id}$.

  2. The functor $g_! : \textit{Mod}(\mathcal{X}_{lisse,{\acute{e}tale}}, \mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}}) \to \textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_{\mathcal{X}})$ has a left derived functor

    \[ Lg_! : D(\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}}) \longrightarrow D(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X}) \]

    which is left adjoint to $g^*$ and such that $g^*Lg_! = \text{id}$.

  3. The functor $g_! : \textit{Ab}(\mathcal{X}_{flat,fppf}) \to \textit{Ab}(\mathcal{X}_{fppf})$ has a left derived functor

    \[ Lg_! : D(\mathcal{X}_{flat, fppf}) \longrightarrow D(\mathcal{X}_{fppf}) \]

    which is left adjoint to $g^{-1}$ and such that $g^{-1}Lg_! = \text{id}$.

  4. The functor $g_! : \textit{Mod}(\mathcal{X}_{flat,fppf}, \mathcal{O}_{\mathcal{X}_{flat,fppf}}) \to \textit{Mod}(\mathcal{X}_{fppf}, \mathcal{O}_{\mathcal{X}})$ has a left derived functor

    \[ Lg_! : D(\mathcal{O}_{\mathcal{X}_{flat, fppf}}) \longrightarrow D(\mathcal{O}_\mathcal {X}) \]

    which is left adjoint to $g^*$ and such that $g^*Lg_! = \text{id}$.

Warning: It is not clear (a priori) that $Lg_!$ on modules agrees with $Lg_!$ on abelian sheaves, see Cohomology on Sites, Remark 21.37.3.

Proof. The existence of the functor $Lg_!$ and adjointness to $g^*$ is Cohomology on Sites, Lemma 21.37.2. (For the case of abelian sheaves use the constant sheaf $\mathbf{Z}$ as the structure sheaves.) Moreover, it is computed on a complex $\mathcal{H}^\bullet $ by taking a suitable left resolution $\mathcal{K}^\bullet \to \mathcal{H}^\bullet $ and applying the functor $g_!$ to $\mathcal{K}^\bullet $. Since $g^{-1}g_!\mathcal{K}^\bullet = \mathcal{K}^\bullet $ by Cohomology of Stacks, Lemmas 102.14.4 and 102.14.2 we see that the final assertion holds in each case. $\square$

Lemma 103.3.2. With assumptions and notation as in Cohomology of Stacks, Lemma 102.15.1. We have

\[ g^{-1} \circ Rf_* = Rf'_* \circ (g')^{-1} \quad \text{and}\quad L(g')_! \circ (f')^{-1} = f^{-1} \circ Lg_! \]

on unbounded derived categories (both for the case of modules and for the case of abelian sheaves).

Proof. Let $\tau = {\acute{e}tale}$ (resp. $\tau = fppf$). Let $\mathcal{F}$ be an abelian sheaf on $\mathcal{X}_\tau $. By Cohomology of Stacks, Lemma 102.15.3 the canonical (base change) map

\[ g^{-1}Rf_*\mathcal{F} \longrightarrow Rf'_*(g')^{-1}\mathcal{F} \]

is an isomorphism. The rest of the proof is formal. Since cohomology of abelian groups and sheaves of modules agree we also conclude that $g^{-1} Rf_*\mathcal{F} = Rf'_* (g')^{-1}\mathcal{F}$ when $\mathcal{F}$ is a sheaf of modules on $\mathcal{X}_\tau $.

Next we show that for $\mathcal{G}$ (either sheaf of modules or abelian groups) on $\mathcal{Y}_{lisse,{\acute{e}tale}}$ (resp. $\mathcal{Y}_{flat,fppf}$) the canonical map

\[ L(g')_!(f')^{-1}\mathcal{G} \to f^{-1}Lg_!\mathcal{G} \]

is an isomorphism. To see this it is enough to prove for any injective sheaf $\mathcal{I}$ on $\mathcal{X}_\tau $ the induced map

\[ \mathop{\mathrm{Hom}}\nolimits (L(g')_!(f')^{-1}\mathcal{G}, \mathcal{I}[n]) \leftarrow \mathop{\mathrm{Hom}}\nolimits (f^{-1}Lg_!\mathcal{G}, \mathcal{I}[n]) \]

is an isomorphism for all $n \in \mathbf{Z}$. (Hom's taken in suitable derived categories.) By the adjointness of $f^{-1}$ and $Rf_*$, the adjointness of $Lg_!$ and $g^{-1}$, and their “primed” versions this follows from the isomorphism $g^{-1} Rf_*\mathcal{I} \to Rf'_* (g')^{-1}\mathcal{I}$ proved above.

In the case of a bounded complex $\mathcal{G}^\bullet $ (of modules or abelian groups) on $\mathcal{Y}_{lisse,{\acute{e}tale}}$ (resp. $\mathcal{Y}_{fppf}$) the canonical map

103.3.2.1
\begin{equation} \label{stacks-perfect-equation-to-show} L(g')_!(f')^{-1}\mathcal{G}^\bullet \to f^{-1}Lg_!\mathcal{G}^\bullet \end{equation}

is an isomorphism as follows from the case of a sheaf by the usual arguments involving truncations and the fact that the functors $L(g')_!(f')^{-1}$ and $f^{-1}Lg_!$ are exact functors of triangulated categories.

Suppose that $\mathcal{G}^\bullet $ is a bounded above complex (of modules or abelian groups) on $\mathcal{Y}_{lisse,{\acute{e}tale}}$ (resp. $\mathcal{Y}_{fppf}$). The canonical map (103.3.2.1) is an isomorphism because we can use the stupid truncations $\sigma _{\geq -n}$ (see Homology, Section 12.15) to write $\mathcal{G}^\bullet $ as a colimit $\mathcal{G}^\bullet = \mathop{\mathrm{colim}}\nolimits \mathcal{G}_ n^\bullet $ of bounded complexes. This gives a distinguished triangle

\[ \bigoplus \nolimits _{n \geq 1} \mathcal{G}_ n^\bullet \to \bigoplus \nolimits _{n \geq 1} \mathcal{G}_ n^\bullet \to \mathcal{G}^\bullet \to \ldots \]

and each of the functors $L(g')_!$, $(f')^{-1}$, $f^{-1}$, $Lg_!$ commutes with direct sums (of complexes).

If $\mathcal{G}^\bullet $ is an arbitrary complex (of modules or abelian groups) on $\mathcal{Y}_{lisse,{\acute{e}tale}}$ (resp. $\mathcal{Y}_{fppf}$) then we use the canonical truncations $\tau _{\leq n}$ (see Homology, Section 12.15) to write $\mathcal{G}^\bullet $ as a colimit of bounded above complexes and we repeat the argument of the paragraph above.

Finally, by the adjointness of $f^{-1}$ and $Rf_*$, the adjointness of $Lg_!$ and $g^{-1}$, and their “primed” versions we conclude that the first identity of the lemma follows from the second in full generality. $\square$

Lemma 103.3.3. Let $\mathcal{X}$ be an algebraic stack. Notation as in Cohomology of Stacks, Lemma 102.14.2.

  1. Let $\mathcal{H}$ be a quasi-coherent $\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}}$-module on the lisse-étale site of $\mathcal{X}$. For all $p \in \mathbf{Z}$ the sheaf $H^ p(Lg_!\mathcal{H})$ is a locally quasi-coherent module with the flat base change property on $\mathcal{X}$.

  2. Let $\mathcal{H}$ be a quasi-coherent $\mathcal{O}_{\mathcal{X}_{flat,fppf}}$-module on the flat-fppf site of $\mathcal{X}$. For all $p \in \mathbf{Z}$ the sheaf $H^ p(Lg_!\mathcal{H})$ is a locally quasi-coherent module with the flat base change property on $\mathcal{X}$.

Proof. Pick a scheme $U$ and a surjective smooth morphism $x : U \to \mathcal{X}$. By Modules on Sites, Definition 18.23.1 there exists an étale (resp. fppf) covering $\{ U_ i \to U\} _{i \in I}$ such that each pullback $f_ i^{-1}\mathcal{H}$ has a global presentation (see Modules on Sites, Definition 18.17.1). Here $f_ i : U_ i \to \mathcal{X}$ is the composition $U_ i \to U \to \mathcal{X}$ which is a morphism of algebraic stacks. (Recall that the pullback “is” the restriction to $\mathcal{X}/f_ i$, see Sheaves on Stacks, Definition 95.9.2 and the discussion following.) After refining the covering we may assume each $U_ i$ is an affine scheme. Since each $f_ i$ is smooth (resp. flat) by Lemma 103.3.2 we see that $f_ i^{-1}Lg_!\mathcal{H} = Lg_{i, !}(f'_ i)^{-1}\mathcal{H}$. Using Cohomology of Stacks, Lemma 102.8.2 we reduce the statement of the lemma to the case where $\mathcal{H}$ has a global presentation and where $\mathcal{X} = (\mathit{Sch}/X)_{fppf}$ for some affine scheme $X = \mathop{\mathrm{Spec}}(A)$.

Say our presentation looks like

\[ \bigoplus \nolimits _{j \in J} \mathcal{O} \longrightarrow \bigoplus \nolimits _{i \in I} \mathcal{O} \longrightarrow \mathcal{H} \longrightarrow 0 \]

where $\mathcal{O} = \mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}}$ (resp. $\mathcal{O} = \mathcal{O}_{\mathcal{X}_{flat,fppf}}$). Note that the site $\mathcal{X}_{lisse,{\acute{e}tale}}$ (resp. $\mathcal{X}_{flat,fppf}$) has a final object, namely $X/X$ which is quasi-compact (see Cohomology on Sites, Section 21.16). Hence we have

\[ \Gamma (\bigoplus \nolimits _{i \in I} \mathcal{O}) = \bigoplus \nolimits _{i \in I} A \]

by Sites, Lemma 7.17.7. Hence the map in the presentation corresponds to a similar presentation

\[ \bigoplus \nolimits _{j \in J} A \longrightarrow \bigoplus \nolimits _{i \in I} A \longrightarrow M \longrightarrow 0 \]

of an $A$-module $M$. Moreover, $\mathcal{H}$ is equal to the restriction to the lisse-étale (resp. flat-fppf) site of the quasi-coherent sheaf $M^ a$ associated to $M$. Choose a resolution

\[ \ldots \to F_2 \to F_1 \to F_0 \to M \to 0 \]

by free $A$-modules. The complex

\[ \ldots \mathcal{O} \otimes _ A F_2 \to \mathcal{O} \otimes _ A F_1 \to \mathcal{O} \otimes _ A F_0 \to \mathcal{H} \to 0 \]

is a resolution of $\mathcal{H}$ by free $\mathcal{O}$-modules because for each object $U/X$ of $\mathcal{X}_{lisse,{\acute{e}tale}}$ (resp. $\mathcal{X}_{flat,fppf}$) the structure morphism $U \to X$ is flat. Hence by construction the value of $Lg_!\mathcal{H}$ is

\[ \ldots \to \mathcal{O}_\mathcal {X} \otimes _ A F_2 \to \mathcal{O}_\mathcal {X} \otimes _ A F_1 \to \mathcal{O}_\mathcal {X} \otimes _ A F_0 \to 0 \to \ldots \]

Since this is a complex of quasi-coherent modules on $\mathcal{X}_{\acute{e}tale}$ (resp. $\mathcal{X}_{fppf}$) it follows from Cohomology of Stacks, Proposition 102.8.1 that $H^ p(Lg_!\mathcal{H})$ is quasi-coherent. $\square$


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