104.3 The lisse-étale and the flat-fppf sites
The section is the analogue of Cohomology of Stacks, Section 103.14 for derived categories.
Lemma 104.3.1. Let $\mathcal{X}$ be an algebraic stack. Notation as in Cohomology of Stacks, Lemmas 103.14.2 and 103.14.4.
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 $g^{-1}Lg_! = \text{id}$.
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 $g^*Lg_! = \text{id}$.
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 $g^{-1}Lg_! = \text{id}$.
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 103.14.4 and 103.14.2 we see that the final assertion holds in each case.
$\square$
Lemma 104.3.2. With assumptions and notation as in Cohomology of Stacks, Lemma 103.15.1. 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 103.15.3 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
104.3.2.1
\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 (104.3.2.1) 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 104.3.3. Let $\mathcal{X}$ be an algebraic stack. Notation as in Cohomology of Stacks, Lemma 103.14.2.
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 $\mathcal{X}$.
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 96.9.2 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 104.3.2 we see that $f_ i^{-1}Lg_!\mathcal{H} = Lg_{i, !}(f'_ i)^{-1}\mathcal{H}$. Using Cohomology of Stacks, Lemma 103.8.2 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 103.8.1 that $H^ p(Lg_!\mathcal{H})$ is quasi-coherent.
$\square$
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