103.15 Functoriality of the lisse-étale and flat-fppf sites
The lisse-étale site is functorial for smooth morphisms of algebraic stacks and the flat-fppf site is functorial for flat morphisms of algebraic stacks. We warn the reader that the lisse-étale and flat-fppf topoi are not functorial with respect to all morphisms of algebraic stacks, see Examples, Section 110.58.
Lemma 103.15.1. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks.
If $f$ is smooth, then $f$ restricts to a continuous and cocontinuous functor $\mathcal{X}_{lisse,{\acute{e}tale}} \to \mathcal{Y}_{lisse,{\acute{e}tale}}$ which gives a morphism of ringed topoi fitting into the following commutative diagram
\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{lisse,{\acute{e}tale}}) \ar[r]_{g'} \ar[d]_{f'} & \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{\acute{e}tale}) \ar[d]^ f \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{Y}_{lisse,{\acute{e}tale}}) \ar[r]^ g & \mathop{\mathit{Sh}}\nolimits (\mathcal{Y}_{\acute{e}tale}) } \]
We have $f'_*(g')^{-1} = g^{-1}f_*$ and $g'_!(f')^{-1} = f^{-1}g_!$.
If $f$ is flat, then $f$ restricts to a continuous and cocontinuous functor $\mathcal{X}_{flat,fppf} \to \mathcal{Y}_{flat,fppf}$ which gives a morphism of ringed topoi fitting into the following commutative diagram
\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{flat,fppf}) \ar[r]_{g'} \ar[d]_{f'} & \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{fppf}) \ar[d]^ f \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{Y}_{flat,fppf}) \ar[r]^ g & \mathop{\mathit{Sh}}\nolimits (\mathcal{Y}_{fppf}) } \]
We have $f'_*(g')^{-1} = g^{-1}f_*$ and $g'_!(f')^{-1} = f^{-1}g_!$.
Proof.
The initial statement comes from the fact that if $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X})$ lies over a scheme $U$ such that $x : U \to \mathcal{X}$ is smooth (resp. flat) and if $f$ is smooth (resp. flat) then $f(x) : U \to \mathcal{Y}$ is smooth (resp. flat), see Morphisms of Stacks, Lemmas 101.33.2 and 101.25.2. The induced functor $\mathcal{X}_{lisse,{\acute{e}tale}} \to \mathcal{Y}_{lisse,{\acute{e}tale}}$ (resp. $\mathcal{X}_{flat,fppf} \to \mathcal{Y}_{flat,fppf}$) is continuous and cocontinuous by our definition of coverings in these categories. Finally, the commutativity of the diagram is a consequence of the fact that the horizontal morphisms are given by the inclusion functors (see Lemma 103.14.2) and Sites, Lemma 7.21.2.
To show that $f'_*(g')^{-1} = g^{-1}f_*$ let $\mathcal{F}$ be a sheaf on $\mathcal{X}_{\acute{e}tale}$ (resp. $\mathcal{X}_{fppf}$). There is a canonical pullback map
\[ g^{-1}f_*\mathcal{F} \longrightarrow f'_*(g')^{-1}\mathcal{F} \]
see Sites, Section 7.45. We claim this map is an isomorphism. To prove this pick an object $y$ of $\mathcal{Y}_{lisse,{\acute{e}tale}}$ (resp. $\mathcal{Y}_{flat,fppf}$). Say $y$ lies over the scheme $V$ such that $y : V \to \mathcal{Y}$ is smooth (resp. flat). Since $g^{-1}$ is the restriction we find that
\[ \left(g^{-1}f_*\mathcal{F}\right)(y) = \Gamma (V \times _{y, \mathcal{Y}} \mathcal{X},\ \text{pr}^{-1}\mathcal{F}) \]
by Sheaves on Stacks, Equation (96.5.0.1). Let $(V \times _{y, \mathcal{Y}} \mathcal{X})' \subset V \times _{y, \mathcal{Y}} \mathcal{X}$ be the full subcategory consisting of objects $z : W \to V \times _{y, \mathcal{Y}} \mathcal{X}$ such that the induced morphism $W \to \mathcal{X}$ is smooth (resp. flat). Denote
\[ \text{pr}' : (V \times _{y, \mathcal{Y}} \mathcal{X})' \longrightarrow \mathcal{X}_{lisse,{\acute{e}tale}} \ (\text{resp. }\mathcal{X}_{flat,fppf}) \]
the restriction of the functor $\text{pr}$ used in the formula above. Exactly the same argument that proves Sheaves on Stacks, Equation (96.5.0.1) shows that for any sheaf $\mathcal{H}$ on $\mathcal{X}_{lisse,{\acute{e}tale}}$ (resp. $\mathcal{X}_{flat,fppf}$) we have
103.15.1.1
\begin{equation} \label{stacks-cohomology-equation-pushforward-lisse-etale} f'_*\mathcal{H}(y) = \Gamma ((V \times _{y, \mathcal{Y}} \mathcal{X})', \ (\text{pr}')^{-1}\mathcal{H}) \end{equation}
Since $(g')^{-1}$ is restriction we see that
\[ \left(f'_*(g')^{-1}\mathcal{F}\right)(y) = \Gamma ((V \times _{y, \mathcal{Y}} \mathcal{X})', \ \text{pr}^{-1}\mathcal{F}|_{(V \times _{y, \mathcal{Y}} \mathcal{X})'}) \]
By Sheaves on Stacks, Lemma 96.23.3 we see that
\[ \Gamma ((V \times _{y, \mathcal{Y}} \mathcal{X})', \ \text{pr}^{-1}\mathcal{F}|_{(V \times _{y, \mathcal{Y}} \mathcal{X})'}) = \Gamma (V \times _{y, \mathcal{Y}} \mathcal{X},\ \text{pr}^{-1}\mathcal{F}) \]
are equal as desired; although we omit the verification of the assumptions of the lemma we note that the fact that $V \to \mathcal{Y}$ is smooth (resp. flat) is used to verify the second condition.
Finally, the equality $g'_!(f')^{-1} = f^{-1}g_!$ follows formally from the equality $f'_*(g')^{-1} = g^{-1}f_*$ by the adjointness of $f^{-1}$ and $f_*$, the adjointness of $g_!$ and $g^{-1}$, and their “primed” versions.
$\square$
Lemma 103.15.2. With assumptions and notation as in Lemma 103.15.1. Let $\mathcal{H}$ be an abelian sheaf on $\mathcal{X}_{lisse,{\acute{e}tale}}$ (resp. $\mathcal{X}_{flat,fppf}$). Then
103.15.2.1
\begin{equation} \label{stacks-cohomology-equation-higher-direct-image-lisse-etale} R^ pf'_*\mathcal{H} = \text{sheaf associated to }y \longmapsto H^ p((V \times _{y, \mathcal{Y}} \mathcal{X})', (\text{pr}')^{-1}\mathcal{H}) \end{equation}
Here $y$ is an object of $\mathcal{Y}_{lisse,{\acute{e}tale}}$ (resp. $\mathcal{Y}_{flat,fppf}$) lying over the scheme $V$ and the notation $(V \times _{y, \mathcal{Y}} \mathcal{X})'$ and $\text{pr}'$ are explained in the proof.
Proof.
As in the proof of Lemma 103.15.1 let $(V \times _{y, \mathcal{Y}} \mathcal{X})' \subset V \times _{y, \mathcal{Y}} \mathcal{X}$ be the full subcategory consisting of objects $(x, \varphi )$ where $x$ is an object of $\mathcal{X}_{lisse,{\acute{e}tale}}$ (resp. $\mathcal{X}_{flat,fppf}$) and $\varphi : f(x) \to y$ is a morphism in $\mathcal{Y}$. By Equation (103.15.1.1) we have
\[ f'_*\mathcal{H}(y) = \Gamma ((V \times _{y, \mathcal{Y}} \mathcal{X})', \ (\text{pr}')^{-1}\mathcal{H}) \]
where $\text{pr}'$ is the projection. For an object $(x, \varphi )$ of $(V \times _{y, \mathcal{Y}} \mathcal{X})'$ we can think of $\varphi $ as a section of $(f')^{-1}h_ y$ over $x$. Thus $(V \times _\mathcal {Y} \mathcal{X})'$ is the localization of the site $\mathcal{X}_{lisse,{\acute{e}tale}}$ (resp. $\mathcal{X}_{flat,fppf}$) at the sheaf of sets $(f')^{-1}h_ y$, see Sites, Lemma 7.30.3. The morphism
\[ \text{pr}' : (V \times _{y, \mathcal{Y}} \mathcal{X})' \to \mathcal{X}_{lisse,{\acute{e}tale}} \ (\text{resp. } \text{pr}' : (V \times _{y, \mathcal{Y}} \mathcal{X})' \to \mathcal{X}_{flat,fppf}) \]
is the localization morphism. In particular, the pullback $(\text{pr}')^{-1}$ preserves injective abelian sheaves, see Cohomology on Sites, Lemma 21.13.3.
Choose an injective resolution $\mathcal{H} \to \mathcal{I}^\bullet $ on $\mathcal{X}_{lisse,{\acute{e}tale}}$ (resp. $\mathcal{X}_{flat,fppf}$). By the formula for pushforward we see that $R^ if'_*\mathcal{H}$ is the sheaf associated to the presheaf which associates to $y$ the cohomology of the complex
\[ \begin{matrix} \Gamma \Big((V \times _{y, \mathcal{Y}} \mathcal{X})', (\text{pr}')^{-1}\mathcal{I}^{i - 1}\Big)
\\ \downarrow
\\ \Gamma \Big((V \times _{y, \mathcal{Y}} \mathcal{X})', (\text{pr}')^{-1}\mathcal{I}^ i\Big)
\\ \downarrow
\\ \Gamma \Big((V \times _{y, \mathcal{Y}} \mathcal{X})', (\text{pr}')^{-1}\mathcal{I}^{i + 1}\Big)
\end{matrix} \]
Since $(\text{pr}')^{-1}$ is exact and preserves injectives the complex $(\text{pr}')^{-1}\mathcal{I}^\bullet $ is an injective resolution of $(\text{pr}')^{-1}\mathcal{H}$ and the proof is complete.
$\square$
Lemma 103.15.3. With assumptions and notation as in Lemma 103.15.1 the canonical (base change) map
\[ g^{-1}Rf_*\mathcal{F} \longrightarrow Rf'_*(g')^{-1}\mathcal{F} \]
is an isomorphism for any abelian sheaf $\mathcal{F}$ on $\mathcal{X}_{\acute{e}tale}$ (resp. $\mathcal{X}_{fppf}$).
Proof.
Comparing the formula for $g^{-1}R^ pf_*\mathcal{F}$ and $R^ pf'_*(g')^{-1}\mathcal{F}$ given in Sheaves on Stacks, Lemma 96.21.2 and Lemma 103.15.2 we see that it suffices to show
\[ H^ p((V \times _{y, \mathcal{Y}} \mathcal{X})', \ \text{pr}^{-1}\mathcal{F}|_{(V \times _{y, \mathcal{Y}} \mathcal{X})'}) = H^ p_\tau (V \times _{y, \mathcal{Y}} \mathcal{X},\ \text{pr}^{-1}\mathcal{F}) \]
where $\tau = {\acute{e}tale}$ (resp. $\tau = fppf$). Here $y$ is an object of $\mathcal{Y}$ lying over a scheme $V$ such that the morphism $y : V \to \mathcal{Y}$ is smooth (resp. flat). This equality follows from Sheaves on Stacks, Lemma 96.23.3. Although we omit the verification of the assumptions of the lemma, we note that the fact that $V \to \mathcal{Y}$ is smooth (resp. flat) is used to verify the second condition.
$\square$
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