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 110.59.
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
Comments (0)