The Stacks project

95.10 Restriction to algebraic spaces

In this section we consider sheaves on categories representable by algebraic spaces. The following lemma is the analogue of Topologies, Lemma 34.4.14 for algebraic spaces.

Lemma 95.10.1. Let $S$ be a scheme. Let $\mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Assume $\mathcal{X}$ is representable by an algebraic space $F$. Then there exists a continuous and cocontinuous functor $ F_{\acute{e}tale}\to \mathcal{X}_{\acute{e}tale}$ which induces a morphism of ringed sites

\[ \pi _ F : (\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X}) \longrightarrow (F_{\acute{e}tale}, \mathcal{O}_ F) \]

and a morphism of ringed topoi

\[ i_ F : (\mathop{\mathit{Sh}}\nolimits (F_{\acute{e}tale}), \mathcal{O}_ F) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{\acute{e}tale}), \mathcal{O}_\mathcal {X}) \]

such that $\pi _ F \circ i_ F = \text{id}$. Moreover $\pi _{F, *} = i_ F^{-1}$.

Proof. Choose an equivalence $j : \mathcal{S}_ F \to \mathcal{X}$, see Algebraic Stacks, Sections 93.7 and 93.8. An object of $F_{\acute{e}tale}$ is a scheme $U$ together with an étale morphism $\varphi : U \to F$. Then $\varphi $ is an object of $\mathcal{S}_ F$ over $U$. Hence $j(\varphi )$ is an object of $\mathcal{X}$ over $U$. In this way $j$ induces a functor $u : F_{\acute{e}tale}\to \mathcal{X}$. It is clear that $u$ is continuous and cocontinuous for the étale topology on $\mathcal{X}$. Since $j$ is an equivalence, the functor $u$ is fully faithful. Also, fibre products and equalizers exist in $F_{\acute{e}tale}$ and $u$ commutes with them because these are computed on the level of underlying schemes in $F_{\acute{e}tale}$. Thus Sites, Lemmas 7.21.5, 7.21.6, and 7.21.7 apply. In particular $u$ defines a morphism of topoi $i_ F : \mathop{\mathit{Sh}}\nolimits (F_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{\acute{e}tale})$ and there exists a left adjoint $i_{F, !}$ of $i_ F^{-1}$ which commutes with fibre products and equalizers.

We claim that $i_{F, !}$ is exact. If this is true, then we can define $\pi _ F$ by the rules $\pi _ F^{-1} = i_{F, !}$ and $\pi _{F, *} = i_ F^{-1}$ and everything is clear. To prove the claim, note that we already know that $i_{F, !}$ is right exact and preserves fibre products. Hence it suffices to show that $i_{F, !}* = *$ where $*$ indicates the final object in the category of sheaves of sets. Let $U$ be a scheme and let $\varphi : U \to F$ be surjective and étale. Set $R = U \times _ F U$. Then

\[ \xymatrix{ h_ R \ar@<1ex>[r] \ar@<-1ex>[r] & h_ U \ar[r] & {*} } \]

is a coequalizer diagram in $\mathop{\mathit{Sh}}\nolimits (F_{\acute{e}tale})$. Using the right exactness of $i_{F, !}$, using $i_{F, !} = (u_ p\ )^\# $, and using Sites, Lemma 7.5.6 we see that

\[ \xymatrix{ h_{u(R)} \ar@<1ex>[r] \ar@<-1ex>[r] & h_{u(U)} \ar[r] & i_{F, !}{*} } \]

is a coequalizer diagram in $\mathop{\mathit{Sh}}\nolimits (F_{\acute{e}tale})$. Using that $j$ is an equivalence and that $F = U/R$ it follows that the coequalizer in $\mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{\acute{e}tale})$ of the two maps $h_{u(R)} \to h_{u(U)}$ is $*$. We omit the proof that these morphisms are compatible with structure sheaves. $\square$

Remark 95.10.2. The constructions in Lemma 95.10.1 are compatible with étale localization. Here is a precise formulation. Let $S$ be a scheme. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Assume $\mathcal{X}$, $\mathcal{Y}$ are representable by algebraic spaces $F$, $G$, and that the induced morphism $f : F \to G$ of algebraic spaces is étale. Denote $f_{small} : F_{\acute{e}tale}\to G_{\acute{e}tale}$ the corresponding morphism of ringed topoi. Then

\[ \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (F_{\acute{e}tale}), \mathcal{O}_ F) \ar[rr]_{f_{small}} \ar[d]_{i_ F} & & (\mathop{\mathit{Sh}}\nolimits (G_{\acute{e}tale}), \mathcal{O}_ G) \ar[d]^{i_ G} \\ (\mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{\acute{e}tale}), \mathcal{O}_\mathcal {X}) \ar[d]_{\pi _ F} \ar[rr]_ f & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{Y}_{\acute{e}tale}), \mathcal{O}_\mathcal {Y}) \ar[d]^{\pi _ G} \\ (\mathop{\mathit{Sh}}\nolimits (F_{\acute{e}tale}), \mathcal{O}_ F) \ar[rr]^{f_{small}} & & (\mathop{\mathit{Sh}}\nolimits (G_{\acute{e}tale}), \mathcal{O}_ G) } \]

is a commutative diagram of ringed topoi. We omit the details.

Assume $\mathcal{X}$ is an algebraic stack represented by the algebraic space $F$. Let $j : \mathcal{S}_ F \to \mathcal{X}$ be an equivalence and denote $u : F_{\acute{e}tale}\to \mathcal{X}_{\acute{e}tale}$ the functor of the proof of Lemma 95.10.1 above. Given a sheaf $\mathcal{F}$ on $\mathcal{X}_{\acute{e}tale}$ we have

\[ \pi _{F, *}\mathcal{F}(U) = i_ F^{-1}\mathcal{F}(U) = \mathcal{F}(u(U)). \]

This is why we often think of $i_ F^{-1}$ as a restriction functor similarly to Definition 95.9.2 and to the restriction of a sheaf on the big étale site of a scheme to the small étale site of a scheme. We often use the notation

95.10.2.1
\begin{equation} \label{stacks-sheaves-equation-restrict} \mathcal{F}|_{F_{\acute{e}tale}} = i_ F^{-1}\mathcal{F} = \pi _{F, *}\mathcal{F} \end{equation}

in this situation.

Lemma 95.10.3. Let $S$ be a scheme. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Assume $\mathcal{X}$, $\mathcal{Y}$ are representable by algebraic spaces $F$, $G$. Denote $f : F \to G$ the induced morphism of algebraic spaces, and $f_{small} : F_{\acute{e}tale}\to G_{\acute{e}tale}$ the corresponding morphism of ringed topoi. Then

\[ \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{\acute{e}tale}), \mathcal{O}_\mathcal {X}) \ar[d]_{\pi _ F} \ar[rr]_ f & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{Y}_{\acute{e}tale}), \mathcal{O}_\mathcal {Y}) \ar[d]^{\pi _ G} \\ (\mathop{\mathit{Sh}}\nolimits (F_{\acute{e}tale}), \mathcal{O}_ F) \ar[rr]^{f_{small}} & & (\mathop{\mathit{Sh}}\nolimits (G_{\acute{e}tale}), \mathcal{O}_ G) } \]

is a commutative diagram of ringed topoi.

Proof. This is similar to Topologies, Lemma 34.4.17 (3) but there is a small snag due to the fact that $F \to G$ may not be representable by schemes. In particular we don't get a commutative diagram of ringed sites, but only a commutative diagram of ringed topoi.

Before we start the proof proper, we choose equivalences $j : \mathcal{S}_ F \to \mathcal{X}$ and $j' : \mathcal{S}_ G \to \mathcal{Y}$ which induce functors $u : F_{\acute{e}tale}\to \mathcal{X}$ and $u' : G_{\acute{e}tale}\to \mathcal{Y}$ as in the proof of Lemma 95.10.1. Because of the 2-functoriality of sheaves on categories fibred in groupoids over $\mathit{Sch}_{fppf}$ (see discussion in Section 95.3) we may assume that $\mathcal{X} = \mathcal{S}_ F$ and $\mathcal{Y} = \mathcal{S}_ G$ and that $f : \mathcal{S}_ F \to \mathcal{S}_ G$ is the functor associated to the morphism $f : F \to G$. Correspondingly we will omit $u$ and $u'$ from the notation, i.e., given an object $U \to F$ of $F_{\acute{e}tale}$ we denote $U/F$ the corresponding object of $\mathcal{X}$. Similarly for $G$.

Let $\mathcal{G}$ be a sheaf on $\mathcal{X}_{\acute{e}tale}$. To prove (2) we compute $\pi _{G, *}f_*\mathcal{G}$ and $f_{small, *}\pi _{F, *}\mathcal{G}$. To do this let $V \to G$ be an object of $G_{\acute{e}tale}$. Then

\[ \pi _{G, *}f_*\mathcal{G}(V) = f_*\mathcal{G}(V/G) = \Gamma \Big( (\mathit{Sch}/V)_{fppf} \times _{\mathcal{Y}} \mathcal{X}, \ \text{pr}^{-1}\mathcal{G}\Big) \]

see (95.5.0.1). The fibre product in the formula is

\[ (\mathit{Sch}/V)_{fppf} \times _{\mathcal{Y}} \mathcal{X} = (\mathit{Sch}/V)_{fppf} \times _{\mathcal{S}_ G} \mathcal{S}_ F = \mathcal{S}_{V \times _ G F} \]

i.e., it is the split category fibred in groupoids associated to the algebraic space $V \times _ G F$. And $\text{pr}^{-1}\mathcal{G}$ is a sheaf on $\mathcal{S}_{V \times _ G F}$ for the étale topology.

In particular, if $V \times _ G F$ is representable, i.e., if it is a scheme, then $\pi _{G, *}f_*\mathcal{G}(V) = \mathcal{G}(V \times _ G F/F)$ and also

\[ f_{small, *}\pi _{F, *}\mathcal{G}(V) = \pi _{F, *}\mathcal{G}(V \times _ G F) = \mathcal{G}(V \times _ G F/F) \]

which proves the desired equality in this special case.

In general, choose a scheme $U$ and a surjective étale morphism $U \to V \times _ G F$. Set $R = U \times _{V \times _ G F} U$. Then $U/V \times _ G F$ and $R/V \times _ G F$ are objects of the fibre product category above. Since $\text{pr}^{-1}\mathcal{G}$ is a sheaf for the étale topology on $\mathcal{S}_{V \times _ G F}$ the diagram

\[ \xymatrix{ \Gamma \Big( (\mathit{Sch}/V)_{fppf} \times _{\mathcal{Y}} \mathcal{X}, \ \text{pr}^{-1}\mathcal{G}\Big) \ar[r] & \text{pr}^{-1}\mathcal{G}(U/V \times _ G F) \ar@<1ex>[r] \ar@<-1ex>[r] & \text{pr}^{-1}\mathcal{G}(R/V \times _ G F) } \]

is an equalizer diagram. Note that $\text{pr}^{-1}\mathcal{G}(U/V \times _ G F) = \mathcal{G}(U/F)$ and $\text{pr}^{-1}\mathcal{G}(R/V \times _ G F) = \mathcal{G}(R/F)$ by the definition of pullbacks. Moreover, by the material in Properties of Spaces, Section 65.18 (especially, Properties of Spaces, Remark 65.18.4 and Lemma 65.18.8) we see that there is an equalizer diagram

\[ \xymatrix{ f_{small, *}\pi _{F, *}\mathcal{G}(V) \ar[r] & \pi _{F, *}\mathcal{G}(U/F) \ar@<1ex>[r] \ar@<-1ex>[r] & \pi _{F, *}\mathcal{G}(R/F) } \]

Since we also have $\pi _{F, *}\mathcal{G}(U/F) = \mathcal{G}(U/F)$ and $\pi _{F, *}\mathcal{G}(U/F) = \mathcal{G}(U/F)$ we obtain a canonical identification $f_{small, *}\pi _{F, *}\mathcal{G}(V) = \pi _{G, *}f_*\mathcal{G}(V)$. We omit the proof that this is compatible with restriction mappings and that it is functorial in $\mathcal{G}$. $\square$

Let $f : \mathcal{X} \to \mathcal{Y}$ and $f : F \to G$ be as in the second part of the lemma above. A consequence of the lemma, using (95.10.2.1), is that

95.10.3.1
\begin{equation} \label{stacks-sheaves-equation-compare-big-small} (f_*\mathcal{F})|_{G_{\acute{e}tale}} = f_{small, *}(\mathcal{F}|_{F_{\acute{e}tale}}) \end{equation}

for any sheaf $\mathcal{F}$ on $\mathcal{X}_{\acute{e}tale}$. Moreover, if $\mathcal{F}$ is a sheaf of $\mathcal{O}$-modules, then (95.10.3.1) is an isomorphism of $\mathcal{O}_ G$-modules on $G_{\acute{e}tale}$.

Finally, suppose that we have a $2$-commutative diagram

\[ \xymatrix{ \mathcal{U} \ar[r]^ a \ar[dr]_ f \drtwocell <\omit>{<-2>\varphi } & \mathcal{V} \ar[d]^ g \\ & \mathcal{X} } \]

of $1$-morphisms of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$, that $\mathcal{F}$ is a sheaf on $\mathcal{X}_{\acute{e}tale}$, and that $\mathcal{U}, \mathcal{V}$ are representable by algebraic spaces $U, V$. Then we obtain a comparison map

95.10.3.2
\begin{equation} \label{stacks-sheaves-equation-comparison-algebraic-spaces} c_\varphi : a_{small}^{-1}(g^{-1}\mathcal{F}|_{V_{\acute{e}tale}}) \longrightarrow f^{-1}\mathcal{F}|_{U_{\acute{e}tale}} \end{equation}

where $a : U \to V$ denotes the morphism of algebraic spaces corresponding to $a$. This is the analogue of (95.9.2.2). We define $c_\varphi $ as the adjoint to the map

\[ g^{-1}\mathcal{F}|_{V_{\acute{e}tale}} \longrightarrow a_{small, *}(f^{-1}\mathcal{F}|_{U_{\acute{e}tale}}) = (a_*f^{-1}\mathcal{F})|_{V_{\acute{e}tale}} \]

(equality by (95.10.3.1)) which is the restriction to $V$ (95.10.2.1) of the map

\[ g^{-1}\mathcal{F} \to a_*a^{-1}g^{-1}\mathcal{F} = a_*f^{-1}\mathcal{F} \]

where the last equality uses the $2$-commutativity of the diagram above. In case $\mathcal{F}$ is a sheaf of $\mathcal{O}_\mathcal {X}$-modules $c_\varphi $ induces a comparison map

95.10.3.3
\begin{equation} \label{stacks-sheaves-equation-comparison-algebraic-spaces-modules} c_\varphi : a_{small}^*(g^*\mathcal{F}|_{V_{\acute{e}tale}}) \longrightarrow f^*\mathcal{F}|_{U_{\acute{e}tale}} \end{equation}

of $\mathcal{O}_{U_{\acute{e}tale}}$-modules. This is the analogue of (95.9.4.1). Note that the properties (1), (2), (3), and (4) of Lemma 95.9.3 hold in this setting as well.


Comments (4)

Comment #1424 by Yogesh more on

minor typo in lemma74.10.1: representably should be representable. Very Minor comment: the line the pi_F refers to the map on topoi induced by the map pi_F on sites stated in the lemma, as usual we use the same notation for both maps, but I had forgotten that and the way it was stated it confused me for a minute since the target and domain of i_F and pi_F don't match up.

Comment #1437 by on

About your minor comment: Yes, this is a valid point, but it would be very costly to adjust the notation everywhere. The notation is already very heavy in this chapter, so let's hope your comment helps those who are confused as well. I fixed the typo here.

Comment #1440 by jojo on

There seems to be a latex problem at the end of this section on the formula after the phrase

"Finally, suppose that we have a 2-commutative diagram"

I get an error message : parse error at or near "\mathcal{U} \ar[r]^a \ar[dr]_f \drtwocell <\omit >{<-2>\varphi} & "

Comment #1444 by on

@#1440 This seems to have been fixed by the edit mentioned in #1437.


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