In this section we discuss the notion of a étale covering of algebraic spaces, and we define the big étale site of an algebraic space. Please compare with Topologies, Section 34.4.
Definition 73.4.1. Let $S$ be a scheme, and let $X$ be an algebraic space over $S$. An étale covering of $X$ is a family of morphisms $\{ f_ i : X_ i \to X\} _{i \in I}$ of algebraic spaces over $S$ such that each $f_ i$ is étale and such that i.e., the morphisms are jointly surjective.
\[ |X| = \bigcup \nolimits _{i \in I} |f_ i|(|X_ i|), \]
This is exactly the same as Topologies, Definition 34.4.1. In particular, if $X$ and all the $X_ i$ are schemes, then we recover the usual notion of a étale covering of schemes.
Lemma 73.4.2. Any Zariski covering is an étale covering.
Proof. This is clear from the definitions and the fact that an open immersion is an étale morphism (this follows from Morphisms, Lemma 29.36.9 via Spaces, Lemma 65.5.8 as immersions are representable). $\square$
Lemma 73.4.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.
If $X' \to X$ is an isomorphism then $\{ X' \to X\} $ is a étale covering of $X$.
If $\{ X_ i \to X\} _{i\in I}$ is a étale covering and for each $i$ we have a étale covering $\{ X_{ij} \to X_ i\} _{j\in J_ i}$, then $\{ X_{ij} \to X\} _{i \in I, j\in J_ i}$ is a étale covering.
If $\{ X_ i \to X\} _{i\in I}$ is a étale covering and $X' \to X$ is a morphism of algebraic spaces then $\{ X' \times _ X X_ i \to X'\} _{i\in I}$ is a étale covering.
Proof. Omitted. $\square$
The following lemma tells us that the sites $(\textit{Spaces}/X)_{\acute{e}tale}$ and $(\textit{Spaces}/X)_{smooth}$ have the same categories of sheaves.
Lemma 73.4.4. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\{ X_ i \to X\} _{i \in I}$ be a smooth covering of $X$. Then there exists an étale covering $\{ U_ j \to X\} _{j \in J}$ of $X$ which refines $\{ X_ i \to X\} _{i \in I}$.
Proof. First choose a scheme $U$ and a surjective étale morphism $U \to X$. For each $i$ choose a scheme $W_ i$ and a surjective étale morphism $W_ i \to X_ i$. Then $\{ W_ i \to X\} _{i \in I}$ is a smooth covering which refines $\{ X_ i \to X\} _{i \in I}$. Hence $\{ W_ i \times _ X U \to U\} _{i \in I}$ is a smooth covering of schemes. By More on Morphisms, Lemma 37.38.7 we can choose an étale covering $\{ U_ j \to U\} $ which refines $\{ W_ i \times _ X U \to U\} $. Then $\{ U_ j \to X\} _{j \in J}$ is an étale covering refining $\{ X_ i \to X\} _{i \in I}$. $\square$
Definition 73.4.5. Let $S$ be a scheme. A big étale site $(\textit{Spaces}/S)_{\acute{e}tale}$ is any site constructed as follows:
Choose a big étale site $(\mathit{Sch}/S)_{\acute{e}tale}$ as in Topologies, Section 34.4.
As underlying category take the category $\textit{Spaces}/S$ of algebraic spaces over $S$ (see discussion in Section 73.2 why this is a set).
Choose any set of coverings as in Sets, Lemma 3.11.1 starting with the category $\textit{Spaces}/S$ and the class of étale coverings of Definition 73.4.1.
Having defined this, we can localize to get the étale site of an algebraic space.
Definition 73.4.6. Let $S$ be a scheme. Let $(\textit{Spaces}/S)_{\acute{e}tale}$ be as in Definition 73.4.5. Let $X$ be an algebraic space over $S$, i.e., an object of $(\textit{Spaces}/S)_{\acute{e}tale}$. Then the big étale site $(\textit{Spaces}/X)_{\acute{e}tale}$ of $X$ is the localization of the site $(\textit{Spaces}/S)_{\acute{e}tale}$ at $X$ introduced in Sites, Section 7.25.
Recall that given an algebraic space $X$ over $S$ as in the definition, we already have defined the small étale sites $X_{spaces, {\acute{e}tale}}$ and $X_{\acute{e}tale}$, see Properties of Spaces, Section 66.18. We will silently identify the corresponding topoi using the inclusion functor $X_{\acute{e}tale}\subset X_{spaces, {\acute{e}tale}}$ (Properties of Spaces, Lemma 66.18.3) and we will call it the small étale topos of $X$. Next, we establish some relationships between the topoi associated to these sites.
Lemma 73.4.7. Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of $(\textit{Spaces}/S)_{\acute{e}tale}$. The inclusion functor $Y_{spaces, {\acute{e}tale}} \to (\textit{Spaces}/X)_{\acute{e}tale}$ is cocontinuous and induces a morphism of topoi For a sheaf $\mathcal{G}$ on $(\textit{Spaces}/X)_{\acute{e}tale}$ we have the formula $(i_ f^{-1}\mathcal{G})(U/Y) = \mathcal{G}(U/X)$. The functor $i_ f^{-1}$ also has a left adjoint $i_{f, !}$ which commutes with fibre products and equalizers.
\[ i_ f : \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}) \longrightarrow \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/X)_{\acute{e}tale}) \]
Proof. Denote the functor $u : Y_{spaces, {\acute{e}tale}} \to (\textit{Spaces}/X)_{\acute{e}tale}$. In other words, given an étale morphism $j : U \to Y$ corresponding to an object of $Y_{spaces, {\acute{e}tale}}$ we set $u(U \to T) = (f \circ j : U \to S)$. The category $Y_{spaces, {\acute{e}tale}}$ has fibre products and equalizers and $u$ commutes with them. It is immediate that $u$ cocontinuous. The functor $u$ is also continuous as $u$ transforms coverings to coverings and commutes with fibre products. Hence the Lemma follows from Sites, Lemmas 7.21.5 and 7.21.6. $\square$
Lemma 73.4.8. Let $S$ be a scheme. Let $X$ be an object of $(\textit{Spaces}/S)_{\acute{e}tale}$. The inclusion functor $X_{spaces, {\acute{e}tale}} \to (\textit{Spaces}/X)_{\acute{e}tale}$ satisfies the hypotheses of Sites, Lemma 7.21.8 and hence induces a morphism of sites and a morphism of topoi such that $\pi _ X \circ i_ X = \text{id}$. Moreover, $i_ X = i_{\text{id}_ X}$ with $i_{\text{id}_ X}$ as in Lemma 73.4.7. In particular the functor $i_ X^{-1} = \pi _{X, *}$ is described by the rule $i_ X^{-1}(\mathcal{G})(U/X) = \mathcal{G}(U/X)$.
\[ \pi _ X : (\textit{Spaces}/X)_{\acute{e}tale}\longrightarrow X_{spaces, {\acute{e}tale}} \]
\[ i_ X : \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \longrightarrow \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/X)_{\acute{e}tale}) \]
Proof. In this case the functor $u : X_{spaces, {\acute{e}tale}} \to (\textit{Spaces}/X)_{\acute{e}tale}$, in addition to the properties seen in the proof of Lemma 73.4.7 above, also is fully faithful and transforms the final object into the final object. The lemma follows from Sites, Lemma 7.21.8. $\square$
Definition 73.4.9. In the situation of Lemma 73.4.8 the functor $i_ X^{-1} = \pi _{X, *}$ is often called the restriction to the small étale site, and for a sheaf $\mathcal{F}$ on the big étale site we often denote $\mathcal{F}|_{X_{\acute{e}tale}}$ this restriction.
With this notation in place we have for a sheaf $\mathcal{F}$ on the big site and a sheaf $\mathcal{G}$ on the small site that
Moreover, we have $(i_{X, *}\mathcal{G})|_{X_{\acute{e}tale}} = \mathcal{G}$ and we have $(\pi _ X^{-1}\mathcal{G})|_{X_{\acute{e}tale}} = \mathcal{G}$.
Lemma 73.4.10. Let $S$ be a scheme. Let $f : Y \to X$ be a morphism in $(\textit{Spaces}/S)_{\acute{e}tale}$. The functor is cocontinuous, and has a continuous right adjoint They induce the same morphism of topoi We have $f_{big}^{-1}(\mathcal{G})(U/Y) = \mathcal{G}(U/X)$. We have $f_{big, *}(\mathcal{F})(U/X) = \mathcal{F}(U \times _ X Y/Y)$. Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with fibre products and equalizers.
\[ u : (\textit{Spaces}/Y)_{\acute{e}tale}\longrightarrow (\textit{Spaces}/X)_{\acute{e}tale}, \quad V/Y \longmapsto V/X \]
\[ v : (\textit{Spaces}/X)_{\acute{e}tale}\longrightarrow (\textit{Spaces}/Y)_{\acute{e}tale}, \quad (U \to X) \longmapsto (U \times _ X Y \to Y). \]
\[ f_{big} : \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/Y)_{\acute{e}tale}) \longrightarrow \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/X)_{\acute{e}tale}) \]
Proof. The functor $u$ is cocontinuous, continuous and commutes with fibre products and equalizers (details omitted; compare with the proof of Lemma 73.4.7). Hence Sites, Lemmas 7.21.5 and 7.21.6 apply and we deduce the formula for $f_{big}^{-1}$ and the existence of $f_{big!}$. Moreover, the functor $v$ is a right adjoint because given $U/Y$ and $V/X$ we have $\mathop{\mathrm{Mor}}\nolimits _ X(u(U), V) = \mathop{\mathrm{Mor}}\nolimits _ Y(U, V \times _ X Y)$ as desired. Thus we may apply Sites, Lemmas 7.22.1 and 7.22.2 to get the formula for $f_{big, *}$. $\square$
Lemma 73.4.11. Let $S$ be a scheme. Let $f : Y \to X$ be a morphism in $(\textit{Spaces}/S)_{\acute{e}tale}$.
We have $i_ f = f_{big} \circ i_ T$ with $i_ f$ as in Lemma 73.4.7 and $i_ T$ as in Lemma 73.4.8.
The functor $X_{spaces, {\acute{e}tale}} \to T_{spaces, {\acute{e}tale}}$, $(U \to X) \mapsto (U \times _ X Y \to Y)$ is continuous and induces a morphism of sites
The corresponding morphism of small étale topoi is denoted
We have $f_{small, *}(\mathcal{F})(U/X) = \mathcal{F}(U \times _ X Y/Y)$.
We have a commutative diagram of morphisms of sites
so that $f_{small} \circ \pi _ Y = \pi _ X \circ f_{big}$ as morphisms of topoi.
We have $f_{small} = \pi _ X \circ f_{big} \circ i_ Y = \pi _ X \circ i_ f$.
\[ f_{spaces, {\acute{e}tale}} : Y_{spaces, {\acute{e}tale}} \longrightarrow X_{spaces, {\acute{e}tale}} \]
\[ f_{small} : \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \]
\[ \xymatrix{ Y_{spaces, {\acute{e}tale}} \ar[d]_{f_{spaces, {\acute{e}tale}}} & (\textit{Spaces}/Y)_{\acute{e}tale}\ar[d]^{f_{big}} \ar[l]^-{\pi _ Y}\\ X_{spaces, {\acute{e}tale}} & (\textit{Spaces}/X)_{\acute{e}tale}\ar[l]_-{\pi _ X} } \]
Proof. The equality $i_ f = f_{big} \circ i_ Y$ follows from the equality $i_ f^{-1} = i_ T^{-1} \circ f_{big}^{-1}$ which is clear from the descriptions of these functors above. Thus we see (1).
The functor $u : X_{spaces, {\acute{e}tale}} \to Y_{spaces, {\acute{e}tale}}$, $u(U \to X) = (U \times _ X Y \to Y)$ was shown to give rise to a morphism of sites and correspong morphism of small étale topoi in Properties of Spaces, Lemma 66.18.8. The description of the pushforward is clear.
Part (3) follows because $\pi _ X$ and $\pi _ Y$ are given by the inclusion functors and $f_{spaces, {\acute{e}tale}}$ and $f_{big}$ by the base change functors $U \mapsto U \times _ X Y$.
Statement (4) follows from (3) by precomposing with $i_ Y$. $\square$
In the situation of the lemma, using the terminology of Definition 73.4.9 we have: for $\mathcal{F}$ a sheaf on the big étale site of $Y$
This equality is clear from the commutativity of the diagram of sites of the lemma, since restriction to the small étale site of $Y$, resp. $X$ is given by $\pi _{Y, *}$, resp. $\pi _{X, *}$. A similar formula involving pullbacks and restrictions is false.
Lemma 73.4.12. Let $S$ be a scheme. Given morphisms $f : X \to Y$, $g : Y \to Z$ in $(\textit{Spaces}/S)_{\acute{e}tale}$ we have $g_{big} \circ f_{big} = (g \circ f)_{big}$ and $g_{small} \circ f_{small} = (g \circ f)_{small}$.
Proof. This follows from the simple description of pushforward and pullback for the functors on the big sites from Lemma 73.4.10. For the functors on the small sites this follows from the description of the pushforward functors in Lemma 73.4.11. $\square$
Lemma 73.4.13. Let $S$ be a scheme. Consider a cartesian diagram in $(\textit{Spaces}/S)_{\acute{e}tale}$. Then $i_ g^{-1} \circ f_{big, *} = f'_{small, *} \circ (i_{g'})^{-1}$ and $g_{big}^{-1} \circ f_{big, *} = f'_{big, *} \circ (g'_{big})^{-1}$.
\[ \xymatrix{ Y' \ar[r]_{g'} \ar[d]_{f'} & Y \ar[d]^ f \\ X' \ar[r]^ g & X } \]
Proof. Since the diagram is cartesian, we have for $U'/X'$ that $U' \times _{X'} Y' = U' \times _ X Y$. Hence both $i_ g^{-1} \circ f_{big, *}$ and $f'_{small, *} \circ (i_{g'})^{-1}$ send a sheaf $\mathcal{F}$ on $(\textit{Spaces}/Y)_{\acute{e}tale}$ to the sheaf $U' \mapsto \mathcal{F}(U' \times _{X'} Y')$ on $X'_{\acute{e}tale}$ (use Lemmas 73.4.7 and 73.4.10). The second equality can be proved in the same manner or can be deduced from the very general Sites, Lemma 7.28.1. $\square$
Remark 73.4.14. The sites $(\textit{Spaces}/X)_{\acute{e}tale}$ and $X_{spaces, {\acute{e}tale}}$ come with structure sheaves. For the small étale site we have seen this in Properties of Spaces, Section 66.21. The structure sheaf $\mathcal{O}$ on the big étale site $(\textit{Spaces}/X)_{\acute{e}tale}$ is defined by assigning to an object $U$ the global sections of the structure sheaf of $U$. This makes sense because after all $U$ is an algebraic space itself hence has a structure sheaf. Since $\mathcal{O}_ U$ is a sheaf on the étale site of $U$, the presheaf $\mathcal{O}$ so defined satisfies the sheaf condition for coverings of $U$, i.e., $\mathcal{O}$ is a sheaf. We can upgrade the morphisms $i_ f$, $\pi _ X$, $i_ X$, $f_{small}$, and $f_{big}$ defined above to morphisms of ringed sites, respectively topoi. Let us deal with these one by one.
In Lemma 73.4.7 denote $\mathcal{O}$ the structure sheaf on $(\textit{Spaces}/X)_{\acute{e}tale}$. We have $(i_ f^{-1}\mathcal{O})(U/Y) = \mathcal{O}_ U(U) = \mathcal{O}_ Y(U)$ by construction. Hence an isomorphism $i_ f^\sharp : i_ f^{-1}\mathcal{O} \to \mathcal{O}_ Y$.
In Lemma 73.4.8 it was noted that $i_ X$ is a special case of $i_ f$ with $f = \text{id}_ X$ hence we are back in case (1).
In Lemma 73.4.8 the morphism $\pi _ X$ satisfies $(\pi _{X, *}\mathcal{O})(U) = \mathcal{O}(U) = \mathcal{O}_ X(U)$. Hence we can use this to define $\pi _ X^\sharp : \mathcal{O}_ X \to \pi _{X, *}\mathcal{O}$.
In Lemma 73.4.11 the extension of $f_{small}$ to a morphism of ringed topoi was discussed in Properties of Spaces, Lemma 66.21.3.
In Lemma 73.4.11 the functor $f_{big}^{-1}$ is simply the restriction via the inclusion functor $(\textit{Spaces}/Y)_{\acute{e}tale}\to (\textit{Spaces}/X)_{\acute{e}tale}$. Let $\mathcal{O}_1$ be the structure sheaf on $(\textit{Spaces}/X)_{\acute{e}tale}$ and let $\mathcal{O}_2$ be the structure sheaf on $(\textit{Spaces}/Y)_{\acute{e}tale}$. We obtain a canonical isomorphism $f_{big}^\sharp : f_{big}^{-1}\mathcal{O}_1 \to \mathcal{O}_2$.
Moreover, with these definitions compositions work out correctly too. We omit giving a detailed statement and proof.
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Comment #1793 by Matthieu Romagny on