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 \input{preamble} % OK, start here. % \begin{document} \title{Properties of Algebraic Spaces} \maketitle \phantomsection \label{section-phantom} \tableofcontents \section{Introduction} \label{section-introduction} \noindent Please see Spaces, Section \ref{spaces-section-introduction} for a brief introduction to algebraic spaces, and please read some of that chapter for our basic definitions and conventions concerning algebraic spaces. In this chapter we start introducing some basic notions and properties of algebraic spaces. A fundamental reference for the case of quasi-separated algebraic spaces is \cite{Kn}. \medskip\noindent The discussion is somewhat awkward at times since we made the design decision to first talk about properties of algebraic spaces by themselves, and only later about properties of morphisms of algebraic spaces. We make an exception for this rule regarding {\it \'etale morphisms} of algebraic spaces, which we introduce in Section \ref{section-etale-morphisms}. But until that section whenever we say a morphism has a certain property, it automatically means the source of the morphism is a scheme (or perhaps the morphism is representable). \medskip\noindent Some of the material in the chapter (especially regarding points) will be improved upon in the chapter on decent algebraic spaces. \section{Conventions} \label{section-conventions} \noindent The standing assumption is that all schemes are contained in a big fppf site $\Sch_{fppf}$. And all rings $A$ considered have the property that $\Spec(A)$ is (isomorphic) to an object of this big site. \medskip\noindent Let $S$ be a scheme and let $X$ be an algebraic space over $S$. In this chapter and the following we will write $X \times_S X$ for the product of $X$ with itself (in the category of algebraic spaces over $S$), instead of $X \times X$. The reason is that we want to avoid confusion when changing base schemes, as in Spaces, Section \ref{spaces-section-change-base-scheme}. \section{Separation axioms} \label{section-separation} \noindent In this section we collect all the absolute'' separation conditions of algebraic spaces. Since in our language any algebraic space is an algebraic space over some definite base scheme, any absolute property of $X$ over $S$ corresponds to a conditions imposed on $X$ viewed as an algebraic space over $\Spec(\mathbf{Z})$. Here is the precise formulation. \begin{definition} \label{definition-separated} (Compare Spaces, Definition \ref{spaces-definition-separated}.) Consider a big fppf site $\Sch_{fppf} = (\Sch/\Spec(\mathbf{Z}))_{fppf}$. Let $X$ be an algebraic space over $\Spec(\mathbf{Z})$. Let $\Delta : X \to X \times X$ be the diagonal morphism. \begin{enumerate} \item We say $X$ is {\it separated} if $\Delta$ is a closed immersion. \item We say $X$ is {\it locally separated}\footnote{In the literature this often refers to quasi-separated and locally separated algebraic spaces.} if $\Delta$ is an immersion. \item We say $X$ is {\it quasi-separated} if $\Delta$ is quasi-compact. \item We say $X$ is {\it Zariski locally quasi-separated}\footnote{ This notion was suggested by B.\ Conrad.} if there exists a Zariski covering $X = \bigcup_{i \in I} X_i$ (see Spaces, Definition \ref{spaces-definition-Zariski-open-covering}) such that each $X_i$ is quasi-separated. \end{enumerate} Let $S$ is a scheme contained in $\Sch_{fppf}$, and let $X$ be an algebraic space over $S$. Then we say $X$ is {\it separated}, {\it locally separated}, {\it quasi-separated}, or {\it Zariski locally quasi-separated} if $X$ viewed as an algebraic space over $\Spec(\mathbf{Z})$ (see Spaces, Definition \ref{spaces-definition-base-change}) has the corresponding property. \end{definition} \noindent It is true that an algebraic space $X$ over $S$ which is separated (in the absolute sense above) is separated over $S$ (and similarly for the other absolute separation properties above). This will be discussed in great detail in Morphisms of Spaces, Section \ref{spaces-morphisms-section-separation-axioms}. We will see in Lemma \ref{lemma-quasi-separated-quasi-compact-pieces} that being Zariski locally separated is independent of the base scheme (hence equivalent to the absolute notion). \begin{lemma} \label{lemma-trivial-implications} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. We have the following implications among the separation axioms of Definition \ref{definition-separated}: \begin{enumerate} \item separated implies all the others, \item quasi-separated implies Zariski locally quasi-separated. \end{enumerate} \end{lemma} \begin{proof} Omitted. \end{proof} \begin{lemma} \label{lemma-characterize-quasi-separated} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The following are equivalent \begin{enumerate} \item $X$ is a quasi-separated algebraic space, \item for $U \to X$, $V \to X$ with $U$, $V$ quasi-compact schemes the fibre product $U \times_X V$ is quasi-compact, \item for $U \to X$, $V \to X$ with $U$, $V$ affine the fibre product $U \times_X V$ is quasi-compact. \end{enumerate} \end{lemma} \begin{proof} Using Spaces, Lemma \ref{spaces-lemma-category-of-spaces-over-smaller-base-scheme} we see that we may assume $S = \Spec(\mathbf{Z})$. Since $U \times_X V = X \times_{X \times X} (U \times V)$ and since $U \times V$ is quasi-compact if $U$ and $V$ are so, we see that (1) implies (2). It is clear that (2) implies (3). Assume (3). Choose a scheme $W$ and a surjective \'etale morphism $W \to X$. Then $W \times W \to X \times X$ is surjective \'etale. Hence it suffices to show that $$j : W \times_X W = X \times_{(X \times X)} (W \times W) \to W \times W$$ is quasi-compact, see Spaces, Lemma \ref{spaces-lemma-descent-representable-transformations-property}. If $U \subset W$ and $V \subset W$ are affine opens, then $j^{-1}(U \times V) = U \times_X V$ is quasi-compact by assumption. Since the affine opens $U \times V$ form an affine open covering of $W \times W$ (Schemes, Lemma \ref{schemes-lemma-affine-covering-fibre-product}) we conclude by Schemes, Lemma \ref{schemes-lemma-quasi-compact-affine}. \end{proof} \begin{lemma} \label{lemma-characterize-separated} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The following are equivalent \begin{enumerate} \item $X$ is a separated algebraic space, \item for $U \to X$, $V \to X$ with $U$, $V$ affine the fibre product $U \times_X V$ is affine and $$\mathcal{O}(U) \otimes_\mathbf{Z} \mathcal{O}(V) \longrightarrow \mathcal{O}(U \times_X V)$$ is surjective. \end{enumerate} \end{lemma} \begin{proof} Using Spaces, Lemma \ref{spaces-lemma-category-of-spaces-over-smaller-base-scheme} we see that we may assume $S = \Spec(\mathbf{Z})$. Since $U \times_X V = X \times_{X \times X} (U \times V)$ and since $U \times V$ is affine if $U$ and $V$ are so, we see that (1) implies (2). Assume (2). Choose a scheme $W$ and a surjective \'etale morphism $W \to X$. Then $W \times W \to X \times X$ is surjective \'etale. Hence it suffices to show that $$j : W \times_X W = X \times_{(X \times X)} (W \times W) \to W \times W$$ is a closed immersion, see Spaces, Lemma \ref{spaces-lemma-descent-representable-transformations-property}. If $U \subset W$ and $V \subset W$ are affine opens, then $j^{-1}(U \times V) = U \times_X V$ is affine by assumption and the map $U \times_X V \to U \times V$ is a closed immersion because the corresponding ring map is surjective. Since the affine opens $U \times V$ form an affine open covering of $W \times W$ (Schemes, Lemma \ref{schemes-lemma-affine-covering-fibre-product}) we conclude by Morphisms, Lemma \ref{morphisms-lemma-closed-immersion}. \end{proof} \section{Points of algebraic spaces} \label{section-points} \noindent As is clear from Spaces, Example \ref{spaces-example-affine-line-translation} a point of an algebraic space should not be defined as a monomorphism from the spectrum of a field. Instead we define them as equivalence classes of morphisms of spectra of fields exactly as explained in Schemes, Section \ref{schemes-section-points}. \medskip\noindent Let $S$ be a scheme. Let $F$ be a presheaf on $(\Sch/S)_{fppf}$. Let $K$ is a field. Consider a morphism $$\Spec(K) \longrightarrow F.$$ By the Yoneda Lemma this is given by an element $p \in F(\Spec(K))$. We say that two such pairs $(\Spec(K), p)$ and $(\Spec(L), q)$ are {\it equivalent} if there exists a third field $\Omega$ and a commutative diagram $$\xymatrix{ \Spec(\Omega) \ar[r] \ar[d] & \Spec(L) \ar[d]^q \\ \Spec(K) \ar[r]^p & F. }$$ In other words, there are field extensions $K \to \Omega$ and $L \to \Omega$ such that $p$ and $q$ map to the same element of $F(\Spec(\Omega))$. We omit the verification that this defines an equivalence relation. \begin{definition} \label{definition-points} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. A {\it point} of $X$ is an equivalence class of morphisms from spectra of fields into $X$. The set of points of $X$ is denoted $|X|$. \end{definition} \noindent Note that if $f : X \to Y$ is a morphism of algebraic spaces over $S$, then there is an induced map $|f| : |X| \to |Y|$ which maps a representative $x : \Spec(K) \to X$ to the representative $f \circ x : \Spec(K) \to Y$. \begin{lemma} \label{lemma-scheme-points} Let $S$ be a scheme. Let $X$ be a scheme over $S$. The points of $X$ as a scheme are in canonical 1-1 correspondence with the points of $X$ as an algebraic space. \end{lemma} \begin{proof} This is Schemes, Lemma \ref{schemes-lemma-characterize-points}. \end{proof} \begin{lemma} \label{lemma-points-cartesian} Let $S$ be a scheme. Let $$\xymatrix{ Z \times_Y X \ar[r] \ar[d] & X \ar[d] \\ Z \ar[r] & Y }$$ be a cartesian diagram of algebraic spaces over $S$. Then the map of sets of points $$|Z \times_Y X| \longrightarrow |Z| \times_{|Y|} |X|$$ is surjective. \end{lemma} \begin{proof} Namely, suppose given fields $K$, $L$ and morphisms $\Spec(K) \to X$, $\Spec(L) \to Z$, then the assumption that they agree as elements of $|Y|$ means that there is a common extension $K \subset M$ and $L \subset M$ such that $\Spec(M) \to \Spec(K) \to X \to Y$ and $\Spec(M) \to \Spec(L) \to Z \to Y$ agree. And this is exactly the condition which says you get a morphism $\Spec(M) \to Z \times_Y X$. \end{proof} \begin{lemma} \label{lemma-characterize-surjective} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $f : T \to X$ be a morphism from a scheme to $X$. The following are equivalent \begin{enumerate} \item $f : T \to X$ is surjective (according to Spaces, Definition \ref{spaces-definition-relative-representable-property}), and \item $|f| : |T| \to |X|$ is surjective. \end{enumerate} \end{lemma} \begin{proof} Assume (1). Let $x : \Spec(K) \to X$ be a morphism from the spectrum of a field into $X$. By assumption the morphism of schemes $\Spec(K) \times_X T \to \Spec(K)$ is surjective. Hence there exists a field extension $K \subset K'$ and a morphism $\Spec(K') \to \Spec(K) \times_X T$ such that the left square in the diagram $$\xymatrix{ \Spec(K') \ar[r] \ar[d] & \Spec(K) \times_X T \ar[d] \ar[r] & T \ar[d] \\ \Spec(K) \ar@{=}[r] & \Spec(K) \ar[r]^-x & X }$$ is commutative. This shows that $|f| : |T| \to |X|$ is surjective. \medskip\noindent Assume (2). Let $Z \to X$ be a morphism where $Z$ is a scheme. We have to show that the morphism of schemes $Z \times_X T \to T$ is surjective, i.e., that $|Z \times_X T| \to |Z|$ is surjective. This follows from (2) and Lemma \ref{lemma-points-cartesian}. \end{proof} \begin{lemma} \label{lemma-points-presentation} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $X = U/R$ be a presentation of $X$, see Spaces, Definition \ref{spaces-definition-presentation}. Then the image of $|R| \to |U| \times |U|$ is an equivalence relation and $|X|$ is the quotient of $|U|$ by this equivalence relation. \end{lemma} \begin{proof} The assumption means that $U$ is a scheme, $p : U \to X$ is a surjective, \'etale morphism, $R = U \times_X U$ is a scheme and defines an \'etale equivalence relation on $U$ such that $X = U/R$ as sheaves. By Lemma \ref{lemma-characterize-surjective} we see that $|U| \to |X|$ is surjective. By Lemma \ref{lemma-points-cartesian} the map $$|R| \longrightarrow |U| \times_{|X|} |U|$$ is surjective. Hence the image of $|R| \to |U| \times |U|$ is exactly the set of pairs $(u_1, u_2) \in |U| \times |U|$ such that $u_1$ and $u_2$ have the same image in $|X|$. Combining these two statements we get the result of the lemma. \end{proof} \begin{lemma} \label{lemma-topology-points} Let $S$ be a scheme. There exists a unique topology on the sets of points of algebraic spaces over $S$ with the following properties: \begin{enumerate} \item for every morphism of algebraic spaces $X \to Y$ over $S$ the map $|X| \to |Y|$ is continuous, and \item for every \'etale morphism $U \to X$ with $U$ a scheme the map of topological spaces $|U| \to |X|$ is continuous and open. \end{enumerate} \end{lemma} \begin{proof} Let $X$ be an algebraic space over $S$. Let $p : U \to X$ be a surjective \'etale morphism where $U$ is a scheme over $S$. We define $W \subset |X|$ is open if and only if $|p|^{-1}(W)$ is an open subset of $|U|$. This is a topology on $|X|$ (it is the quotient topology on $|X|$, see Topology, Lemma \ref{topology-lemma-quotient}). \medskip\noindent Let us prove that the topology is independent of the choice of the presentation. To do this it suffices to show that if $U'$ is a scheme, and $U' \to X$ is an \'etale morphism, then the map $|U'| \to |X|$ (with topology on $|X|$ defined using $U \to X$ as above) is open and continuous; which in addition will prove that (2) holds. Set $U'' = U \times_X U'$, so that we have the commutative diagram $$\xymatrix{ U'' \ar[r] \ar[d] & U' \ar[d] \\ U \ar[r] & X }$$ As $U \to X$ and $U' \to X$ are \'etale we see that both $U'' \to U$ and $U'' \to U'$ are \'etale morphisms of schemes. Moreover, $U'' \to U'$ is surjective. Hence we get a commutative diagram of maps of sets $$\xymatrix{ |U''| \ar[r] \ar[d] & |U'| \ar[d] \\ |U| \ar[r] & |X| }$$ The lower horizontal arrow is surjective (see Lemma \ref{lemma-characterize-surjective} or Lemma \ref{lemma-points-presentation}) and continuous by definition of the topology on $|X|$. The top horizontal arrow is surjective, continuous, and open by Morphisms, Lemma \ref{morphisms-lemma-etale-open}. The left vertical arrow is continuous and open (by Morphisms, Lemma \ref{morphisms-lemma-etale-open} again.) Hence it follows formally that the right vertical arrow is continuous and open. \medskip\noindent To finish the proof we prove (1). Let $a : X \to Y$ be a morphism of algebraic spaces. According to Spaces, Lemma \ref{spaces-lemma-lift-morphism-presentations} we can find a diagram $$\xymatrix{ U \ar[d]_p \ar[r]_\alpha & V \ar[d]^q \\ X \ar[r]^a & Y }$$ where $U$ and $V$ are schemes, and $p$ and $q$ are surjective and \'etale. This gives rise to the diagram $$\xymatrix{ |U| \ar[d]_p \ar[r]_\alpha & |V| \ar[d]^q \\ |X| \ar[r]^a & |Y| }$$ where all but the lower horizontal arrows are known to be continuous and the two vertical arrows are surjective and open. It follows that the lower horizontal arrow is continuous as desired. \end{proof} \begin{definition} \label{definition-topological-space} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The underlying {\it topological space} of $X$ is the set of points $|X|$ endowed with the topology constructed in Lemma \ref{lemma-topology-points}. \end{definition} \noindent It turns out that this topological space carries the same information as the small Zariski site $X_{Zar}$ of Spaces, Definition \ref{spaces-definition-small-Zariski-site}. \begin{lemma} \label{lemma-open-subspaces} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \begin{enumerate} \item The rule $X' \mapsto |X'|$ defines an inclusion preserving bijection between open subspaces $X'$ (see Spaces, Definition \ref{spaces-definition-immersion}) of $X$, and opens of the topological space $|X|$. \item A family $\{X_i \subset X\}_{i \in I}$ of open subspaces of $X$ is a Zariski covering (see Spaces, Definition \ref{spaces-definition-Zariski-open-covering}) if and only if $|X| = \bigcup |X_i|$. \end{enumerate} In other words, the small Zariski site $X_{Zar}$ of $X$ is canonically identified with a site associated to the topological space $|X|$ (see Sites, Example \ref{sites-example-site-topological}). \end{lemma} \begin{proof} In order to prove (1) let us construct the inverse of the rule. Namely, suppose that $W \subset |X|$ is open. Choose a presentation $X = U/R$ corresponding to the surjective \'etale map $p : U \to X$ and \'etale maps $s, t : R \to U$. By construction we see that $|p|^{-1}(W)$ is an open of $U$. Denote $W' \subset U$ the corresponding open subscheme. It is clear that $R' = s^{-1}(W') = t^{-1}(W')$ is a Zariski open of $R$ which defines an \'etale equivalence relation on $W'$. By Spaces, Lemma \ref{spaces-lemma-finding-opens} the morphism $X' = W'/R' \to X$ is an open immersion. Hence $X'$ is an algebraic space by Spaces, Lemma \ref{spaces-lemma-representable-over-space}. By construction $|X'| = W$, i.e., $X'$ is a subspace of $X$ corresponding to $W$. Thus (1) is proved. \medskip\noindent To prove (2), note that if $\{X_i \subset X\}_{i \in I}$ is a collection of open subspaces, then it is a Zariski covering if and only if the $U = \bigcup U \times_X X_i$ is an open covering. This follows from the definition of a Zariski covering and the fact that the morphism $U \to X$ is surjective as a map of presheaves on $(\Sch/S)_{fppf}$. On the other hand, we see that $|X| = \bigcup |X_i|$ if and only if $U = \bigcup U \times_X X_i$ by Lemma \ref{lemma-points-presentation} (and the fact that the projections $U \times_X X_i \to X_i$ are surjective and \'etale). Thus the equivalence of (2) follows. \end{proof} \begin{lemma} \label{lemma-factor-through-open-subspace} Let $S$ be a scheme. Let $X$, $Y$ be algebraic spaces over $S$. Let $X' \subset X$ be an open subspace. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. Then $f$ factors through $X'$ if and only if $|f| : |Y| \to |X|$ factors through $|X'| \subset |X|$. \end{lemma} \begin{proof} By Spaces, Lemma \ref{spaces-lemma-base-change-immersions} we see that $Y' = Y \times_X X' \to Y$ is an open immersion. If $|f|(|Y|) \subset |X'|$, then clearly $|Y'| = |Y|$. Hence $Y' = Y$ by Lemma \ref{lemma-open-subspaces}. \end{proof} \begin{lemma} \label{lemma-etale-image-open} Let $S$ be a scheme. Let $X$ be an algebraic spaces over $S$. Let $U$ be a scheme and let $f : U \to X$ be an \'etale morphism. Let $X' \subset X$ be the open subspace corresponding to the open $|f|(|U|) \subset |X|$ via Lemma \ref{lemma-open-subspaces}. Then $f$ factors through a surjective \'etale morphism $f' : U \to X'$. Moreover, if $R = U \times_X U$, then $R = U \times_{X'} U$ and $X'$ has the presentation $X' = U/R$. \end{lemma} \begin{proof} The existence of the factorization follows from Lemma \ref{lemma-factor-through-open-subspace}. The morphism $f'$ is surjective according to Lemma \ref{lemma-characterize-surjective}. To see $f'$ is \'etale, suppose that $T \to X'$ is a morphism where $T$ is a scheme. Then $T \times_X U = T \times_{X'} U$ as $X" \to X$ is a monomorphism of sheaves. Thus the projection $T \times_{X'} U \to T$ is \'etale as we assumed $f$ \'etale. We have $U \times_X U = U \times_{X'} U$ as $X' \to X$ is a monomorphism. Then $X' = U/R$ follows from Spaces, Lemma \ref{spaces-lemma-space-presentation}. \end{proof} \begin{lemma} \label{lemma-points-monomorphism} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Consider the map $$\{\Spec(k) \to X \text{ monomorphism}\} \longrightarrow |X|$$ This map is injective. \end{lemma} \begin{proof} Suppose that $\varphi_i : \Spec(k_i) \to X$ are monomorphisms for $i = 1, 2$. If $\varphi_1$ and $\varphi_2$ define the same point of $|X|$, then we see that the scheme $$Y = \Spec(k_1) \times_{\varphi_1, X, \varphi_2} \Spec(k_2)$$ is nonempty. Since the base change of a monomorphism is a monomorphism this means that the projection morphisms $Y \to \Spec(k_i)$ are monomorphisms. Hence $\Spec(k_1) = Y = \Spec(k_2)$ as schemes over $X$, see Schemes, Lemma \ref{schemes-lemma-mono-towards-spec-field}. We conclude that $\varphi_1 = \varphi_2$, which proves the lemma. \end{proof} \noindent We will see in Decent Spaces, Lemma \ref{decent-spaces-lemma-decent-points-monomorphism} that this map is a bijection when $X$ is decent. \section{Quasi-compact spaces} \label{section-quasi-compact} \begin{definition} \label{definition-quasi-compact} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. We say $X$ is {\it quasi-compact} if there exists a surjective \'etale morphism $U \to X$ with $U$ quasi-compact. \end{definition} \begin{lemma} \label{lemma-quasi-compact-space} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Then $X$ is quasi-compact if and only if $|X|$ is quasi-compact. \end{lemma} \begin{proof} Choose a scheme $U$ and an \'etale surjective morphism $U \to X$. We will use Lemma \ref{lemma-characterize-surjective}. If $U$ is quasi-compact, then since $|U| \to |X|$ is surjective we conclude that $|X|$ is quasi-compact. If $|X|$ is quasi-compact, then since $|U| \to |X|$ is open we see that there exists a quasi-compact open $U' \subset U$ such that $|U'| \to |X|$ is surjective (and still \'etale). Hence we win. \end{proof} \begin{lemma} \label{lemma-finite-disjoint-quasi-compact} A finite disjoint union of quasi-compact algebraic spaces is a quasi-compact algebraic space. \end{lemma} \begin{proof} This is clear from Lemma \ref{lemma-quasi-compact-space} and the corresponding topological fact. \end{proof} \begin{example} \label{example-quasi-compact-not-very-reasonable} The space $\mathbf{A}^1_{\mathbf{Q}}/\mathbf{Z}$ is a quasi-compact algebraic space. \end{example} \begin{lemma} \label{lemma-space-locally-quasi-compact} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Every point of $|X|$ has a fundamental system of open quasi-compact neighbourhoods. In particular $|X|$ is locally quasi-compact in the sense of Topology, Definition \ref{topology-definition-locally-quasi-compact}. \end{lemma} \begin{proof} This follows formally from the fact that there exists a scheme $U$ and a surjective, open, continuous map $U \to |X|$ of topological spaces. To be a bit more precise, if $u \in U$ maps to $x \in |X|$, then the images of the affine neighbourhoods of $u$ will give a fundamental system of quasi-compact open neighbourhoods of $x$. \end{proof} \section{Special coverings} \label{section-special-coverings} \noindent In this section we collect some straightforward lemmas on the existence of \'etale surjective coverings of algebraic spaces. \begin{lemma} \label{lemma-cover-by-union-affines} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. There exists a surjective \'etale morphism $U \to X$ where $U$ is a disjoint union of affine schemes. We may in addition assume each of these affines maps into an affine open of $S$. \end{lemma} \begin{proof} Let $V \to X$ be a surjective \'etale morphism. Let $V = \bigcup_{i \in I} V_i$ be a Zariski open covering such that each $V_i$ maps into an affine open of $S$. Then set $U = \coprod_{i \in I} V_i$ with induced morphism $U \to V \to X$. This is \'etale and surjective as a composition of \'etale and surjective representable transformations of functors (via the general principle Spaces, Lemma \ref{spaces-lemma-composition-representable-transformations-property} and Morphisms, Lemmas \ref{morphisms-lemma-composition-surjective} and \ref{morphisms-lemma-composition-etale}). \end{proof} \begin{lemma} \label{lemma-union-of-quasi-compact} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. There exists a Zariski covering $X = \bigcup X_i$ such that each algebraic space $X_i$ has a surjective \'etale covering by an affine scheme. We may in addition assume each $X_i$ maps into an affine open of $S$. \end{lemma} \begin{proof} By Lemma \ref{lemma-cover-by-union-affines} we can find a surjective \'etale morphism $U = \coprod U_i \to X$, with $U_i$ affine and mapping into an affine open of $S$. Let $X_i \subset X$ be the open subspace of $X$ such that $U_i \to X$ factors through an \'etale surjective morphism $U_i \to X_i$, see Lemma \ref{lemma-etale-image-open}. Since $U = \bigcup U_i$ we see that $X = \bigcup X_i$. As $U_i \to X_i$ is surjective it follows that $X_i \to S$ maps into an affine open of $S$. \end{proof} \begin{lemma} \label{lemma-quasi-compact-affine-cover} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Then $X$ is quasi-compact if and only if there exists an \'etale surjective morphism $U \to X$ with $U$ an affine scheme. \end{lemma} \begin{proof} If there exists an \'etale surjective morphism $U \to X$ with $U$ affine then $X$ is quasi-compact by Definition \ref{definition-quasi-compact}. Conversely, if $X$ is quasi-compact, then $|X|$ is quasi-compact. Let $U = \coprod_{i \in I} U_i$ be a disjoint union of affine schemes with an \'etale and surjective map $\varphi : U \to X$ (Lemma \ref{lemma-cover-by-union-affines}). Then $|X| = \bigcup \varphi(|U_i|)$ and by quasi-compactness there is a finite subset $i_1, \ldots, i_n$ such that $|X| = \bigcup \varphi(|U_{i_j}|)$. Hence $U_{i_1} \cup \ldots \cup U_{i_n}$ is an affine scheme with a finite surjective morphism towards $X$. \end{proof} \noindent The following lemma will be obsoleted by the discussion of separated morphisms in the chapter on morphisms of algebraic spaces. \begin{lemma} \label{lemma-separated-cover} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a separated scheme and $U \to X$ \'etale. Then $U \to X$ is separated, and $R = U \times_X U$ is a separated scheme. \end{lemma} \begin{proof} Let $X' \subset X$ be the open subscheme such that $U \to X$ factors through an \'etale surjection $U \to X'$, see Lemma \ref{lemma-etale-image-open}. If $U \to X'$ is separated, then so is $U \to X$, see Spaces, Lemma \ref{spaces-lemma-composition-representable-transformations-property} (as the open immersion $X' \to X$ is separated by Spaces, Lemma \ref{spaces-lemma-representable-transformations-property-implication} and Schemes, Lemma \ref{schemes-lemma-immersions-monomorphisms}). Moreover, since $U \times_{X'} U = U \times_X U$ it suffices to prove the result after replacing $X$ by $X'$, i.e., we may assume $U \to X$ surjective. Consider the commutative diagram $$\xymatrix{ R = U \times_X U \ar[r] \ar[d] & U \ar[d] \\ U \ar[r] & X }$$ In the proof of Spaces, Lemma \ref{spaces-lemma-properties-diagonal} we have seen that $j : R \to U \times_S U$ is separated. The morphism of schemes $U \to S$ is separated as $U$ is a separated scheme, see Schemes, Lemma \ref{schemes-lemma-compose-after-separated}. Hence $U \times_S U \to U$ is separated as a base change, see Schemes, Lemma \ref{schemes-lemma-separated-permanence}. Hence the scheme $U \times_S U$ is separated (by the same lemma). Since $j$ is separated we see in the same way that $R$ is separated. Hence $R \to U$ is a separated morphism (by Schemes, Lemma \ref{schemes-lemma-compose-after-separated} again). Thus by Spaces, Lemma \ref{spaces-lemma-representable-morphisms-spaces-property} and the diagram above we conclude that $U \to X$ is separated. \end{proof} \begin{lemma} \label{lemma-quasi-separated} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If there exists a quasi-separated scheme $U$ and a surjective \'etale morphism $U \to X$ such that either of the projections $U \times_X U \to U$ is quasi-compact, then $X$ is quasi-separated. \end{lemma} \begin{proof} We may think of $X$ as an algebraic space over $\mathbf{Z}$. Consider the cartesian diagram $$\xymatrix{ U \times_X U \ar[r] \ar[d]_j & X \ar[d]^\Delta \\ U \times U \ar[r] & X \times X }$$ Since $U$ is quasi-separated the projection $U \times U \to U$ is quasi-separated (as a base change of a quasi-separated morphism of schemes, see Schemes, Lemma \ref{schemes-lemma-separated-permanence}). Hence the assumption in the lemma implies $j$ is quasi-compact by Schemes, Lemma \ref{schemes-lemma-quasi-compact-permanence}. By Spaces, Lemma \ref{spaces-lemma-representable-morphisms-spaces-property} we see that $\Delta$ is quasi-compact as desired. \end{proof} \begin{lemma} \label{lemma-quasi-separated-quasi-compact-pieces} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The following are equivalent \begin{enumerate} \item $X$ is Zariski locally quasi-separated over $S$, \item $X$ is Zariski locally quasi-separated, \item there exists a Zariski open covering $X = \bigcup X_i$ such that for each $i$ there exists an affine scheme $U_i$ and a quasi-compact surjective \'etale morphism $U_i \to X_i$, and \item there exists a Zariski open covering $X = \bigcup X_i$ such that for each $i$ there exists an affine scheme $U_i$ which maps into an affine open of $S$ and a quasi-compact surjective \'etale morphism $U_i \to X_i$. \end{enumerate} \end{lemma} \begin{proof} Assume $U_i \to X_i \subset X$ are as in (3). To prove (4) choose for each $i$ a finite affine open covering $U_i = U_{i1} \cup \ldots \cup U_{in_i}$ such that each $U_{ij}$ maps into an affine open of $S$. The compositions $U_{ij} \to U_i \to X_i$ are \'etale and quasi-compact (see Spaces, Lemma \ref{spaces-lemma-composition-representable-transformations-property}). Let $X_{ij} \subset X_i$ be the open subspace corresponding to the image of $|U_{ij}| \to |X_i|$, see Lemma \ref{lemma-etale-image-open}. Note that $U_{ij} \to X_{ij}$ is quasi-compact as $X_{ij} \subset X_i$ is a monomorphism and as $U_{ij} \to X$ is quasi-compact. Then $X = \bigcup X_{ij}$ is a covering as in (4). The implication (4) $\Rightarrow$ (3) is immediate. \medskip\noindent Assume (4). To show that $X$ is Zariski locally quasi-separated over $S$ it suffices to show that $X_i$ is quasi-separated over $S$. Hence we may assume there exists an affine scheme $U$ mapping into an affine open of $S$ and a quasi-compact surjective \'etale morphism $U \to X$. Consider the fibre product square $$\xymatrix{ U \times_X U \ar[r] \ar[d] & U \times_S U \ar[d] \\ X \ar[r]^-{\Delta_{X/S}} & X \times_S X }$$ The right vertical arrow is surjective \'etale (see Spaces, Lemma \ref{spaces-lemma-product-representable-transformations-property}) and $U \times_S U$ is affine (as $U$ maps into an affine open of $S$, see Schemes, Section \ref{schemes-section-fibre-products}), and $U \times_X U$ is quasi-compact because the projection $U \times_X U \to U$ is quasi-compact as a base change of $U \to X$. It follows from Spaces, Lemma \ref{spaces-lemma-representable-morphisms-spaces-property} that $\Delta_{X/S}$ is quasi-compact as desired. \medskip\noindent Assume (1). To prove (3) there is an immediate reduction to the case where $X$ is quasi-separated over $S$. By Lemma \ref{lemma-union-of-quasi-compact} we can find a Zariski open covering $X = \bigcup X_i$ such that each $X_i$ maps into an affine open of $S$, and such that there exist affine schemes $U_i$ and surjective \'etale morphisms $U_i \to X_i$. Since $U_i \to S$ maps into an affine open of $S$ we see that $U_i \times_S U_i$ is affine, see Schemes, Section \ref{schemes-section-fibre-products}. As $X$ is quasi-separated over $S$, the morphisms $$R_i = U_i \times_{X_i} U_i = U_i \times_X U_i \longrightarrow U_i \times_S U_i$$ as base changes of $\Delta_{X/S}$ are quasi-compact. Hence we conclude that $R_i$ is a quasi-compact scheme. This in turn implies that each projection $R_i \to U_i$ is quasi-compact. Hence, applying Spaces, Lemma \ref{spaces-lemma-representable-morphisms-spaces-property} to the covering $U_i \to X_i$ and the morphism $U_i \to X_i$ we conclude that the morphisms $U_i \to X_i$ are quasi-compact as desired. \medskip\noindent At this point we see that (1), (3), and (4) are equivalent. Since (3) does not refer to the base scheme we conclude that these are also equivalent with (2). \end{proof} \noindent The following lemma will turn out to be quite useful. \begin{lemma} \label{lemma-finite-fibres-presentation} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a scheme. Let $\varphi : U \to X$ be an \'etale morphism such that the projections $R = U \times_X U \to U$ are quasi-compact; for example if $\varphi$ is quasi-compact. Then the fibres of $$|U| \to |X| \quad\text{and}\quad |R| \to |X|$$ are finite. \end{lemma} \begin{proof} Denote $R = U \times_X U$, and $s, t : R \to U$ the projections. Let $u \in U$ be a point, and let $x \in |X|$ be its image. The fibre of $|U| \to |X|$ over $x$ is equal to $s(t^{-1}(\{u\}))$ by Lemma \ref{lemma-points-cartesian}, and the fibre of $|R| \to |X|$ over $x$ is $t^{-1}(s(t^{-1}(\{u\})))$. Since $t : R \to U$ is \'etale and quasi-compact, it has finite fibres (as its fibres are disjoint unions of spectra of fields by Morphisms, Lemma \ref{morphisms-lemma-etale-over-field} and quasi-compact). Hence we win. \end{proof} \section{Properties of Spaces defined by properties of schemes} \label{section-types-properties} \noindent Any \'etale local property of schemes gives rise to a corresponding property of algebraic spaces via the following lemma. \begin{lemma} \label{lemma-type-property} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{P}$ be a property of schemes which is local in the \'etale topology, see Descent, Definition \ref{descent-definition-property-local}. The following are equivalent \begin{enumerate} \item for some scheme $U$ and surjective \'etale morphism $U \to X$ the scheme $U$ has property $\mathcal{P}$, and \item for every scheme $U$ and every \'etale morphism $U \to X$ the scheme $U$ has property $\mathcal{P}$. \end{enumerate} If $X$ is representable this is equivalent to $\mathcal{P}(X)$. \end{lemma} \begin{proof} The implication (2) $\Rightarrow$ (1) is immediate. For the converse, choose a surjective \'etale morphism $U \to X$ with $U$ a scheme that has $\mathcal{P}$ and let $V$ be an \'etale $X$-scheme. Then $U \times_X V \rightarrow V$ is an \'etale surjection of schemes, so $V$ inherits $\mathcal{P}$ from $U \times_X V$, which in turn inherits $\mathcal{P}$ from $U$ (see discussion following Descent, Definition \ref{descent-definition-property-local}). The last claim is clear from (1) and Descent, Definition \ref{descent-definition-property-local}. \end{proof} \begin{definition} \label{definition-type-property} Let $\mathcal{P}$ be a property of schemes which is local in the \'etale topology. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. We say $X$ {\it has property $\mathcal{P}$} if any of the equivalent conditions of Lemma \ref{lemma-type-property} hold. \end{definition} \begin{remark} \label{remark-list-properties-local-etale-topology} Here is a list of properties which are local for the \'etale topology (keep in mind that the fpqc, fppf, syntomic, and smooth topologies are stronger than the \'etale topology): \begin{enumerate} \item locally Noetherian, see Descent, Lemma \ref{descent-lemma-Noetherian-local-fppf}, \item Jacobson, see Descent, Lemma \ref{descent-lemma-Jacobson-local-fppf}, \item locally Noetherian and $(S_k)$, see Descent, Lemma \ref{descent-lemma-Sk-local-syntomic}, \item Cohen-Macaulay, see Descent, Lemma \ref{descent-lemma-CM-local-syntomic}, \item reduced, see Descent, Lemma \ref{descent-lemma-reduced-local-smooth}, \item normal, see Descent, Lemma \ref{descent-lemma-normal-local-smooth}, \item locally Noetherian and $(R_k)$, see Descent, Lemma \ref{descent-lemma-Rk-local-smooth}, \item regular, see Descent, Lemma \ref{descent-lemma-regular-local-smooth}, \item Nagata, see Descent, Lemma \ref{descent-lemma-Nagata-local-smooth}. \end{enumerate} \end{remark} \noindent Any \'etale local property of germs of schemes gives rise to a corresponding property of algebraic spaces. Here is the obligatory lemma. \begin{lemma} \label{lemma-local-source-target-at-point} Let $\mathcal{P}$ be a property of germs of schemes which is \'etale local, see Descent, Definition \ref{descent-definition-local-at-point}. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$ be a point of $X$. Consider \'etale morphisms $a : U \to X$ where $U$ is a scheme. The following are equivalent \begin{enumerate} \item for any $U \to X$ as above and $u \in U$ with $a(u) = x$ we have $\mathcal{P}(U, u)$, and \item for some $U \to X$ as above and $u \in U$ with $a(u) = x$ we have $\mathcal{P}(U, u)$. \end{enumerate} If $X$ is representable, then this is equivalent to $\mathcal{P}(X, x)$. \end{lemma} \begin{proof} Omitted. \end{proof} \begin{definition} \label{definition-property-at-point} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$. Let $\mathcal{P}$ be a property of germs of schemes which is \'etale local. We say $X$ {\it has property $\mathcal{P}$ at $x$} if any of the equivalent conditions of Lemma \ref{lemma-local-source-target-at-point} hold. \end{definition} \begin{remark} \label{remark-list-properties-local-ring-local-etale-topology} Let $P$ be a property of local rings. Assume that for any \'etale ring map $A \to B$ and $\mathfrak q$ is a prime of $B$ lying over the prime $\mathfrak p$ of $A$, then $P(A_\mathfrak p) \Leftrightarrow P(B_\mathfrak q)$. Then we obtain an \'etale local property of germs $(U, u)$ of schemes by setting $\mathcal{P}(U, u) = P(\mathcal{O}_{U, u})$. In this situation we will use the terminology the local ring of $X$ at $x$ has $P$'' to mean $X$ has property $\mathcal{P}$ at $x$. Here is a list of such properties $P$: \begin{enumerate} \item Noetherian, see More on Algebra, Lemma \ref{more-algebra-lemma-Noetherian-etale-extension}, \item dimension $d$, see More on Algebra, Lemma \ref{more-algebra-lemma-dimension-etale-extension}, \item regular, see More on Algebra, Lemma \ref{more-algebra-lemma-regular-etale-extension}, \item discrete valuation ring, follows from (2), (3), and Algebra, Lemma \ref{algebra-lemma-characterize-dvr}, \item reduced, see More on Algebra, Lemma \ref{more-algebra-lemma-henselization-reduced}, \item normal, see More on Algebra, Lemma \ref{more-algebra-lemma-henselization-normal}, \item Noetherian and depth $k$, see More on Algebra, Lemma \ref{more-algebra-lemma-henselization-depth}, \item Noetherian and Cohen-Macaulay, see More on Algebra, Lemma \ref{more-algebra-lemma-henselization-CM}, \end{enumerate} There are more properties for which this holds, for example G-ring and Nagata. If we every need these we will add them here as well as references to detailed proofs of the corresponding algebra facts. \end{remark} \section{Dimension at a point} \label{section-dimension} \noindent We can use Descent, Lemma \ref{descent-lemma-dimension-at-point-local} to define the dimension of an algebraic space $X$ at a point $x$. This will give us a different notion than the topological one (i.e., the dimension of $|X|$ at $x$). \begin{definition} \label{definition-dimension-at-point} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$ be a point of $X$. We define the {\it dimension of $X$ at $x$} to be the element $\dim_x(X) \in \{0, 1, 2, \ldots, \infty\}$ such that $\dim_x(X) = \dim_u(U)$ for any (equivalently some) pair $(a : U \to X, u)$ consisting of an \'etale morphism $a : U \to X$ from a scheme to $X$ and a point $u \in U$ with $a(u) = x$. See Definition \ref{definition-property-at-point}, Lemma \ref{lemma-local-source-target-at-point}, and Descent, Lemma \ref{descent-lemma-dimension-at-point-local}. \end{definition} \noindent Warning: It is {\bf not} the case that $\dim_x(X) = \dim_x(|X|)$ in general. A counter example is the algebraic space $X$ of Spaces, Example \ref{spaces-example-infinite-product}. Namely, in this example we have $\dim_x(X) = 0$ and $\dim_x(|X|) = 1$ (this holds for any $x \in |X|$). In particular, it also means that the dimension of $X$ (as defined below) is different from the dimension of $|X|$. \begin{definition} \label{definition-dimension} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The {\it dimension} $\dim(X)$ of $X$ is defined by the rule $$\dim(X) = \sup\nolimits_{x \in |X|} \dim_x(X)$$ \end{definition} \noindent By Properties, Lemma \ref{properties-lemma-dimension} we see that this is the usual notion if $X$ is a scheme. There is another integer that measures the dimension of a scheme at a point, namely the dimension of the local ring. This invariant is compatible with \'etale morphisms also, see Section \ref{section-dimension-local-ring}. \section{Dimension of local rings} \label{section-dimension-local-ring} \noindent The dimension of the local ring of an algebraic space is a well defined concept. \begin{lemma} \label{lemma-pre-dimension-local-ring} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$ be a point. Let $d \in \{0, 1, 2, \ldots, \infty\}$. The following are equivalent \begin{enumerate} \item for some scheme $U$ and \'etale morphism $a : U \to X$ and point $u \in U$ with $a(u) = x$ we have $\dim(\mathcal{O}_{U, u}) = d$, \item for any scheme $U$, any \'etale morphism $a : U \to X$, and any point $u \in U$ with $a(u) = x$ we have $\dim(\mathcal{O}_{U, u}) = d$. \end{enumerate} If $X$ is a scheme, this is equivalent to $\dim(\mathcal{O}_{X, x}) = d$. \end{lemma} \begin{proof} Combine Lemma \ref{lemma-local-source-target-at-point} and Descent, Lemma \ref{descent-lemma-dimension-local-ring-local}. \end{proof} \begin{definition} \label{definition-dimension-local-ring} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$ be a point. The {\it dimension of the local ring of $X$ at $x$} is the element $d \in \{0, 1, 2, \ldots, \infty\}$ satisfying the equivalent conditions of Lemma \ref{lemma-pre-dimension-local-ring}. In this case we will also say {\it $x$ is a point of codimension $d$ on $X$}. \end{definition} \noindent Besides the lemma below we also point the reader to Lemmas \ref{lemma-dimension-local-ring} and \ref{lemma-dimension-decent-invariant-under-etale}. \begin{lemma} \label{lemma-dimension} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The following quantities are equal: \begin{enumerate} \item The dimension of $X$. \item The supremum of the dimensions of the local rings of $X$. \item The supremum of $\dim_x(X)$ for $x \in |X|$. \end{enumerate} \end{lemma} \begin{proof} The numbers in (1) and (3) are equal by Definition \ref{definition-dimension}. Let $U \to X$ be a surjective \'etale morphism from a scheme $U$. The supremum of $\dim_x(X)$ for $x \in |X|$ is the same as the supremum of $\dim_u(U)$ for points $u$ of $U$ by definition. This is the same as the supremum of $\dim(\mathcal{O}_{U, u})$ by Properties, Lemma \ref{properties-lemma-dimension}. This in turn is the same as (2) by definition. \end{proof} \section{Generic points} \label{section-generic-points} \noindent Let $T$ be a topological space. According to the second edition of EGA I, a {\it maximal point of $T$} is a generic point of an irreducible component of $T$. If $T = |X|$ is the topological space associated to an algebraic space $X$, there are at least two notions of maximal points: we can look at maximal points of $T$ viewed as a topological space, or we can look at images of maximal points of $U$ where $U \to X$ is an \'etale morphism and $U$ is a scheme. The second notion corresponds to the set of points of codimension $0$ (Lemma \ref{lemma-codimension-0-points}). The codimension $0$ points are easier to work with for general algebraic spaces; the two notions agree for quasi-separated and more generally decent algebraic spaces (Decent Spaces, Lemma \ref{decent-spaces-lemma-decent-generic-points}). \begin{lemma} \label{lemma-codimension-0-points} Let $S$ be a scheme and let $X$ be an algebraic space over $S$. Let $x \in |X|$. Consider \'etale morphisms $a : U \to X$ where $U$ is a scheme. The following are equivalent \begin{enumerate} \item $x$ is a point of codimension $0$ on $X$, \item for some $U \to X$ as above and $u \in U$ with $a(u) = x$, the point $u$ is the generic point of an irreducible component of $U$, and \item for any $U \to X$ as above and any $u \in U$ mapping to $x$, the point $u$ is the generic point of an irreducible component of $U$. \end{enumerate} If $X$ is representable, this is equivalent to $x$ being a generic point of an irreducible component of $|X|$. \end{lemma} \begin{proof} Observe that a point $u$ of a scheme $U$ is a generic point of an irreducible component of $U$ if and only if $\dim(\mathcal{O}_{U, u}) = 0$ (Properties, Lemma \ref{properties-lemma-generic-point}). Hence this follows from the definition of the codimension of a point on $X$ (Definition \ref{definition-dimension-local-ring}). \end{proof} \begin{lemma} \label{lemma-codimension-0-points-dense} Let $S$ be a scheme and let $X$ be an algebraic space over $S$. The set of codimension $0$ points of $X$ is dense in $|X|$. \end{lemma} \begin{proof} If $U$ is a scheme, then the set of generic points of irreducible components is dense in $U$ (holds for any quasi-sober topological space). Thus if $U \to X$ is a surjective \'etale morphism, then the set of codimension $0$ points of $X$ is the image of a dense subset of $|U|$ (Lemma \ref{lemma-codimension-0-points}). Since $|X|$ has the quotient topology for $|U| \to |X|$ we conclude. \end{proof} \section{Reduced spaces} \label{section-reduced} \noindent We have already defined reduced algebraic spaces in Section \ref{section-types-properties}. Here we just prove some simple lemmas regarding reduced algebraic spaces. \begin{lemma} \label{lemma-reduced-space} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The following are equivalent \begin{enumerate} \item $X$ is reduced, \item for every $x \in |X|$ the local ring of $X$ at $x$ is reduced (Remark \ref{remark-list-properties-local-ring-local-etale-topology}). \end{enumerate} In this case $\Gamma(X, \mathcal{O}_X)$ is a reduced ring and if $f \in \Gamma(X, \mathcal{O}_X)$ has $X = V(f)$, then $f = 0$. \end{lemma} \begin{proof} The equivalence of (1) and (2) follows from Properties, Lemma \ref{properties-lemma-characterize-reduced} applied to affine schemes \'etale over $X$. The final statements follow the cited lemma and fact that $\Gamma(X, \mathcal{O}_X)$ is a subring of $\Gamma(U, \mathcal{O}_U)$ for some reduced scheme $U$ \'etale over $X$. \end{proof} \begin{lemma} \label{lemma-subspace-induced-topology} Let $S$ be a scheme. Let $Z \to X$ be an immersion of algebraic spaces. Then $|Z| \to |X|$ is a homeomorphism of $|Z|$ onto a locally closed subset of $|X|$. \end{lemma} \begin{proof} Let $U$ be a scheme and $U \to X$ a surjective \'etale morphism. Then $Z \times_X U \to U$ is an immersion of schemes, hence gives a homeomorphism of $|Z \times_X U|$ with a locally closed subset $T'$ of $|U|$. By Lemma \ref{lemma-points-cartesian} the subset $T'$ is the inverse image of the image $T$ of $|Z| \to |X|$. The map $|Z| \to |X|$ is injective because the transformation of functors $Z \to X$ is injective, see Spaces, Section \ref{spaces-section-Zariski}. By Topology, Lemma \ref{topology-lemma-open-morphism-quotient-topology} we see that $T$ is locally closed in $|X|$. Moreover, the continuous map $|Z| \to T$ is a homeomorphism as the map $|Z \times_X U| \to T'$ is a homeomorphism and $|Z \times_Y U| \to |Z|$ is submersive. \end{proof} \noindent The following lemma will help us construct (locally) closed subspaces. \begin{lemma} \label{lemma-subspaces-presentation} Let $S$ be a scheme. Let $j : R \to U \times_S U$ be an \'etale equivalence relation. Let $X = U/R$ be the associated algebraic space (Spaces, Theorem \ref{spaces-theorem-presentation}). There is a canonical bijection $$R\text{-invariant locally closed subschemes }Z'\text{ of }U \leftrightarrow \text{locally closed subspaces }Z\text{ of }X$$ Moreover, if $Z \to X$ is closed (resp.\ open) if and only if $Z' \to U$ is closed (resp.\ open). \end{lemma} \begin{proof} Denote $\varphi : U \to X$ the canonical map. The bijection sends $Z \to X$ to $Z' = Z \times_X U \to U$. It is immediate from the definition that $Z' \to U$ is an immersion, resp.\ closed immersion, resp.\ open immersion if $Z \to X$ is so. It is also clear that $Z'$ is $R$-invariant (see Groupoids, Definition \ref{groupoids-definition-invariant-open}). \medskip\noindent Conversely, assume that $Z' \to U$ is an immersion which is $R$-invariant. Let $R'$ be the restriction of $R$ to $Z'$, see Groupoids, Definition \ref{groupoids-definition-restrict-groupoid}. Since $R' = R \times_{s, U} Z' = Z' \times_{U, t} R$ in this case we see that $R'$ is an \'etale equivalence relation on $Z'$. By Spaces, Theorem \ref{spaces-theorem-presentation} we see $Z = Z'/R'$ is an algebraic space. By construction we have $U \times_X Z = Z'$, so $U \times_X Z \to Z$ is an immersion. Note that the property immersion'' is preserved under base change and fppf local on the base (see Spaces, Section \ref{spaces-section-lists}). Moreover, immersions are separated and locally quasi-finite (see Schemes, Lemma \ref{schemes-lemma-immersions-monomorphisms} and Morphisms, Lemma \ref{morphisms-lemma-immersion-locally-quasi-finite}). Hence by More on Morphisms, Lemma \ref{more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend} immersions satisfy descent for fppf covering. This means all the hypotheses of Spaces, Lemma \ref{spaces-lemma-morphism-sheaves-with-P-effective-descent-etale} are satisfied for $Z \to X$, $\mathcal{P}=$immersion'', and the \'etale surjective morphism $U \to X$. We conclude that $Z \to X$ is representable and an immersion, which is the definition of a subspace (see Spaces, Definition \ref{spaces-definition-immersion}). \medskip\noindent It is clear that these constructions are inverse to each other and we win. \end{proof} \begin{lemma} \label{lemma-reduced-closed-subspace} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $T \subset |X|$ be a closed subset. There exists a unique closed subspace $Z \subset X$ with the following properties: (a) we have $|Z| = T$, and (b) $Z$ is reduced. \end{lemma} \begin{proof} Let $U \to X$ be a surjective \'etale morphism, where $U$ is a scheme. Set $R = U \times_X U$, so that $X = U/R$, see Spaces, Lemma \ref{spaces-lemma-space-presentation}. As usual we denote $s, t : R \to U$ the two projection morphisms. By Lemma \ref{lemma-points-presentation} we see that $T$ corresponds to a closed subset $T' \subset |U|$ such that $s^{-1}(T') = t^{-1}(T')$. Let $Z' \subset U$ be the reduced induced scheme structure on $T'$. In this case the fibre products $Z' \times_{U, t} R$ and $Z' \times_{U, s} R$ are closed subschemes of $R$ (Schemes, Lemma \ref{schemes-lemma-base-change-immersion}) which are \'etale over $Z'$ (Morphisms, Lemma \ref{morphisms-lemma-base-change-etale}), and hence reduced (because being reduced is local in the \'etale topology, see Remark \ref{remark-list-properties-local-etale-topology}). Since they have the same underlying topological space (see above) we conclude that $Z' \times_{U, t} R = Z' \times_{U, s} R$. Thus we can apply Lemma \ref{lemma-subspaces-presentation} to obtain a closed subspace $Z \subset X$ whose pullback to $U$ is $Z'$. By construction $|Z| = T$ and $Z$ is reduced. This proves existence. We omit the proof of uniqueness. \end{proof} \begin{lemma} \label{lemma-map-into-reduction} Let $S$ be a scheme. Let $X$, $Y$ be algebraic spaces over $S$. Let $Z \subset X$ be a closed subspace. Assume $Y$ is reduced. A morphism $f : Y \to X$ factors through $Z$ if and only if $f(|Y|) \subset |Z|$. \end{lemma} \begin{proof} Assume $f(|Y|) \subset |Z|$. Choose a diagram $$\xymatrix{ V \ar[d]_b \ar[r]_h & U \ar[d]^a \\ Y \ar[r]^f & X }$$ where $U$, $V$ are schemes, and the vertical arrows are surjective and \'etale. The scheme $V$ is reduced, see Lemma \ref{lemma-type-property}. Hence $h$ factors through $a^{-1}(Z)$ by Schemes, Lemma \ref{schemes-lemma-map-into-reduction}. So $a \circ h$ factors through $Z$. As $Z \subset X$ is a subsheaf, and $V \to Y$ is a surjection of sheaves on $(\Sch/S)_{fppf}$ we conclude that $X \to Y$ factors through $Z$. \end{proof} \begin{definition} \label{definition-reduced-induced-space} Let $S$ be a scheme, and let $X$ be an algebraic space over $S$. Let $Z \subset |X|$ be a closed subset. An {\it algebraic space structure on $Z$} is given by a closed subspace $Z'$ of $X$ with $|Z'|$ equal to $Z$. The {\it reduced induced algebraic space structure} on $Z$ is the one constructed in Lemma \ref{lemma-reduced-closed-subspace}. The {\it reduction $X_{red}$ of $X$} is the reduced induced algebraic space structure on $|X|$. \end{definition} \section{The schematic locus} \label{section-schematic} \noindent Every algebraic space has a largest open subspace which is a scheme; this is more or less clear but we also write out the proof below. Of course this subspace may be empty, for example if $X = \mathbf{A}^1_{\mathbf{Q}}/\mathbf{Z}$ (the universal counter example). On the other hand, if $X$ is for example quasi-separated, then this largest open subscheme is actually dense in $X$! \begin{lemma} \label{lemma-subscheme} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. There exists a largest open subspace $X' \subset X$ which is a scheme. \end{lemma} \begin{proof} Let $U \to X$ be an \'etale surjective morphism, where $U$ is a scheme. Let $R = U \times_X U$. The open subspaces of $X$ correspond $1 - 1$ with open subschemes of $U$ which are $R$-invariant. Hence there is a set of them. Let $X_i$, $i \in I$ be the set of open subspaces of $X$ which are schemes, i.e., are representable. Consider the open subspace $X' \subset X$ whose underlying set of points is the open $\bigcup |X_i|$ of $|X|$. By Lemma \ref{lemma-characterize-surjective} we see that $$\coprod X_i \longrightarrow X'$$ is a surjective map of sheaves on $(\Sch/S)_{fppf}$. But since each $X_i \to X'$ is representable by open immersions we see that in fact the map is surjective in the Zariski topology. Namely, if $T \to X'$ is a morphism from a scheme into $X'$, then $X_i \times_X' T$ is an open subscheme of $T$. Hence we can apply Schemes, Lemma \ref{schemes-lemma-glue-functors} to see that $X'$ is a scheme. \end{proof} \noindent In the rest of this section we say that an open subspace $X'$ of an algebraic space $X$ is {\it dense} if the corresponding open subset $|X'| \subset |X|$ is dense. \begin{lemma} \label{lemma-quasi-separated-finite-etale-cover-dense-open-scheme} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If there exists a finite, \'etale, surjective morphism $U \to X$ where $U$ is a quasi-separated scheme, then there exists a dense open subspace $X'$ of $X$ which is a scheme. More precisely, every point $x \in |X|$ of codimension $0$ in $X$ is contained in $X'$. \end{lemma} \begin{proof} Let $X' \subset X$ be the maximal open subspace which is a scheme (Lemma \ref{lemma-subscheme}). Let $x \in |X|$ be a point of codimension $0$ on $X$. By Lemma \ref{lemma-codimension-0-points-dense} it suffices to show $x \in X'$. Let $U \to X$ be as in the statement of the lemma. Write $R = U \times_X U$ and denote $s, t : R \to U$ the projections as usual. Note that $s, t$ are surjective, finite and \'etale. By Lemma \ref{lemma-finite-fibres-presentation} the fibre of $|U| \to |X|$ over $x$ is finite, say $\{\eta_1, \ldots, \eta_n\}$. By Lemma \ref{lemma-codimension-0-points} each $\eta_i$ is the generic point of an irreducible component of $U$. By Properties, Lemma \ref{properties-lemma-maximal-points-affine} we can find an affine open $W \subset U$ containing $\{\eta_1, \ldots, \eta_n\}$ (this is where we use that $U$ is quasi-separated). By Groupoids, Lemma \ref{groupoids-lemma-find-invariant-affine} we may assume that $W$ is $R$-invariant. Since $W \subset U$ is an $R$-invariant affine open, the restriction $R_W$ of $R$ to $W$ equals $R_W = s^{-1}(W) = t^{-1}(W)$ (see Groupoids, Definition \ref{groupoids-definition-invariant-open} and discussion following it). In particular the maps $R_W \to W$ are finite \'etale also. It follows that $R_W$ is affine. Thus we see that $W/R_W$ is a scheme, by Groupoids, Proposition \ref{groupoids-proposition-finite-flat-equivalence}. On the other hand, $W/R_W$ is an open subspace of $X$ by Spaces, Lemma \ref{spaces-lemma-finding-opens} and it contains $x$ by construction. \end{proof} \noindent We will improve the following proposition to the case of decent algebraic spaces in Decent Spaces, Theorem \ref{decent-spaces-theorem-decent-open-dense-scheme}. \begin{proposition} \label{proposition-locally-quasi-separated-open-dense-scheme} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If $X$ is Zariski locally quasi-separated (for example if $X$ is quasi-separated), then there exists a dense open subspace of $X$ which is a scheme. More precisely, every point $x \in |X|$ of codimension $0$ on $X$ is contained in $X'$. \end{proposition} \begin{proof} The question is local on $X$ by Lemma \ref{lemma-subscheme}. Thus by Lemma \ref{lemma-quasi-separated-quasi-compact-pieces} we may assume that there exists an affine scheme $U$ and a surjective, quasi-compact, \'etale morphism $U \to X$. Moreover $U \to X$ is separated (Lemma \ref{lemma-separated-cover}). Set $R = U \times_X U$ and denote $s, t : R \to U$ the projections as usual. Then $s, t$ are surjective, quasi-compact, separated, and \'etale. Hence $s, t$ are also quasi-finite and have finite fibres (Morphisms, Lemmas \ref{morphisms-lemma-etale-locally-quasi-finite}, \ref{morphisms-lemma-quasi-finite-locally-quasi-compact}, and \ref{morphisms-lemma-quasi-finite}). By Morphisms, Lemma \ref{morphisms-lemma-generically-finite} for every $\eta \in U$ which is the generic point of an irreducible component of $U$, there exists an open neighbourhood $V \subset U$ of $\eta$ such that $s^{-1}(V) \to V$ is finite. By Descent, Lemma \ref{descent-lemma-descending-property-finite} being finite is fpqc (and in particular \'etale) local on the target. Hence we may apply More on Groupoids, Lemma \ref{more-groupoids-lemma-property-invariant} which says that the largest open $W \subset U$ over which $s$ is finite is $R$-invariant. By the above $W$ contains every generic point of an irreducible component of $U$. The restriction $R_W$ of $R$ to $W$ equals $R_W = s^{-1}(W) = t^{-1}(W)$ (see Groupoids, Definition \ref{groupoids-definition-invariant-open} and discussion following it). By construction $s_W, t_W : R_W \to W$ are finite \'etale. Consider the open subspace $X' = W/R_W \subset X$ (see Spaces, Lemma \ref{spaces-lemma-finding-opens}). By construction the inclusion map $X' \to X$ induces a bijection on points of codimension $0$. This reduces us to Lemma \ref{lemma-quasi-separated-finite-etale-cover-dense-open-scheme}. \end{proof} \section{Obtaining a scheme} \label{section-getting-a-scheme} \noindent We have used in the previous section that the quotient $U/R$ of an affine scheme $U$ by an equivalence relation $R$ is a scheme if the morphisms $s, t : R \to U$ are finite \'etale. This is a special case of the following result. \begin{proposition} \label{proposition-finite-flat-equivalence-global} Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume \begin{enumerate} \item $s, t : R \to U$ finite locally free, \item $j = (t, s)$ is an equivalence, and \item for a dense set of points $u \in U$ the $R$-equivalence class $t(s^{-1}(\{u\}))$ is contained in an affine open of $U$. \end{enumerate} Then there exists a finite locally free morphism $U \to M$ of schemes over $S$ such that $R = U \times_M U$ and such that $M$ represents the quotient sheaf $U/R$ in the fppf topology. \end{proposition} \begin{proof} By assumption (3) and Groupoids, Lemma \ref{groupoids-lemma-find-invariant-affine} we can find an open covering $U = \bigcup U_i$ such that each $U_i$ is an $R$-invariant affine open of $U$. Set $R_i = R|_{U_i}$. Consider the fppf sheaves $F = U/R$ and $F_i = U_i/R_i$. By Spaces, Lemma \ref{spaces-lemma-finding-opens} the morphisms $F_i \to F$ are representable and open immersions. By Groupoids, Proposition \ref{groupoids-proposition-finite-flat-equivalence} the sheaves $F_i$ are representable by affine schemes. If $T$ is a scheme and $T \to F$ is a morphism, then $V_i = F_i \times_F T$ is open in $T$ and we claim that $T = \bigcup V_i$. Namely, fppf locally on $T$ we can lift $T \to F$ to a morphism $f : T \to U$ and in that case $f^{-1}(U_i) \subset V_i$. Hence we conclude that $F$ is representable by a scheme, see Schemes, Lemma \ref{schemes-lemma-glue-functors}. \end{proof} \noindent For example, if $U$ is isomorphic to a locally closed subscheme of an affine scheme or isomorphic to a locally closed subscheme of $\text{Proj}(A)$ for some graded ring $A$, then the third assumption holds by Properties, Lemma \ref{properties-lemma-ample-finite-set-in-affine}. In particular we can apply this to free actions of finite groups and finite group schemes on quasi-affine or quasi-projective schemes. For example, the quotient $X/G$ of a quasi-projective variety $X$ by a free action of a finite group $G$ is a scheme. Here is a detailed statement. \begin{lemma} \label{lemma-quotient-scheme} Let $S$ be a scheme. Let $G \to S$ be a group scheme. Let $X \to S$ be a morphism of schemes. Let $a : G \times_S X \to X$ be an action. Assume that \begin{enumerate} \item $G \to S$ is finite locally free, \item the action $a$ is free, \item $X \to S$ is affine, or quasi-affine, or projective, or quasi-projective, or $X$ is isomorphic to an open subscheme of an affine scheme or isomorphic to an open subscheme of $\text{Proj}(A)$ for some graded ring $A$. \end{enumerate} Then the fppf quotient sheaf $X/G$ is a scheme. \end{lemma} \begin{proof} Since the action is free the morphism $j = (a, \text{pr}) : G \times_S X \to X \times_S X$ is a monomorphism and hence an equivalence relation, see Groupoids, Lemma \ref{groupoids-lemma-free-action}. The maps $s, t : G \times_S X \to X$ are finite locally free as we've assumed that $G \to S$ is finite locally free. To conclude it now suffices to prove the last assumption of Proposition \ref{proposition-finite-flat-equivalence-global} holds. Since the action of $G$ is over $S$ it suffices to prove that any finite set of points in a fibre of $X \to S$ is contained in an affine open of $X$. If $X$ is isomorphic to an open subscheme of an affine scheme or isomorphic to an open subscheme of $\text{Proj}(A)$ for some graded ring $A$ this follows from Properties, Lemma \ref{properties-lemma-ample-finite-set-in-affine}. In the remaining cases, we may replace $S$ by an affine open and we get back to the case we just dealt with. Some details omitted. \end{proof} \begin{lemma} \label{lemma-quotient-separated} Notation and assumptions as in Proposition \ref{proposition-finite-flat-equivalence-global}. Then \begin{enumerate} \item if $U$ is quasi-separated over $S$, then $U/R$ is quasi-separated over $S$, \item if $U$ is quasi-separated, then $U/R$ is quasi-separated, \item if $U$ is separated over $S$, then $U/R$ is separated over $S$, \item if $U$ is separated, then $U/R$ is separated, and \item add more here. \end{enumerate} Similar results hold in the setting of Lemma \ref{lemma-quotient-scheme}. \end{lemma} \begin{proof} Since $M$ represents the quotient sheaf we have a cartesian diagram $$\xymatrix{ R \ar[r]_-j \ar[d] & U \times_S U \ar[d] \\ M \ar[r] & M \times_S M }$$ of schemes. Since $U \times_S U \to M \times_S M$ is surjective finite locally free, to show that $M \to M \times_S M$ is quasi-compact, resp.\ a closed immersion, it suffices to show that $j : R \to U \times_S U$ is quasi-compact, resp.\ a closed immersion, see Descent, Lemmas \ref{descent-lemma-descending-property-quasi-compact} and \ref{descent-lemma-descending-property-closed-immersion}. Since $j : R \to U \times_S U$ is a morphism over $U$ and since $R$ is finite over $U$, we see that $j$ is quasi-compact as soon as the projection $U \times_S U \to U$ is quasi-separated (Schemes, Lemma \ref{schemes-lemma-quasi-compact-permanence}). Since $j$ is a monomorphism and locally of finite type, we see that $j$ is a closed immersion as soon as it is proper (\'Etale Morphisms, Lemma \ref{etale-lemma-characterize-closed-immersion}) which will be the case as soon as the projection $U \times_S U \to U$ is separated (Morphisms, Lemma \ref{morphisms-lemma-image-proper-scheme-closed}). This proves (1) and (3). To prove (2) and (4) we replace $S$ by $\Spec(\mathbf{Z})$, see Definition \ref{definition-separated}. Since Lemma \ref{lemma-quotient-scheme} is proved through an application of Proposition \ref{proposition-finite-flat-equivalence-global} the final statement is clear too. \end{proof} \section{Points on quasi-separated spaces} \label{section-points-quasi-separated} \noindent Points can behave very badly on algebraic spaces in the generality introduced in the Stacks project. However, for quasi-separated spaces their behaviour is mostly like the behaviour of points on schemes. We prove a few results on this in this section; the chapter on decent spaces contains many more results on this, see for example Decent Spaces, Section \ref{decent-spaces-section-points}. \begin{lemma} \label{lemma-quasi-separated-sober} Let $S$ be a scheme. Let $X$ be a Zariski locally quasi-separated algebraic space over $S$. Then the topological space $|X|$ is sober (see Topology, Definition \ref{topology-definition-generic-point}). \end{lemma} \begin{proof} Combining Topology, Lemma \ref{topology-lemma-sober-local} and Lemma \ref{lemma-quasi-separated-quasi-compact-pieces} we see that we may assume that there exists an affine scheme $U$ and a surjective, quasi-compact, \'etale morphism $U \to X$. Set $R = U \times_X U$ with projection maps $s, t : R \to U$. Applying Lemma \ref{lemma-finite-fibres-presentation} we see that the fibres of $s, t$ are finite. It follows all the assumptions of Topology, Lemma \ref{topology-lemma-quotient-kolmogorov} are met, and we conclude that $|X|$ is Kolmogorov\footnote{ Actually we use here also Schemes, Lemma \ref{schemes-lemma-scheme-sober} (soberness schemes), Morphisms, Lemmas \ref{morphisms-lemma-etale-flat} and \ref{morphisms-lemma-generalizations-lift-flat} (generalizations lift along \'etale morphisms), Lemma \ref{lemma-points-presentation} (points on an algebraic space in terms of a presentation), and Lemma \ref{lemma-topology-points} (openness quotient map).}. \medskip\noindent It remains to show that every irreducible closed subset $T \subset |X|$ has a generic point. By Lemma \ref{lemma-reduced-closed-subspace} there exists a closed subspace $Z \subset X$ with $|Z| = |T|$. Note that $U \times_X Z \to Z$ is a quasi-compact, surjective, \'etale morphism from an affine scheme to $Z$, hence $Z$ is Zariski locally quasi-separated by Lemma \ref{lemma-quasi-separated-quasi-compact-pieces}. By Proposition \ref{proposition-locally-quasi-separated-open-dense-scheme} we see that there exists an open dense subspace $Z' \subset Z$ which is a scheme. This means that $|Z'| \subset T$ is open dense. Hence the topological space $|Z'|$ is irreducible, which means that $Z'$ is an irreducible scheme. By Schemes, Lemma \ref{schemes-lemma-scheme-sober} we conclude that $|Z'|$ is the closure of a single point $\eta \in |Z'| \subset T$ and hence also $T = \overline{\{\eta\}}$, and we win. \end{proof} \begin{lemma} \label{lemma-quasi-compact-quasi-separated-spectral} Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. The topological space $|X|$ is a spectral space. \end{lemma} \begin{proof} By Topology, Definition \ref{topology-definition-spectral-space} we have to check that $|X|$ is sober, quasi-compact, has a basis of quasi-compact opens, and the intersection of any two quasi-compact opens is quasi-compact. By Lemma \ref{lemma-quasi-separated-sober} we see that $|X|$ is sober. By Lemma \ref{lemma-quasi-compact-space} we see that $|X|$ is quasi-compact. By Lemma \ref{lemma-quasi-compact-affine-cover} there exists an affine scheme $U$ and a surjective \'etale morphism $f : U \to X$. Since $|f| : |U| \to |X|$ is open and continuous and since $|U|$ has a basis of quasi-compact opens, we conclude that $|X|$ has a basis of quasi-compact opens. Finally, suppose that $A, B \subset |X|$ are quasi-compact open. Then $A = |X'|$ and $B = |X''|$ for some open subspaces $X', X'' \subset X$ (Lemma \ref{lemma-open-subspaces}) and we can choose affine schemes $V$ and $W$ and surjective \'etale morphisms $V \to X'$ and $W \to X''$ (Lemma \ref{lemma-quasi-compact-affine-cover}). Then $A \cap B$ is the image of $|V \times_X W| \to |X|$ (Lemma \ref{lemma-points-cartesian}). Since $V \times_X W$ is quasi-compact as $X$ is quasi-separated (Lemma \ref{lemma-characterize-quasi-separated}) we conclude that $A \cap B$ is quasi-compact and the proof is finished. \end{proof} \noindent The following lemma can be used to prove that an algebraic space is isomorphic to the spectrum of a field. \begin{lemma} \label{lemma-point-like-spaces} Let $S$ be a scheme. Let $k$ be a field. Let $X$ be an algebraic space over $S$ and assume that there exists a surjective \'etale morphism $\Spec(k) \to X$. If $X$ is quasi-separated, then $X \cong \Spec(k')$ where $k' \subset k$ is a finite separable extension. \end{lemma} \begin{proof} Set $R = \Spec(k) \times_X \Spec(k)$, so that we have a fibre product diagram $$\xymatrix{ R \ar[r]_-s \ar[d]_-t & \Spec(k) \ar[d] \\ \Spec(k) \ar[r] & X }$$ By Spaces, Lemma \ref{spaces-lemma-space-presentation} we know $X = \Spec(k)/R$ is the quotient sheaf. Because $\Spec(k) \to X$ is \'etale, the morphisms $s$ and $t$ are \'etale. Hence $R = \coprod_{i \in I} \Spec(k_i)$ is a disjoint union of spectra of fields, and both $s$ and $t$ induce finite separable field extensions $s, t : k \subset k_i$, see Morphisms, Lemma \ref{morphisms-lemma-etale-over-field}. Because $$R = \Spec(k) \times_X \Spec(k) = (\Spec(k) \times_S \Spec(k)) \times_{X \times_S X, \Delta} X$$ and since $\Delta$ is quasi-compact by assumption we conclude that $R \to \Spec(k) \times_S \Spec(k)$ is quasi-compact. Hence $R$ is quasi-compact as $\Spec(k) \times_S \Spec(k)$ is affine. We conclude that $I$ is finite. This implies that $s$ and $t$ are finite locally free morphisms. Hence by Groupoids, Proposition \ref{groupoids-proposition-finite-flat-equivalence} we conclude that $\Spec(k)/R$ is represented by $\Spec(k')$, with $k' \subset k$ finite locally free where $$k' = \{x \in k \mid s_i(x) = t_i(x)\text{ for all }i \in I\}$$ It is easy to see that $k'$ is a field. \end{proof} \begin{remark} \label{remark-cannot-decide-yet} Lemma \ref{lemma-point-like-spaces} holds for decent algebraic spaces, see Decent Spaces, Lemma \ref{decent-spaces-lemma-decent-point-like-spaces}. In fact a decent algebraic space with one point is a scheme, see Decent Spaces, Lemma \ref{decent-spaces-lemma-when-field}. This also holds when $X$ is locally separated, because a locally separated algebraic space is decent, see Decent Spaces, Lemma \ref{decent-spaces-lemma-locally-separated-decent}. \end{remark} \section{\'Etale morphisms of algebraic spaces} \label{section-etale-morphisms} \noindent This section really belongs in the chapter on morphisms of algebraic spaces, but we need the notion of an algebraic space \'etale over another in order to define the small \'etale site of an algebraic space. Thus we need to do some preliminary work on \'etale morphisms from schemes to algebraic spaces, and \'etale morphisms between algebraic spaces. For more about \'etale morphisms of algebraic spaces, see Morphisms of Spaces, Section \ref{spaces-morphisms-section-etale}. \begin{lemma} \label{lemma-etale-over-space} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$, $U'$ be schemes over $S$. \begin{enumerate} \item If $U \to U'$ is an \'etale morphism of schemes, and if $U' \to X$ is an \'etale morphism from $U'$ to $X$, then the composition $U \to X$ is an \'etale morphism from $U$ to $X$. \item If $\varphi : U \to X$ and $\varphi' : U' \to X$ are \'etale morphisms towards $X$, and if $\chi : U \to U'$ is a morphism of schemes such that $\varphi = \varphi' \circ \chi$, then $\chi$ is an \'etale morphism of schemes. \item If $\chi : U \to U'$ is a surjective \'etale morphism of schemes and $\varphi' : U' \to X$ is a morphism such that $\varphi = \varphi' \circ \chi$ is \'etale, then $\varphi'$ is \'etale. \end{enumerate} \end{lemma} \begin{proof} Recall that our definition of an \'etale morphism from a scheme into an algebraic space comes from Spaces, Definition \ref{spaces-definition-relative-representable-property} via the fact that any morphism from a scheme into an algebraic space is representable. \medskip\noindent Part (1) of the lemma follows from this, the fact that \'etale morphisms are preserved under composition (Morphisms, Lemma \ref{morphisms-lemma-composition-etale}) and Spaces, Lemmas \ref{spaces-lemma-composition-representable-transformations-property} and \ref{spaces-lemma-morphism-schemes-gives-representable-transformation-property} (which are formal). \medskip\noindent To prove part (2) choose a scheme $W$ over $S$ and a surjective \'etale morphism $W \to X$. Consider the base change $\chi_W : W \times_X U \to W \times_X U'$ of $\chi$. As $W \times_X U$ and $W \times_X U'$ are \'etale over $W$, we conclude that $\chi_W$ is \'etale, by Morphisms, Lemma \ref{morphisms-lemma-etale-permanence-two}. On the other hand, in the commutative diagram $$\xymatrix{ W \times_X U \ar[r] \ar[d] & W \times_X U' \ar[d] \\ U \ar[r] & U' }$$ the two vertical arrows are \'etale and surjective. Hence by Descent, Lemma \ref{descent-lemma-syntomic-smooth-etale-permanence} we conclude that $U \to U'$ is \'etale. \medskip\noindent To prove part (2) choose a scheme $W$ over $S$ and a morphism $W \to X$. As above we consider the diagram $$\xymatrix{ W \times_X U \ar[r] \ar[d] & W \times_X U' \ar[d] \ar[r] & W \ar[d] \\ U \ar[r] & U' \ar[r] & X }$$ Now we know that $W \times_X U \to W \times_X U'$ is surjective \'etale (as a base change of $U \to U'$) and that $W \times_X U \to W$ is \'etale. Thus $W \times_X U' \to W$ is \'etale by Descent, Lemma \ref{descent-lemma-syntomic-smooth-etale-permanence}. By definition this means that $\varphi'$ is \'etale. \end{proof} \begin{definition} \label{definition-etale} Let $S$ be a scheme. A morphism $f : X \to Y$ between algebraic spaces over $S$ is called {\it \'etale} if and only if for every \'etale morphism $\varphi : U \to X$ where $U$ is a scheme, the composition $f \circ \varphi$ is \'etale also. \end{definition} \noindent If $X$ and $Y$ are schemes, then this agree with the usual notion of an \'etale morphism of schemes. In fact, whenever $X \to Y$ is a representable morphism of algebraic spaces, then this agrees with the notion defined via Spaces, Definition \ref{spaces-definition-relative-representable-property}. This follows by combining Lemma \ref{lemma-etale-local} below and Spaces, Lemma \ref{spaces-lemma-representable-morphisms-spaces-property}. \begin{lemma} \label{lemma-etale-local} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \begin{enumerate} \item $f$ is \'etale, \item there exists a surjective \'etale morphism $\varphi : U \to X$, where $U$ is a scheme, such that the composition $f \circ \varphi$ is \'etale (as a morphism of algebraic spaces), \item there exists a surjective \'etale morphism $\psi : V \to Y$, where $V$ is a scheme, such that the base change $V \times_X Y \to V$ is \'etale (as a morphism of algebraic spaces), \item there exists a commutative diagram $$\xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y }$$ where $U$, $V$ are schemes, the vertical arrows are \'etale, and the left vertical arrow is surjective such that the horizontal arrow is \'etale. \end{enumerate} \end{lemma} \begin{proof} Let us prove that (4) implies (1). Assume a diagram as in (4) given. Let $W \to X$ be an \'etale morphism with $W$ a scheme. Then we see that $W \times_X U \to U$ is \'etale. Hence $W \times_X U \to V$ is \'etale as the composition of the \'etale morphisms of schemes $W \times_X U \to U$ and $U \to V$. Therefore $W \times_X U \to Y$ is \'etale by Lemma \ref{lemma-etale-over-space} (1). Since also the projection $W \times_X U \to W$ is surjective and \'etale, we conclude from Lemma \ref{lemma-etale-over-space} (3) that $W \to Y$ is \'etale. \medskip\noindent Let us prove that (1) implies (4). Assume (1). Choose a commutative diagram $$\xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y }$$ where $U \to X$ and $V \to Y$ are surjective and \'etale, see Spaces, Lemma \ref{spaces-lemma-lift-morphism-presentations}. By assumption the morphism $U \to Y$ is \'etale, and hence $U \to V$ is \'etale by Lemma \ref{lemma-etale-over-space} (2). \medskip\noindent We omit the proof that (2) and (3) are also equivalent to (1). \end{proof} \begin{lemma} \label{lemma-composition-etale} The composition of two \'etale morphisms of algebraic spaces is \'etale. \end{lemma} \begin{proof} This is immediate from the definition. \end{proof} \begin{lemma} \label{lemma-base-change-etale} The base change of an \'etale morphism of algebraic spaces by any morphism of algebraic spaces is \'etale. \end{lemma} \begin{proof} Let $X \to Y$ be an \'etale morphism of algebraic spaces over $S$. Let $Z \to Y$ be a morphism of algebraic spaces. Choose a scheme $U$ and a surjective \'etale morphism $U \to X$. Choose a scheme $W$ and a surjective \'etale morphism $W \to Z$. Then $U \to Y$ is \'etale, hence in the diagram $$\xymatrix{ W \times_Y U \ar[d] \ar[r] & W \ar[d] \\ Z \times_Y X \ar[r] & Z }$$ the top horizontal arrow is \'etale. Moreover, the left vertical arrow is surjective and \'etale (verification omitted). Hence we conclude that the lower horizontal arrow is \'etale by Lemma \ref{lemma-etale-local}. \end{proof} \begin{lemma} \label{lemma-etale-permanence} Let $S$ be a scheme. Let $X, Y, Z$ be algebraic spaces. Let $g : X \to Z$, $h : Y \to Z$ be \'etale morphisms and let $f : X \to Y$ be a morphism such that $h \circ f = g$. Then $f$ is \'etale. \end{lemma} \begin{proof} Choose a commutative diagram $$\xymatrix{ U \ar[d] \ar[r]_\chi & V \ar[d] \\ X \ar[r] & Y }$$ where $U \to X$ and $V \to Y$ are surjective and \'etale, see Spaces, Lemma \ref{spaces-lemma-lift-morphism-presentations}. By assumption the morphisms $\varphi : U \to X \to Z$ and $\psi : V \to Y \to Z$ are \'etale. Moreover, $\psi \circ \chi = \varphi$ by our assumption on $f, g, h$. Hence $U \to V$ is \'etale by Lemma \ref{lemma-etale-over-space} part (2). \end{proof} \begin{lemma} \label{lemma-etale-open} Let $S$ be a scheme. If $X \to Y$ is an \'etale morphism of algebraic spaces over $S$, then the associated map $|X| \to |Y|$ of topological spaces is open. \end{lemma} \begin{proof} This is clear from the diagram in Lemma \ref{lemma-etale-local} and Lemma \ref{lemma-topology-points}. \end{proof} \noindent Finally, here is a fun lemma. It is not true that an algebraic space with an \'etale morphism towards a scheme is a scheme, see Spaces, Example \ref{spaces-example-non-representable-descent}. But it is true if the target is the spectrum of a field. \begin{lemma} \label{lemma-etale-over-field-scheme} Let $S$ be a scheme. Let $X \to \Spec(k)$ be \'etale morphism over $S$, where $k$ is a field. Then $X$ is a scheme. \end{lemma} \begin{proof} Let $U$ be an affine scheme, and let $U \to X$ be an \'etale morphism. By Definition \ref{definition-etale} we see that $U \to \Spec(k)$ is an \'etale morphism. Hence $U = \coprod_{i = 1, \ldots, n} \Spec(k_i)$ is a finite disjoint union of spectra of finite separable extensions $k_i$ of $k$, see Morphisms, Lemma \ref{morphisms-lemma-etale-over-field}. The $R = U \times_X U \to U \times_{\Spec(k)} U$ is a monomorphism and $U \times_{\Spec(k)} U$ is also a finite disjoint union of spectra of finite separable extensions of $k$. Hence by Schemes, Lemma \ref{schemes-lemma-mono-towards-spec-field} we see that $R$ is similarly a finite disjoint union of spectra of finite separable extensions of $k$. This $U$ and $R$ are affine and both projections $R \to U$ are finite locally free. Hence $U/R$ is a scheme by Groupoids, Proposition \ref{groupoids-proposition-finite-flat-equivalence}. By Spaces, Lemma \ref{spaces-lemma-finding-opens} it is also an open subspace of $X$. By Lemma \ref{lemma-subscheme} we conclude that $X$ is a scheme. \end{proof} \section{Spaces and fpqc coverings} \label{section-fpqc} \noindent Let $S$ be a scheme. An algebraic space over $S$ is defined as a sheaf in the fppf topology with additional properties. Hence it is not immediately clear that it satisfies the sheaf property for the fpqc topology (see Topologies, Definition \ref{topologies-definition-sheaf-property-fpqc}). In this section we give Gabber's argument showing this is true. However, when we say that the algebraic space $X$ satisfies the sheaf property for the fpqc topology we really only consider fpqc coverings $\{f_i : T_i \to T\}_{i \in I}$ such that $T, T_i$ are objects of the big site $(\Sch/S)_{fppf}$ (as per our conventions, see Section \ref{section-conventions}). \begin{proposition}[Gabber] \label{proposition-sheaf-fpqc} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Then $X$ satisfies the sheaf property for the fpqc topology. \end{proposition} \begin{proof} Since $X$ is a sheaf for the Zariski topology it suffices to show the following. Given a surjective flat morphism of affines $f : T' \to T$ we have: $X(T)$ is the equalizer of the two maps $X(T') \to X(T' \times_T T')$. See Topologies, Lemma \ref{topologies-lemma-sheaf-property-fpqc} (there is a little argument omitted here because the lemma cited is formulated for functors defined on the category of all schemes). \medskip\noindent Let $a, b : T \to X$ be two morphisms such that $a \circ f = b \circ f$. We have to show $a = b$. Consider the fibre product $$E = X \times_{\Delta_{X/S}, X \times_S X, (a, b)} T.$$ By Spaces, Lemma \ref{spaces-lemma-properties-diagonal} the morphism $\Delta_{X/S}$ is a representable monomorphism. Hence $E \to T$ is a monomorphism of schemes. Our assumption that $a \circ f = b \circ f$ implies that $T' \to T$ factors (uniquely) through $E$. Consider the commutative diagram $$\xymatrix{ T' \times_T E \ar[r] \ar[d] & E \ar[d] \\ T' \ar[r] \ar@/^5ex/[u] \ar[ru] & T }$$ Since the projection $T' \times_T E \to T'$ is a monomorphism with a section we conclude it is an isomorphism. Hence we conclude that $E \to T$ is an isomorphism by Descent, Lemma \ref{descent-lemma-descending-property-isomorphism}. This means $a = b$ as desired. \medskip\noindent Next, let $c : T' \to X$ be a morphism such that the two compositions $T' \times_T T' \to T' \to X$ are the same. We have to find a morphism $a : T \to X$ whose composition with $T' \to T$ is $c$. Choose an affine scheme $U$ and an \'etale morphism $U \to X$ such that the image of $|U| \to |X|$ contains the image of $|c| : |T'| \to |X|$. This is possible by Lemmas \ref{lemma-topology-points} and \ref{lemma-cover-by-union-affines}, the fact that a finite union of affines is affine, and the fact that $|T'|$ is quasi-compact (small argument omitted). Since $U \to X$ is separated (Lemma \ref{lemma-separated-cover}), we see that $$V = U \times_{X, c} T' \longrightarrow T'$$ is a surjective, \'etale, separated morphism of schemes (to see that it is surjective use Lemma \ref{lemma-points-cartesian} and our choice of $U \to X$). The fact that $c \circ \text{pr}_0 = c \circ \text{pr}_1$ means that we obtain a descent datum on $V/T'/T$ (Descent, Definition \ref{descent-definition-descent-datum}) because \begin{align*} V \times_{T'} (T' \times_T T') & = U \times_{X, c \circ \text{pr}_0} (T' \times_T T') \\ & = (T' \times_T T') \times_{c \circ \text{pr}_1, X} U \\ & = (T' \times_T T') \times_{T'} V \end{align*} The morphism $V \to T'$ is ind-quasi-affine by More on Morphisms, Lemma \ref{more-morphisms-lemma-etale-separated-ind-quasi-affine} (because \'etale morphisms are locally quasi-finite, see Morphisms, Lemma \ref{morphisms-lemma-etale-locally-quasi-finite}). By More on Groupoids, Lemma \ref{more-groupoids-lemma-ind-quasi-affine} the descent datum is effective. Say $W \to T$ is a morphism such that there is an isomorphism $\alpha : T' \times_T W \to V$ compatible with the given descent datum on $V$ and the canonical descent datum on $T' \times_T W$. Then $W \to T$ is surjective and \'etale (Descent, Lemmas \ref{descent-lemma-descending-property-surjective} and \ref{descent-lemma-descending-property-etale}). Consider the composition $$b' : T' \times_T W \longrightarrow V = U \times_{X, c} T' \longrightarrow U$$ The two compositions $b' \circ (\text{pr}_0, 1), b' \circ (\text{pr}_1, 1) : (T' \times_T T') \times_T W \to T' \times_T W \to U$ agree by our choice of $\alpha$ and the corresponding property of $c$ (computation omitted). Hence $b'$ descends to a morphism $b : W \to U$ by Descent, Lemma \ref{descent-lemma-fpqc-universal-effective-epimorphisms}. The diagram $$\xymatrix{ T' \times_T W \ar[r] \ar[d] & W \ar[r]_b & U \ar[d] \\ T' \ar[rr]^c & & X }$$ is commutative. What this means is that we have proved the existence of $a$ \'etale locally on $T$, i.e., we have an $a' : W \to X$. However, since we have proved uniqueness in the first paragraph, we find that this \'etale local solution satisfies the glueing condition, i.e., we have $\text{pr}_0^*a' = \text{pr}_1^*a'$ as elements of $X(W \times_T W)$. Since $X$ is an \'etale sheaf we find a unique $a \in X(T)$ restricting to $a'$ on $W$. \end{proof} \section{The \'etale site of an algebraic space} \label{section-etale-site} \noindent In this section we define the small \'etale site of an algebraic space. This is the analogue of the small \'etale site $S_\etale$ of a scheme. Lemma \ref{lemma-etale-over-space} implies that in the definition below any morphism between objects of the \'etale site of $X$ is \'etale, and that any scheme \'etale over an object of $X_\etale$ is also an object of $X_\etale$. \begin{definition} \label{definition-etale-site} Let $S$ be a scheme. Let $\Sch_{fppf}$ be a big fppf site containing $S$, and let $\Sch_\etale$ be the corresponding big \'etale site (i.e., having the same underlying category). Let $X$ be an algebraic space over $S$. The {\it small \'etale site $X_\etale$} of $X$ is defined as follows: \begin{enumerate} \item An object of $X_\etale$ is a morphism $\varphi : U \to X$ where $U \in \Ob((\Sch/S)_\etale)$ is a scheme and $\varphi$ is an \'etale morphism, \item a morphism $(\varphi : U \to X) \to (\varphi' : U' \to X)$ is given by a morphism of schemes $\chi : U \to U'$ such that $\varphi = \varphi' \circ \chi$, and \item a family of morphisms $\{(U_i \to X) \to (U \to X)\}_{i \in I}$ of $X_\etale$ is a covering if and only if $\{U_i \to U\}_{i \in I}$ is a covering of $(\Sch/S)_\etale$. \end{enumerate} \end{definition} \noindent A consequence of our choice is that the \'etale site of an algebraic space in general does not have a final object! On the other hand, if $X$ happens to be a scheme, then the definition above agrees with Topologies, Definition \ref{topologies-definition-big-small-etale}. \medskip\noindent There are several other choices we could have made here. For example we could have considered all {\it algebraic spaces} $U$ which are \'etale over $X$, or we could have considered all {\it affine schemes} $U$ which are \'etale over $X$. We decided not to do so, since we like to think of plain old schemes as the fundamental objects of algebraic geometry. On the other hand, we do need these notions also, since the small \'etale site of an algebraic space is not sufficiently flexible, especially when discussing functoriality of the small \'etale site, see Lemma \ref{lemma-functoriality-etale-site} below. \begin{definition} \label{definition-spaces-etale-site} Let $S$ be a scheme. Let $\Sch_{fppf}$ be a big fppf site containing $S$, and let $\Sch_\etale$ be the corresponding big \'etale site (i.e., having the same underlying category). Let $X$ be an algebraic space over $S$. The site {\it $X_{spaces, \etale}$} of $X$ is defined as follows: \begin{enumerate} \item An object of $X_{spaces, \etale}$ is a morphism $\varphi : U \to X$ where $U$ is an algebraic space over $S$ and $\varphi$ is an \'etale morphism of algebraic spaces over $S$, \item a morphism $(\varphi : U \to X) \to (\varphi' : U' \to X)$ of $X_{spaces, \etale}$ is given by a morphism of algebraic spaces $\chi : U \to U'$ such that $\varphi = \varphi' \circ \chi$, and \item a family of morphisms $\{\varphi_i : (U_i \to X) \to (U \to X)\}_{i \in I}$ of $X_{spaces, \etale}$ is a covering if and only if $|U| = \bigcup \varphi_i(|U_i|)$. \end{enumerate} (As usual we choose a set of coverings of this type, including at least the coverings in $X_\etale$, as in Sets, Lemma \ref{sets-lemma-coverings-site} to turn $X_{spaces, \etale}$ into a site.) \end{definition} \noindent Since the identity morphism of $X$ is \'etale it is clear that $X_{spaces, \etale}$ does have a final object. Let us show right away that the corresponding topos equals the small \'etale topos of $X$. \begin{lemma} \label{lemma-compare-etale-sites} The functor $$X_\etale \longrightarrow X_{spaces, \etale}, \quad U/X \longmapsto U/X$$ is a special cocontinuous functor (Sites, Definition \ref{sites-definition-special-cocontinuous-functor}) and hence induces an equivalence of topoi $\Sh(X_\etale) \to \Sh(X_{spaces, \etale})$. \end{lemma} \begin{proof} We have to show that the functor satisfies the assumptions (1) -- (5) of Sites, Lemma \ref{sites-lemma-equivalence}. It is clear that the functor is continuous and cocontinuous, which proves assumptions (1) and (2). Assumptions (3) and (4) hold simply because the functor is fully faithful. Assumption (5) holds, because an algebraic space by definition has a covering by a scheme. \end{proof} \begin{remark} \label{remark-explain-equivalence} Let us explain the meaning of Lemma \ref{lemma-compare-etale-sites}. Let $S$ be a scheme, and let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a sheaf on the small \'etale site $X_\etale$ of $X$. The lemma says that there exists a unique sheaf $\mathcal{F}'$ on $X_{spaces, \etale}$ which restricts back to $\mathcal{F}$ on the subcategory $X_\etale$. If $U \to X$ is an \'etale morphism of algebraic spaces, then how do we compute $\mathcal{F}'(U)$? Well, by definition of an algebraic space there exists a scheme $U'$ and a surjective \'etale morphism $U' \to U$. Then $\{U' \to U\}$ is a covering in $X_{spaces, \etale}$ and hence we get an equalizer diagram $$\xymatrix{ \mathcal{F}'(U) \ar[r] & \mathcal{F}(U') \ar@<1ex>[r] \ar@<-1ex>[r] & \mathcal{F}(U' \times_U U'). }$$ Note that $U' \times_U U'$ is a scheme, and hence we may write $\mathcal{F}$ and not $\mathcal{F}'$. Thus we see how to compute $\mathcal{F}'$ when given the sheaf $\mathcal{F}$. \end{remark} \begin{lemma} \label{lemma-alternative} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $X_{affine, \etale}$ denote the full subcategory of $X_\etale$ whose objects are those $U/X \in \Ob(X_\etale)$ with $U$ affine. A covering of $X_{affine, \etale}$ will be a standard \'etale covering, see Topologies, Definition \ref{topologies-definition-standard-etale}. Then restriction $$\mathcal{F} \longmapsto \mathcal{F}|_{X_{affine, \etale}}$$ defines an equivalence of topoi $\Sh(S_\etale) \cong \Sh(S_{affine, \etale})$. \end{lemma} \begin{proof} This you can show directly from the definitions, and is a good exercise. But it also follows immediately from Sites, Lemma \ref{sites-lemma-equivalence} by checking that the inclusion functor $X_{affine, \etale} \to X_\etale$ is a special cocontinuous functor as in Sites, Definition \ref{sites-definition-special-cocontinuous-functor}. \end{proof} \begin{definition} \label{definition-etale-topos} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The {\it \'etale topos} of $X$, or more precisely the {\it small \'etale topos} of $X$ is the category $\Sh(X_\etale)$ of sheaves of sets on $X_\etale$. \end{definition} \noindent By Lemma \ref{lemma-compare-etale-sites} we have $\Sh(X_\etale) = \Sh(X_{spaces, \etale})$, so we can also think of this as the category of sheaves of sets on $X_{spaces, \etale}$. Similarly, by Lemma \ref{lemma-alternative} we see that $\Sh(X_\etale) = \Sh(X_{affine, \etale})$. It turns out that the topos is functorial with respect to morphisms of algebraic spaces. Here is a precise statement. \begin{lemma} \label{lemma-functoriality-etale-site} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. \begin{enumerate} \item The continuous functor $$Y_{spaces, \etale} \longrightarrow X_{spaces, \etale}, \quad V \longmapsto X \times_Y V$$ induces a morphism of sites $$f_{spaces, \etale} : X_{spaces, \etale} \to Y_{spaces, \etale}.$$ \item The rule $f \mapsto f_{spaces, \etale}$ is compatible with compositions, in other words $(f \circ g)_{spaces, \etale} = f_{spaces, \etale} \circ g_{spaces, \etale}$ (see Sites, Definition \ref{sites-definition-composition-morphisms-sites}). \item The morphism of topoi associated to $f_{spaces, \etale}$ induces, via Lemma \ref{lemma-compare-etale-sites}, a morphism of topoi $f_{small} : \Sh(X_\etale) \to \Sh(Y_\etale)$ whose construction is compatible with compositions. \item If $f$ is a representable morphism of algebraic spaces, then $f_{small}$ comes from a morphism of sites $X_\etale \to Y_\etale$, corresponding to the continuous functor $V \mapsto X \times_Y V$. \end{enumerate} \end{lemma} \begin{proof} Let us show that the functor described in (1) satisfies the assumptions of Sites, Proposition \ref{sites-proposition-get-morphism}. Thus we have to show that $Y_{spaces, \etale}$ has a final object (namely $Y$) and that the functor transforms this into a final object in $X_{spaces, \etale}$ (namely $X$). This is clear as $X \times_Y Y = X$ in any category. Next, we have to show that $Y_{spaces, \etale}$ has fibre products. This is true since the category of algebraic spaces has fibre products, and since $V \times_Y V'$ is \'etale over $Y$ if $V$ and $V'$ are \'etale over $Y$ (see Lemmas \ref{lemma-composition-etale} and \ref{lemma-base-change-etale} above). OK, so the proposition applies and we see that we get a morphism of sites as described in (1). \medskip\noindent Part (2) you get by unwinding the definitions. Part (3) is clear by using the equivalences for $X$ and $Y$ from Lemma \ref{lemma-compare-etale-sites} above. Part (4) follows, because if $f$ is representable, then the functors above fit into a commutative diagram $$\xymatrix{ X_\etale \ar[r] & X_{spaces, \etale} \\ Y_\etale \ar[r] \ar[u] & Y_{spaces, \etale} \ar[u] }$$ of categories. \end{proof} \noindent We can do a little bit better than the lemma above in describing the relationship between sheaves on $X$ and sheaves on $Y$. Namely, we can formulate this in turns of $f$-maps, compare Sheaves, Definition \ref{sheaves-definition-f-map}, as follows. \begin{definition} \label{definition-f-map} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a sheaf of sets on $X_\etale$ and let $\mathcal{G}$ be a sheaf of sets on $Y_\etale$. An {\it $f$-map $\varphi : \mathcal{G} \to \mathcal{F}$} is a collection of maps $\varphi_{(U, V, g)} : \mathcal{G}(V) \to \mathcal{F}(U)$ indexed by commutative diagrams $$\xymatrix{ U \ar[d]_g \ar[r] & X \ar[d]^f \\ V \ar[r] & Y }$$ where $U \in X_\etale$, $V \in Y_\etale$ such that whenever given an extended diagram $$\xymatrix{ U' \ar[r] \ar[d]_{g'} & U \ar[d]_g \ar[r] & X \ar[d]^f \\ V' \ar[r] & V \ar[r] & Y }$$ with $V' \to V$ and $U' \to U$ \'etale morphisms of schemes the diagram $$\xymatrix{ \mathcal{G}(V) \ar[rr]_{\varphi_{(U, V, g)}} \ar[d]_{\text{restriction of }\mathcal{G}} & & \mathcal{F}(U) \ar[d]^{\text{restriction of }\mathcal{F}} \\ \mathcal{G}(V') \ar[rr]^{\varphi_{(U', V', g')}} & & \mathcal{F}(U') }$$ commutes. \end{definition} \begin{lemma} \label{lemma-f-map} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a sheaf of sets on $X_\etale$ and let $\mathcal{G}$ be a sheaf of sets on $Y_\etale$. There are canonical bijections between the following three sets: \begin{enumerate} \item The set of maps $\mathcal{G} \to f_{small, *}\mathcal{F}$. \item The set of maps $f_{small}^{-1}\mathcal{G} \to \mathcal{F}$. \item The set of $f$-maps $\varphi : \mathcal{G} \to \mathcal{F}$. \end{enumerate} \end{lemma} \begin{proof} Note that (1) and (2) are the same because the functors $f_{small, *}$ and $f_{small}^{-1}$ are a pair of adjoint functors. Suppose that $\alpha : f_{small}^{-1}\mathcal{G} \to \mathcal{F}$ is a map of sheaves on $Y_\etale$. Let a diagram $$\xymatrix{ U \ar[d]_g \ar[r]_{j_U} & X \ar[d]^f \\ V \ar[r]^{j_V} & Y }$$ as in Definition \ref{definition-f-map} be given. By the commutativity of the diagram we also get a map $g_{small}^{-1}(j_V)^{-1}\mathcal{G} \to (j_U)^{-1}\mathcal{F}$ (compare Sites, Section \ref{sites-section-localize} for the description of the localization functors). Hence we certainly get a map $\varphi_{(V, U, g)} : \mathcal{G}(V) = (j_V)^{-1}\mathcal{G}(V) \to (j_U)^{-1}\mathcal{F}(U) = \mathcal{F}(U)$. We omit the verification that this rule is compatible with further restrictions and defines an $f$-map from $\mathcal{G}$ to $\mathcal{F}$. \medskip\noindent Conversely, suppose that we are given an $f$-map $\varphi = (\varphi_{(U, V, g)})$. Let $\mathcal{G}'$ (resp.\ $\mathcal{F}'$) denote the extension of $\mathcal{G}$ (resp.\ $\mathcal{F}$) to $Y_{spaces, \etale}$ (resp.\ $X_{spaces, \etale}$), see Lemma \ref{lemma-compare-etale-sites}. Then we have to construct a map of sheaves $$\mathcal{G}' \longrightarrow (f_{spaces, \etale})_*\mathcal{F}'$$ To do this, let $V \to Y$ be an \'etale morphism of algebraic spaces. We have to construct a map of sets $$\mathcal{G}'(V) \to \mathcal{F}'(X \times_Y V)$$ Choose an \'etale surjective morphism $V' \to V$ with $V'$ a scheme, and after that choose an \'etale surjective morphism $U' \to X \times_U V'$ with $U'$ a scheme. We get a morphism of schemes $g' : U' \to V'$ and also a morphism of schemes $$g'' : U' \times_{X \times_Y V} U' \longrightarrow V' \times_V V'$$ Consider the following diagram $$\xymatrix{ \mathcal{F}'(X \times_Y V) \ar[r] & \mathcal{F}(U') \ar@<1ex>[r] \ar@<-1ex>[r] & \mathcal{F}(U' \times_{X \times_Y V} U') \\ \mathcal{G}'(X \times_Y V) \ar[r] \ar@{..>}[u] & \mathcal{G}(V') \ar@<1ex>[r] \ar@<-1ex>[r] \ar[u]_{\varphi_{(U', V', g')}} & \mathcal{G}(V' \times_V V') \ar[u]_{\varphi_{(U'', V'', g'')}} }$$ The compatibility of the maps $\varphi_{...}$ with restriction shows that the two right squares commute. The definition of coverings in $X_{spaces, \etale}$ shows that the horizontal rows are equalizer diagrams. Hence we get the dotted arrow. We leave it to the reader to show that these arrows are compatible with the restriction mappings. \end{proof} \noindent If the morphism of algebraic spaces $X \to Y$ is \'etale, then the morphism of topoi $\Sh(X_\etale) \to \Sh(Y_\etale)$ is a localization. Here is a statement. \begin{lemma} \label{lemma-etale-morphism-topoi} Let $S$ be a scheme, and let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is \'etale. In this case there is a functor $$j : X_\etale \to Y_\etale, \quad (\varphi : U \to X) \mapsto (f \circ \varphi : U \to Y)$$ which is cocontinuous. The morphism of topoi $f_{small}$ is the morphism of topoi associated to $j$, see Sites, Lemma \ref{sites-lemma-cocontinuous-morphism-topoi}. Moreover, $j$ is continuous as well, hence Sites, Lemma \ref{sites-lemma-when-shriek} applies. In particular $f_{small}^{-1}\mathcal{G}(U) = \mathcal{G}(jU)$ for all sheaves $\mathcal{G}$ on $Y_\etale$. \end{lemma} \begin{proof} Note that by our very definition of an \'etale morphism of algebraic spaces (Definition \ref{definition-etale}) it is indeed the case that the rule given defines a functor $j$ as indicated. It is clear that $j$ is cocontinuous and continuous, simply because a covering $\{U_i \to U\}$ of $j(\varphi : U \to X)$ in $Y_\etale$ is the same thing as a covering of $(\varphi : U \to X)$ in $X_\etale$. It remains to show that $j$ induces the same morphism of topoi as $f_{small}$. To see this we consider the diagram $$\xymatrix{ X_\etale \ar[r] \ar[d]^j & X_{spaces, \etale} \ar@/_/[d]_{j_{spaces}} \\ Y_\etale \ar[r] & Y_{spaces, \etale} \ar@/_/[u]_{v : V \mapsto X \times_Y V} }$$ of categories. Here the functor $j_{spaces}$ is the obvious extension of $j$ to the category $X_{spaces, \etale}$. Thus the inner square is commutative. In fact $j_{spaces}$ can be identified with the localization functor $j_X : Y_{spaces, \etale}/X \to Y_{spaces, \etale}$ discussed in Sites, Section \ref{sites-section-localize}. Hence, by Sites, Lemma \ref{sites-lemma-localize-given-products} the cocontinuous functor $j_{spaces}$ and the functor $v$ of the diagram induce the same morphism of topoi. By Sites, Lemma \ref{sites-lemma-composition-cocontinuous} the commutativity of the inner square (consisting of cocontinuous functors between sites) gives a commutative diagram of associated morphisms of topoi. Hence, by the construction of $f_{small}$ in Lemma \ref{lemma-functoriality-etale-site} we win. \end{proof} \noindent The lemma above says that the pullback of $\mathcal{G}$ via an \'etale morphism $f : X \to Y$ of algebraic spaces is simply the restriction of $\mathcal{G}$ to the category $X_\etale$. We will often use the short hand \begin{equation} \label{equation-restrict} \mathcal{G}|_{X_\etale} = f_{small}^{-1}\mathcal{G} \end{equation} to indicate this. Note that the functor $j : X_\etale \to Y_\etale$ of the lemma in this situation is faithful, but not fully faithful in general. We will discuss this in a more technical fashion in Section \ref{section-localize}. \begin{lemma} \label{lemma-pushforward-etale-base-change} Let $S$ be a scheme. Let $$\xymatrix{ X' \ar[r] \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y }$$ be a cartesian square of algebraic spaces over $S$. Let $\mathcal{F}$ be a sheaf on $X_\etale$. If $g$ is \'etale, then \begin{enumerate} \item $f'_{small, *}(\mathcal{F}|_{X'}) = (f_{small, *}\mathcal{F})|_{Y'}$ in $\Sh(Y'_\etale)$\footnote{Also $(f')_{small}^{-1}(\mathcal{G}|_{Y'}) = (f_{small}^{-1}\mathcal{G})|_{X'}$ because of commutativity of the diagram and (\ref{equation-restrict})}, and \item if $\mathcal{F}$ is an abelian sheaf, then $R^if'_{small, *}(\mathcal{F}|_{X'}) = (R^if_{small, *}\mathcal{F})|_{Y'}$. \end{enumerate} \end{lemma} \begin{proof} Consider the following diagram of functors $$\xymatrix{ X'_{spaces, \etale} \ar[r]_j & X_{spaces, \etale} \\ Y'_{spaces, \etale} \ar[r]^j \ar[u]^{V' \mapsto V' \times_{Y'} X'} & Y_{spaces, \etale} \ar[u]_{V \mapsto V \times_Y X} }$$ The horizontal arrows are localizations and the vertical arrows induce morphisms of sites. Hence the last statement of Sites, Lemma \ref{sites-lemma-localize-morphism} gives (1). To see (2) apply (1) to an injective resolution of $\mathcal{F}$ and use that restriction is exact and preserves injectives (see Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cohomology-of-open}). \end{proof} \noindent The following lemma says that you can think of a sheaf on the small \'etale site of an algebraic space as a compatible collection of sheaves on the small \'etale sites of schemes \'etale over the space. Please note that all the comparison mappings $c_f$ in the lemma are isomorphisms, which is compatible with Topologies, Lemma \ref{topologies-lemma-characterize-sheaf-big-etale} and the fact that all morphisms between objects of $X_\etale$ are \'etale. \begin{lemma} \label{lemma-characterize-sheaf-small-etale} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. A sheaf $\mathcal{F}$ on $X_\etale$ is given by the following data: \begin{enumerate} \item for every $U \in \Ob(X_\etale)$ a sheaf $\mathcal{F}_U$ on $U_\etale$, \item for every $f : U' \to U$ in $X_\etale$ an isomorphism $c_f : f_{small}^{-1}\mathcal{F}_U \to \mathcal{F}_{U'}$. \end{enumerate} These data are subject to the condition that given any $f : U' \to U$ and $g : U'' \to U'$ in $X_\etale$ the composition $g_{small}^{-1}c_f \circ c_g$ is equal to $c_{f \circ g}$. \end{lemma} \begin{proof} Given a sheaf $\mathcal{F}$ on $X_\etale$ and an object $\varphi : U \to X$ of $X_\etale$ we set $\mathcal{F}_U = \varphi_{small}^{-1}\mathcal{F}$. If $\varphi' : U' \to X$ is a second object of $X_\etale$, and $f : U' \to U$ is a morphism between them, then the isomorphism $c_f$ comes from the fact that $f_{small}^{-1} \circ \varphi_{small}^{-1} = (\varphi')^{-1}_{small}$, see Lemma \ref{lemma-functoriality-etale-site}. The condition on the transitivity of the isomorphisms $c_f$ follows from the functoriality of the small \'etale sites also; verification omitted. \medskip\noindent Conversely, suppose we are given a collection of data $(\mathcal{F}_U, c_f)$ as in the lemma. In this case we simply define $\mathcal{F}$ by the rule $U \mapsto \mathcal{F}_U(U)$. Details omitted. \end{proof} \noindent Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $X = U/R$ be a presentation of $X$ coming from any surjective \'etale morphism $\varphi : U \to X$, see Spaces, Definition \ref{spaces-definition-presentation}. In particular, we obtain a groupoid $(U, R, s, t, c, e, i)$ such that $j = (t, s) : R \to U \times_S U$, see Groupoids, Lemma \ref{groupoids-lemma-equivalence-groupoid}. \begin{lemma} \label{lemma-descent-sheaf} With $S$, $\varphi : U \to X$, and $(U, R, s, t, c, e, i)$ as above. For any sheaf $\mathcal{F}$ on $X_\etale$ the sheaf\footnote{In this lemma and its proof we write simply $\varphi^{-1}$ instead of $\varphi_{small}^{-1}$ and similarly for all the other pullbacks.} $\mathcal{G} = \varphi^{-1}\mathcal{F}$ comes equipped with a canonical isomorphism $$\alpha : t^{-1}\mathcal{G} \longrightarrow s^{-1}\mathcal{G}$$ such that the diagram $$\xymatrix{ & \text{pr}_1^{-1}t^{-1}\mathcal{G} \ar[r]_-{\text{pr}_1^{-1}\alpha} & \text{pr}_1^{-1}s^{-1}\mathcal{G} \ar@{=}[rd] & \\ \text{pr}_0^{-1}s^{-1}\mathcal{G} \ar@{=}[ru] & & & c^{-1}s^{-1}\mathcal{G} \\ & \text{pr}_0^{-1}t^{-1}\mathcal{G} \ar[lu]^{\text{pr}_0^{-1}\alpha} \ar@{=}[r] & c^{-1}t^{-1}\mathcal{G} \ar[ru]_{c^{-1}\alpha} }$$ is a commutative. The functor $\mathcal{F} \mapsto (\mathcal{G}, \alpha)$ defines an equivalence of categories between sheaves on $X_\etale$ and pairs $(\mathcal{G}, \alpha)$ as above. \end{lemma} \begin{proof}[First proof of Lemma \ref{lemma-descent-sheaf}] Let $\mathcal{C} = X_{spaces, \etale}$. By Lemma \ref{lemma-etale-morphism-topoi} and its proof we have $U_{spaces, \etale} = \mathcal{C}/U$ and the pullback functor $\varphi^{-1}$ is just the restriction functor. Moreover, $\{U \to X\}$ is a covering of the site $\mathcal{C}$ and $R = U \times_X U$. The isomorphism $\alpha$ is just the canonical identification $$\left(\mathcal{F}|_{\mathcal{C}/U}\right)|_{\mathcal{C}/U \times_X U} = \left(\mathcal{F}|_{\mathcal{C}/U}\right)|_{\mathcal{C}/U \times_X U}$$ and the commutativity of the diagram is the cocycle condition for glueing data. Hence this lemma is a special case of glueing of sheaves, see Sites, Section \ref{sites-section-glueing-sheaves}. \end{proof} \begin{proof}[Second proof of Lemma \ref{lemma-descent-sheaf}] The existence of $\alpha$ comes from the fact that $\varphi \circ t = \varphi \circ s$ and that pullback is functorial in the morphism, see Lemma \ref{lemma-functoriality-etale-site}. In exactly the same way, i.e., by functoriality of pullback, we see that the isomorphism $\alpha$ fits into the commutative diagram. The construction $\mathcal{F} \mapsto (\varphi^{-1}\mathcal{F}, \alpha)$ is clearly functorial in the sheaf $\mathcal{F}$. Hence we obtain the functor. \medskip\noindent Conversely, suppose that $(\mathcal{G}, \alpha)$ is a pair. Let $V \to X$ be an object of $X_\etale$. In this case the morphism $V' = U \times_X V \to V$ is a surjective \'etale morphism of schemes, and hence $\{V' \to V\}$ is an \'etale covering of $V$. Set $\mathcal{G}' = (V' \to V)^{-1}\mathcal{G}$. Since $R = U \times_X U$ with $t = \text{pr}_0$ and $s = \text{pr}_0$ we see that $V' \times_V V' = R \times_X V$ with projection maps $s', t' : V' \times_V V' \to V'$ equal to the pullbacks of $t$ and $s$. Hence $\alpha$ pulls back to an isomorphism $\alpha' : (t')^{-1}\mathcal{G}' \to (s')^{-1}\mathcal{G}'$. Having said this we simply define $$\xymatrix{ \mathcal{F}(V) \ar@{=}[r] & \text{Equalizer}(\mathcal{G}(V') \ar@<1ex>[r] \ar@<-1ex>[r] & \mathcal{G}(V' \times_V V'). }$$ We omit the verification that this defines a sheaf. To see that $\mathcal{G}(V) = \mathcal{F}(V)$ if there exists a morphism $V \to U$ note that in this case the equalizer is $H^0(\{V' \to V\}, \mathcal{G}) = \mathcal{G}(V)$. \end{proof} \section{Points of the small \'etale site} \label{section-points-small-etale-site} \noindent This section is the analogue of \'Etale Cohomology, Section \ref{etale-cohomology-section-stalks}. \begin{definition} \label{definition-geometric-point} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \begin{enumerate} \item A {\it geometric point} of $X$ is a morphism $\overline{x} : \Spec(k) \to X$, where $k$ is an algebraically closed field. We often abuse notation and write $\overline{x} = \Spec(k)$. \item For every geometric point $\overline{x}$ we have the corresponding image'' point $x \in |X|$. We say that $\overline{x}$ is a {\it geometric point lying over $x$}. \end{enumerate} \end{definition} \noindent It turns out that we can take stalks of sheaves on $X_\etale$ at geometric point exactly in the same way as was done in the case of the small \'etale site of a scheme. In order to do this we define the notion of an \'etale neighbourhood as follows. \begin{definition} \label{definition-etale-neighbourhood} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\overline{x}$ be a geometric point of $X$. \begin{enumerate} \item An {\it \'etale neighborhood} of $\overline{x}$ of $X$ is a commutative diagram $$\xymatrix{ & U \ar[d]^\varphi \\ {\bar x} \ar[r]^{\bar x} \ar[ur]^{\bar u} & X }$$ where $\varphi$ is an \'etale morphism of algebraic spaces over $S$. We will use the notation $\varphi : (U, \overline{u}) \to (X, \overline{x})$ to indicate this situation. \item A {\it morphism of \'etale neighborhoods} $(U, \overline{u}) \to (U', \overline{u}')$ is an $X$-morphism $h : U \to U'$ such that $\overline{u}' = h \circ \overline{u}$. \end{enumerate} \end{definition} \noindent Note that we allow $U$ to be an algebraic space. When we take stalks of a sheaf on $X_\etale$ we have to restrict to those $U$ which are in $X_\etale$, and so in this case we will only consider the case where $U$ is a scheme. Alternately we can work with the site $X_{space, \etale}$ and consider all \'etale neighbourhoods. And there won't be any difference because of the last assertion in the following lemma. \begin{lemma} \label{lemma-cofinal-etale} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\overline{x}$ be a geometric point of $X$. The category of \'etale neighborhoods is cofiltered. More precisely: \begin{enumerate} \item Let $(U_i, \overline{u}_i)_{i = 1, 2}$ be two \'etale neighborhoods of $\overline{x}$ in $X$. Then there exists a third \'etale neighborhood $(U, \overline{u})$ and morphisms $(U, \overline{u}) \to (U_i, \overline{u}_i)$, $i = 1, 2$. \item Let $h_1, h_2: (U, \overline{u}) \to (U', \overline{u}')$ be two morphisms between \'etale neighborhoods of $\overline{s}$. Then there exist an \'etale neighborhood $(U'', \overline{u}'')$ and a morphism $h : (U'', \overline{u}'') \to (U, \overline{u})$ which equalizes $h_1$ and $h_2$, i.e., such that $h_1 \circ h = h_2 \circ h$. \end{enumerate} Moreover, given any \'etale neighbourhood $(U, \overline{u}) \to (X, \overline{x})$ there exists a morphism of \'etale neighbourhoods $(U', \overline{u}') \to (U, \overline{u})$ where $U'$ is a scheme. \end{lemma} \begin{proof} For part (1), consider the fibre product $U = U_1 \times_X U_2$. It is \'etale over both $U_1$ and $U_2$ because \'etale morphisms are preserved under base change and composition, see Lemmas \ref{lemma-base-change-etale} and \ref{lemma-composition-etale}. The map $\overline{u} \to U$ defined by $(\overline{u}_1, \overline{u}_2)$ gives it the structure of an \'etale neighborhood mapping to both $U_1$ and $U_2$. \medskip\noindent For part (2), define $U''$ as the fibre product $$\xymatrix{ U'' \ar[r] \ar[d] & U \ar[d]^{(h_1, h_2)} \\ U' \ar[r]^-\Delta & U' \times_X U'. }$$ Since $\overline{u}$ and $\overline{u}'$ agree over $X$ with $\overline{x}$, we see that $\overline{u}'' = (\overline{u}, \overline{u}')$ is a geometric point of $U''$. In particular $U'' \not = \emptyset$. Moreover, since $U'$ is \'etale over $X$, so is the fibre product $U'\times_X U'$ (as seen above in the case of $U_1 \times_X U_2$). Hence the vertical arrow $(h_1, h_2)$ is \'etale by Lemma \ref{lemma-etale-permanence}. Therefore $U''$ is \'etale over $U'$ by base change, and hence also \'etale over $X$ (because compositions of \'etale morphisms are \'etale). Thus $(U'', \overline{u}'')$ is a solution to the problem posed by (2). \medskip\noindent To see the final assertion, choose any surjective \'etale morphism $U' \to U$ where $U'$ is a scheme. Then $U' \times_U \overline{u}$ is a scheme surjective and \'etale over $\overline{u} = \Spec(k)$ with $k$ algebraically closed. It follows (see Morphisms, Lemma \ref{morphisms-lemma-etale-over-field}) that $U' \times_U \overline{u} \to \overline{u}$ has a section which gives us the desired $\overline{u}'$. \end{proof} \begin{lemma} \label{lemma-geometric-lift-to-usual} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\overline{x} : \Spec(k) \to X$ be a geometric point of $X$ lying over $x \in |X|$. Let $\varphi : U \to X$ be an \'etale morphism of algebraic spaces and let $u \in |U|$ with $\varphi(u) = x$. Then there exists a geometric point $\overline{u} : \Spec(k) \to U$ lying over $u$ with $\overline{x} = f \circ \overline{u}$. \end{lemma} \begin{proof} Choose an affine scheme $U'$ with $u' \in U'$ and an \'etale morphism $U' \to U$ which maps $u'$ to $u$. If we can prove the lemma for $(U', u') \to (X, x)$ then the lemma follows. Hence we may assume that $U$ is a scheme, in particular that $U \to X$ is representable. Then look at the cartesian diagram $$\xymatrix{ \Spec(k) \times_{\overline{x}, X, \varphi} U \ar[d]_{\text{pr}_1} \ar[r]_-{\text{pr}_2} & U \ar[d]^\varphi \\ \Spec(k) \ar[r]^-{\overline{x}} & X }$$ The projection $\text{pr}_1$ is the base change of an \'etale morphisms so it is \'etale, see Lemma \ref{lemma-base-change-etale}. Therefore, the scheme $\Spec(k) \times_{\overline{x}, X, \varphi} U$ is a disjoint union of finite separable extensions of $k$, see Morphisms, Lemma \ref{morphisms-lemma-etale-over-field}. But $k$ is algebraically closed, so all these extensions are trivial, so $\Spec(k) \times_{\overline{x}, X, \varphi} U$ is a disjoint union of copies of $\Spec(k)$ and each of these corresponds to a geometric point $\overline{u}$ with $f \circ \overline{u} = \overline{x}$. By Lemma \ref{lemma-points-cartesian} the map $$|\Spec(k) \times_{\overline{x}, X, \varphi} U| \longrightarrow |\Spec(k)| \times_{|X|} |U|$$ is surjective, hence we can pick $\overline{u}$ to lie over $u$. \end{proof} \begin{lemma} \label{lemma-geometric-lift-to-cover} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\overline{x}$ be a geometric point of $X$. Let $(U, \overline{u})$ an \'etale neighborhood of $\overline{x}$. Let $\{\varphi_i : U_i \to U\}_{i \in I}$ be an \'etale covering in $X_{spaces, \etale}$. Then there exist $i \in I$ and $\overline{u}_i : \overline{x} \to U_i$ such that $\varphi_i : (U_i, \overline{u}_i) \to (U, \overline{u})$ is a morphism of \'etale neighborhoods. \end{lemma} \begin{proof} Let $u \in |U|$ be the image of $\overline{u}$. As $|U| = \bigcup_{i \in I} \varphi_i(|U_i|)$ there exists an $i$ and a point $u_i \in U_i$ mapping to $x$. Apply Lemma \ref{lemma-geometric-lift-to-usual} to $(U_i, u_i) \to (U, u)$ and $\overline{u}$ to get the desired geometric point. \end{proof} \begin{definition} \label{definition-stalk} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a presheaf on $X_\etale$. Let $\overline{x}$ be a geometric point of $X$. The {\it stalk} of $\mathcal{F}$ at $\overline{x}$ is $$\mathcal{F}_{\bar x} = \colim_{(U, \overline{u})} \mathcal{F}(U)$$ where $(U, \overline{u})$ runs over all \'etale neighborhoods of $\overline{x}$ in $X$ with $U \in \Ob(X_\etale)$. \end{definition} \noindent By Lemma \ref{lemma-cofinal-etale}, this colimit is over a filtered index category, namely the opposite of the category of \'etale neighborhoods in $X_\etale$. More precisely Lemma \ref{lemma-cofinal-etale} says the opposite of the category of all \'etale neighbourhoods is filtered, and the full subcategory of those which are in $X_\etale$ is a cofinal subcategory hence also filtered. \medskip\noindent This means an element of $\mathcal{F}_{\overline{x}}$ can be thought of as a triple $(U, \overline{u}, \sigma)$ where $U \in \Ob(X_\etale)$ and $\sigma \in \mathcal{F}(U)$. Two triples $(U, \overline{u}, \sigma)$, $(U', \overline{u}', \sigma')$ define the same element of the stalk if there exists a third \'etale neighbourhood $(U'', \overline{u}'')$, $U'' \in \Ob(X_\etale)$ and morphisms of \'etale neighbourhoods $h : (U'', \overline{u}'') \to (U, \overline{u})$, $h' : (U'', \overline{u}'') \to (U', \overline{u}')$ such that $h^*\sigma = (h')^*\sigma'$ in $\mathcal{F}(U'')$. See Categories, Section \ref{categories-section-directed-colimits}. \medskip\noindent This also implies that if $\mathcal{F}'$ is the sheaf on $X_{spaces, \etale}$ corresponding to $\mathcal{F}$ on $X_\etale$, then \begin{equation} \label{equation-stalk-spaces-etale} \mathcal{F}_{\overline{x}} = \colim_{(U, \overline{u})} \mathcal{F}'(U) \end{equation} where now the colimit is over all the \'etale neighbourhoods of $\overline{x}$. We will often jump between the point of view of using $X_\etale$ and $X_{spaces, \etale}$ without further mention. \medskip\noindent In particular this means that if $\mathcal{F}$ is a presheaf of abelian groups, rings, etc then $\mathcal{F}_{\overline{x}}$ is an abelian group, ring, etc simply by the usual way of defining the group structure on a directed colimit of abelian groups, rings, etc. \begin{lemma} \label{lemma-stalk-gives-point} \begin{slogan} A geometric point of an algebraic space gives a point of its \'etale topos. \end{slogan} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\overline{x}$ be a geometric point of $X$. Consider the functor $$u : X_\etale \longrightarrow \textit{Sets}, \quad U \longmapsto |U_{\overline{x}}|$$ Then $u$ defines a point $p$ of the site $X_\etale$ (Sites, Definition \ref{sites-definition-point}) and its associated stalk functor $\mathcal{F} \mapsto \mathcal{F}_p$ (Sites, Equation \ref{sites-equation-stalk}) is the functor $\mathcal{F} \mapsto \mathcal{F}_{\overline{x}}$ defined above. \end{lemma} \begin{proof} In the proof of Lemma \ref{lemma-geometric-lift-to-cover} we have seen that the scheme $U_{\overline{x}}$ is a disjoint union of schemes isomorphic to $\overline{x}$. Thus we can also think of $|U_{\overline{x}}|$ as the set of geometric points of $U$ lying over $\overline{x}$, i.e., as the collection of morphisms $\overline{u} : \overline{x} \to U$ fitting into the diagram of Definition \ref{definition-geometric-point}. From this it follows that $u(X)$ is a singleton, and that $u(U \times_V W) = u(U) \times_{u(V)} u(W)$ whenever $U \to V$ and $W \to V$ are morphisms in $X_\etale$. And, given a covering $\{U_i \to U\}_{i \in I}$ in $X_\etale$ we see that $\coprod u(U_i) \to u(U)$ is surjective by Lemma \ref{lemma-geometric-lift-to-cover}. Hence Sites, Proposition \ref{sites-proposition-point-limits} applies, so $p$ is a point of the site $X_\etale$. Finally, the our functor $\mathcal{F} \mapsto \mathcal{F}_{\overline{s}}$ is given by exactly the same colimit as the functor $\mathcal{F} \mapsto \mathcal{F}_p$ associated to $p$ in Sites, Equation \ref{sites-equation-stalk} which proves the final assertion. \end{proof} \begin{lemma} \label{lemma-stalk-exact} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\overline{x}$ be a geometric point of $X$. \begin{enumerate} \item The stalk functor $\textit{PAb}(X_\etale) \to \textit{Ab}$, $\mathcal{F} \mapsto \mathcal{F}_{\overline{x}}$ is exact. \item We have $(\mathcal{F}^\#)_{\overline{x}} = \mathcal{F}_{\overline{x}}$ for any presheaf of sets $\mathcal{F}$ on $X_\etale$. \item The functor $\textit{Ab}(X_\etale) \to \textit{Ab}$, $\mathcal{F} \mapsto \mathcal{F}_{\overline{x}}$ is exact. \item Similarly the functors $\textit{PSh}(X_\etale) \to \textit{Sets}$ and $\Sh(X_\etale) \to \textit{Sets}$ given by the stalk functor $\mathcal{F} \mapsto \mathcal{F}_{\overline{x}}$ are exact (see Categories, Definition \ref{categories-definition-exact}) and commute with arbitrary colimits. \end{enumerate} \end{lemma} \begin{proof} This result follows from the general material in Modules on Sites, Section \ref{sites-modules-section-stalks}. This is true because $\mathcal{F} \mapsto \mathcal{F}_{\overline{x}}$ comes from a point of the small \'etale site of $X$, see Lemma \ref{lemma-stalk-gives-point}. See the proof of \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-stalk-exact} for a direct proof of some of these statements in the setting of the small \'etale site of a scheme. \end{proof} \noindent We will see below that the stalk functor $\mathcal{F} \mapsto \mathcal{F}_{\overline{x}}$ is really the pullback along the morphism $\overline{x}$. In that sense the following lemma is a generalization of the lemma above. \begin{lemma} \label{lemma-stalk-pullback} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. \begin{enumerate} \item The functor $f_{small}^{-1} : \textit{Ab}(Y_\etale) \to \textit{Ab}(X_\etale)$ is exact. \item The functor $f_{small}^{-1} : \Sh(Y_\etale) \to \Sh(X_\etale)$ is exact, i.e., it commutes with finite limits and colimits, see Categories, Definition \ref{categories-definition-exact}. \item For any \'etale morphism $V \to Y$ of algebraic spaces we have $f_{small}^{-1}h_V = h_{X \times_Y V}$. \item Let $\overline{x} \to X$ be a geometric point. Let $\mathcal{G}$ be a sheaf on $Y_\etale$. Then there is a canonical identification $$(f_{small}^{-1}\mathcal{G})_{\overline{x}} = \mathcal{G}_{\overline{y}}.$$ where $\overline{y} = f \circ \overline{x}$. \end{enumerate} \end{lemma} \begin{proof} Recall that $f_{small}$ is defined via $f_{spaces, small}$ in Lemma \ref{lemma-functoriality-etale-site}. Parts (1), (2) and (3) are general consequences of the fact that $f_{spaces, \etale} : X_{spaces, \etale} \to Y_{spaces, \etale}$ is a morphism of sites, see Sites, Definition \ref{sites-definition-morphism-sites} for (2), Modules on Sites, Lemma \ref{sites-modules-lemma-flat-pullback-exact} for (1), and Sites, Lemma \ref{sites-lemma-pullback-representable-sheaf} for (3). \medskip\noindent Proof of (4). This statement is a special case of Sites, Lemma \ref{sites-lemma-point-morphism-sites} via Lemma \ref{lemma-stalk-gives-point}. We also provide a direct proof. Note that by Lemma \ref{lemma-stalk-exact}. taking stalks commutes with sheafification. Let $\mathcal{G}'$ be the sheaf on $Y_{spaces, \etale}$ whose restriction to $Y_\etale$ is $\mathcal{G}$. Recall that $f_{spaces, \etale}^{-1}\mathcal{G}'$ is the sheaf associated to the presheaf $$U \longrightarrow \colim_{U \to X \times_Y V} \mathcal{G}'(V),$$ see Sites, Sections \ref{sites-section-continuous-functors} and \ref{sites-section-functoriality-PSh}. Thus we have \begin{align*} (f_{spaces, \etale}^{-1}\mathcal{G}')_{\overline{x}} & = \colim_{(U, \overline{u})} f_{spaces, \etale}^{-1}\mathcal{G}'(U) \\ & = \colim_{(U, \overline{u})} \colim_{a : U \to X \times_Y V} \mathcal{G}'(V) \\ & = \colim_{(V, \overline{v})} \mathcal{G}'(V) \\ & = \mathcal{G}'_{\overline{y}} \end{align*} in the third equality the pair $(U, \overline{u})$ and the map $a : U \to X \times_Y V$ corresponds to the pair $(V, a \circ \overline{u})$. Since the stalk of $\mathcal{G}'$ (resp.\ $f_{spaces, \etale}^{-1}\mathcal{G}'$) agrees with the stalk of $\mathcal{G}$ (resp.\ $f_{small}^{-1}\mathcal{G}$), see Equation (\ref{equation-stalk-spaces-etale}) the result follows. \end{proof} \begin{remark} \label{remark-stalk-pullback} This remark is the analogue of \'Etale Cohomology, Remark \ref{etale-cohomology-remark-stalk-pullback}. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\overline{x} : \Spec(k) \to X$ be a geometric point of $X$. By \'Etale Cohomology, Theorem \ref{etale-cohomology-theorem-equivalence-sheaves-point} the category of sheaves on $\Spec(k)_\etale$ is equivalent to the category of sets (by taking a sheaf to its global sections). Hence it follows from Lemma \ref{lemma-stalk-pullback} part (4) applied to the morphism $\overline{x}$ that the functor $$\Sh(X_\etale) \longrightarrow \textit{Sets}, \quad \mathcal{F} \longmapsto \mathcal{F}_{\overline{x}}$$ is isomorphic to the functor $$\Sh(X_\etale) \longrightarrow \Sh(\Spec(k)_\etale) = \textit{Sets}, \quad \mathcal{F} \longmapsto \overline{x}^*\mathcal{F}$$ Hence we may view the stalk functors as pullback functors along geometric morphisms (and not just some abstract morphisms of topoi as in the result of Lemma \ref{lemma-stalk-gives-point}). \end{remark} \begin{remark} \label{remark-map-stalks} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$. We claim that for any pair of geometric points $\overline{x}$ and $\overline{x}'$ lying over $x$ the stalk functors are isomorphic. By definition of $|X|$ we can find a third geometric point $\overline{x}''$ so that there exists a commutative diagram $$\xymatrix{ \overline{x}'' \ar[r] \ar[d] \ar[rd]^{\overline{x}''} & \overline{x}' \ar[d]^{\overline{x}'} \\ \overline{x} \ar[r]^{\overline{x}} & X. }$$ Since the stalk functor $\mathcal{F} \mapsto \mathcal{F}_{\overline{x}}$ is given by pullback along the morphism $\overline{x}$ (and similarly for the others) we conclude by functoriality of pullbacks. \end{remark} \noindent The following theorem says that the small \'etale site of an algebraic space has enough points. \begin{theorem} \label{theorem-exactness-stalks} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. A map $a : \mathcal{F} \to \mathcal{G}$ of sheaves of sets is injective (resp.\ surjective) if and only if the map on stalks $a_{\overline{x}} : \mathcal{F}_{\overline{x}} \to \mathcal{G}_{\overline{x}}$ is injective (resp.\ surjective) for all geometric points of $X$. A sequence of abelian sheaves on $X_\etale$ is exact if and only if it is exact on all stalks at geometric points of $S$. \end{theorem}