Spaces/Stacks for the \'etale topology

	Brief explanation why this is the same thing as for the fppf
	topology.

algebraic.tex

@@ -1289,11 +1289,8 @@ \section{Algebraic stacks}
 in the fppf topology, whereas in many references the \'etale topology is
 used. It somehow seems to us that the fppf topology is the natural topology
 to work with. In the end the resulting $2$-category of algebraic stacks
-ends up being ``the same''. Namely, allthough the actual
-stacks $\mathcal{X}$ being considered may be different, in the end the
-category of algebraic stacks defined using sheaves in the \'etale topology
-is equivalent the the category we define here. This will be clear later
-when we introduce presentations (insert future reference here).
+ends up being the same. This is explained in
+Criteria for Representability, Section \ref{criteria-section-stacks-etale}.
 
 \medskip\noindent
 The second is that we only require the diagonal map of $\mathcal{X}$ to be

bootstrap.tex

@@ -1725,7 +1725,83 @@ \section{Applications}
 
 
 
+\section{Algebraic spaces in the \'etale topology}
+\label{section-spaces-etale}
 
+\noindent
+Let $S$ be a scheme. Instead of working with sheaves over
+the big fppf site $(\Sch/S)_{fppf}$ we could work with sheaves
+over the big \'etale site $(\Sch/S)_{\acute{e}tale}$. All of the material in
+Algebraic Spaces, Sections \ref{spaces-section-representable} and
+\ref{spaces-section-representable-properties}
+makes sense for sheaves over $(\Sch/S)_{\acute{e}tale}$.
+Thus we get a second notion of algebraic spaces by working in the
+\'etale topology. This notion is (a priori) weaker then the notion introduced
+in Algebraic Spaces, Definition \ref{spaces-definition-algebraic-space}
+since a sheaf in the fppf topology is certainly a sheaf in the \'etale
+topology. However, the notions are equivalent as is shown by the following
+lemma.
+
+\begin{lemma}
+\label{lemma-spaces-etale}
+Denote the common underlying category of $\Sch_{fppf}$
+and $\Sch_{\acute{e}tale}$ by $\Sch_\alpha$ (see
+Topologies, Remark \ref{topologies-remark-choice-sites}). Let $S$ be an object
+of $\Sch_\alpha$. 
+$$
+F : (\Sch_\alpha/S)^{opp} \longrightarrow \textit{Sets}
+$$
+be a presheaf with the following properties:
+\begin{enumerate}
+\item $F$ is a sheaf for the \'etale topology,
+\item the diagonal $\Delta : F \to F \times F$ is representable, and
+\item there exists $U \in \Ob(\Sch_\alpha/S)$
+and $U \to F$ which is surjective and \'etale.
+\end{enumerate}
+Then $F$ is an algebraic space in the sense of
+Algebraic Spaces, Definition \ref{spaces-definition-algebraic-space}.
+\end{lemma}
+
+\begin{proof}
+Note that properties (2) and (3) of the lemma and the corresponding
+properties (2) and (3) of
+Algebraic Spaces, Definition \ref{spaces-definition-algebraic-space}
+are independent of the topology. This is true because these properties
+involve only the notion of a fibre product of presheaves, maps of
+presheaves, the notion of a representable transformation of functors,
+and what it means for such a transformation to be surjective and \'etale.
+Thus all we have to prove is that an \'etale sheaf $F$ with properties
+(2) and (3) is also an fppf sheaf.
+
+\medskip\noindent
+To do this, let $R = U \times_F U$. By (2) the presheaf $R$ is representable
+by a scheme and by (3) the projections $R \to U$ are \'etale. Thus
+$j : R \to U \times_S U$ is an \'etale equivalence relation. Moreover
+the map $U \to F$ identifies $F$ as the quotient of $U$ by $R$ for the
+\'etale topology (follows exactly as in the proof of
+Algebraic Spaces, Lemma \ref{spaces-lemma-space-presentation}).
+Next, let $U/R$ denote the quotient sheaf in the fppf topology
+which is an algebraic space by
+Spaces, Theorem \ref{spaces-theorem-presentation}.
+Thus we have morphisms (tranformations of functors)
+$$
+U \to F \to U/R.
+$$
+By the aforementioned
+Spaces, Theorem \ref{spaces-theorem-presentation}
+the composition is representable, surjective, and \'etale. Hence for any
+scheme $T$ and morphism $T \to U/R$ the fibre product $V = T \times_{U/R} U$
+is a scheme surjective and \'etale over $T$. In other words, $\{V \to U\}$
+is an \'etale covering. This proves that $U \to U/R$ is surjective as
+a map of sheaves in the \'etale topology. It follows that
+$F \to U/R$ is surjective as a map of sheaves in the \'etale topology.
+On the other hand, the map $F \to U/R$ is injective (as a map of presheaves)
+since $R = U \times_{U/R} U$ again by
+Spaces, Theorem \ref{spaces-theorem-presentation}.
+It follows that $F \to U/R$ is an isomorphism of \'etale sheaves, see
+Sites, Lemma \ref{sites-lemma-mono-epi-sheaves}
+which concludes the proof.
+\end{proof}
 
 
 

criteria.tex

@@ -3096,6 +3096,133 @@ \section{When is a quotient stack algebraic?}
 
 
 
+\section{Algebraic stacks in the \'etale topology}
+\label{section-stacks-etale}
+
+\noindent
+Let $S$ be a scheme. Instead of working with stacks in groupoids over
+the big fppf site $(\Sch/S)_{fppf}$ we could work with stacks in groupoids
+over the big \'etale site $(\Sch/S)_{\acute{e}tale}$. All of the material in
+Algebraic Stacks, Sections
+\ref{algebraic-section-representable},
+\ref{algebraic-section-2-yoneda},
+\ref{algebraic-section-representable-morphism},
+\ref{algebraic-section-split},
+\ref{algebraic-section-representable-by-algebraic-spaces},
+\ref{algebraic-section-morphisms-representable-by-algebraic-spaces},
+\ref{algebraic-section-representable-properties}, and
+\ref{algebraic-section-stacks}
+makes sense for categories fibred in groupouds over $(\Sch/S)_{\acute{e}tale}$.
+Thus we get a second notion of an algebraic stack by working in the
+\'etale topology. This notion is (a priori) weaker then the notion introduced
+in Algebraic Stacks, Definition \ref{algebraic-definition-algebraic-stack}
+since a stack in the fppf topology is certainly a stack in the \'etale
+topology. However, the notions are equivalent as is shown by the following
+lemma.
+
+\begin{lemma}
+\label{lemma-stacks-etale}
+Denote the common underlying category of $\Sch_{fppf}$
+and $\Sch_{\acute{e}tale}$ by $\Sch_\alpha$ (see
+Sheaves on Stacks, Section \ref{stacks-sheaves-section-sheaves} and
+Topologies, Remark \ref{topologies-remark-choice-sites}). Let $S$ be an object
+of $\Sch_\alpha$. Let
+$$
+p : \mathcal{X} \to \Sch_\alpha/S
+$$
+be a category fibred in groupoids with the following properties:
+\begin{enumerate}
+\item $\mathcal{X}$ is a stack in groupoids over $(\Sch/S)_{\acute{e}tale}$,
+\item the diagonal $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$
+is representable by algebraic spaces\footnote{Here we can either mean
+sheaves in the \'etale topology whose diagonal is representable and which
+have an \'etale surjective covering by a scheme or algebraic spaces as
+defined in
+Algebraic Spaces, Definition \ref{spaces-definition-algebraic-space}.
+Namely, by Bootstrap, Lemma \ref{bootstrap-lemma-spaces-etale}
+there is no difference.}, and
+\item there exists $U \in \Ob(\Sch_\alpha/S)$
+and a $1$-morphism $(\Sch/U)_{\acute{e}tale} \to \mathcal{X}$
+which is surjective and smooth.
+\end{enumerate}
+Then $\mathcal{X}$ is an algebraic stack in the sense of
+Algebraic Stacks, Definition \ref{algebraic-definition-algebraic-stack}.
+\end{lemma}
+
+\begin{proof}
+Note that properties (2) and (3) of the lemma and the corresponding
+properties (2) and (3) of
+Algebraic Stacks, Definition \ref{algebraic-definition-algebraic-stack}
+are independent of the topology. This is true because these properties
+involve only the notion of a $2$-fibre product of categories fibred in
+groupoids, $1$- and $2$-morphisms of categories fibred in groupoids, the
+notion of a $1$-morphism of categories fibred in groupoids representable
+by algebraic spaces, and what it means for such a $1$-morphism to be
+surjective and smooth.
+Thus all we have to prove is that an \'etale stack in groupoids
+$\mathcal{X}$ with properties (2) and (3) is also an fppf stack in groupoids.
+
+\medskip\noindent
+Using (2) let $R$ be an algebraic space representing
+$$
+(\Sch_\alpha/U) \times_\mathcal{X} (\Sch_\alpha/U)
+$$
+By (3) the projections $s, t : R \to U$ are smooth. Exactly as in the proof of
+Algebraic Stacks, Lemma \ref{algebraic-lemma-map-space-into-stack}
+there exists a groupoid in spaces $(U, R, s, t, c)$ and a canonical
+fully faithful $1$-morphism $[U/R]_{\acute{e}tale} \to \mathcal{X}$
+where $[U/R]_{\acute{e}tale}$ is the \'etale stackification of presheaf
+in groupoids
+$$
+T \longmapsto (U(T), R(T), s(T), t(T), c(T))
+$$
+Claim: If $V \to T$ is a surjective smooth morphism from an algebraic space
+$V$ to a scheme $T$, then there exists an \'etale covering $\{T_i \to T\}$
+refining the covering $\{V \to T\}$. This follows from
+More on Morphisms, Lemma \ref{more-morphisms-lemma-etale-dominates-smooth}
+or the more general
+Sheaves on Stacks, Lemma
+\ref{stacks-sheaves-lemma-surjective-flat-locally-finite-presentation}.
+Using the claim and arguing exactly as in
+Algebraic Stacks, Lemma \ref{algebraic-lemma-stack-presentation}
+it follows that $[U/R]_{\acute{e}tale} \to \mathcal{X}$ is an
+equivalence.
+
+\medskip\noindent
+Next, let $[U/R]$ denote the quotient stack in the fppf topology
+which is an algebraic stack by
+Algebraic Stacks, Theorem
+\ref{algebraic-theorem-smooth-groupoid-gives-algebraic-stack}.
+Thus we have $1$-morphisms
+$$
+U \to [U/R]_{\acute{e}tale} \to [U/R].
+$$
+Both $U \to [U/R]_{\acute{e}tale} \cong \mathcal{X}$ and
+$U \to [U/R]$ are surjective and smooth (the first by assumption
+and the second by the theorem) and in both cases the
+fibre product $U \times_\mathcal{X} U$ and $U \times_{[U/R]} U$
+is representable by $R$. Hence the $1$-morphism
+$[U/R]_{\acute{e}tale} \to [U/R]$ is fully faithful (since morphisms
+in the quotient stacks are given by morphisms into $R$, see
+Groupoids in Spaces, Section
+\ref{spaces-groupoids-section-explicit-quotient-stacks}).
+
+\medskip\noindent
+Finally, for any scheme $T$ and morphism $t : T \to [U/R]$ the fibre product
+$V = T \times_{U/R} U$ is an algebraic space surjective and smooth over $T$.
+By the claim above there exists an \'etale covering $\{T_i \to T\}_{i \in I}$
+and morphisms $T_i \to V$ over $T$. This proves that the object
+$t$ of $[U/R]$ over $T$ comes \'etale locally from $U$. We conclude that
+$[U/R]_{\acute{e}tale} \to [U/R]$ is an equivalence of stacks in
+groupoids over $(\Sch/S)_{\acute{e}tale}$ by
+Stacks, Lemma \ref{stacks-lemma-characterize-essentially-surjective-when-ff}.
+This concludes the proof.
+\end{proof}
+
+
+
+
+
 
 
 

spaces.tex

@@ -762,15 +762,12 @@ \section{Algebraic spaces}
 of algebraic spaces satisfy the sheaf condition for the fppf coverings
 (and even for fpqc coverings). Also, one of the reasons that algebraic
 spaces have been so useful is via Michael Artin's results on algebraic spaces.
-Built into his method is a condition which garantees the result is
+Built into his method is a condition which guarantees the result is
 locally of finite presentation over $S$.
 Combined it somehow seems to us that the fppf topology
-is the natural topology to work with. In the end the resulting category
-of algebraic spaces ends up being ``the same''. Namely, allthough the actual
-sheaves $F$ being considered may be different, in the end the
-category of algebraic spaces defined using sheaves in the \'etale topology
-is equivalent the the category we define here. This will be clear later
-when we introduce presentations (insert future reference here).
+is the natural topology to work with. In the end the category
+of algebraic spaces ends up being the same. See
+Bootstrap, Section \ref{bootstrap-section-spaces-etale}.
 
 \medskip\noindent
 The second is that we only require the diagonal map for $F$ to be