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Tag 02WW

Chapter 56: Algebraic Spaces > Section 56.10: Algebraic spaces and equivalence relations

Theorem 56.10.5. Let $S$ be a scheme. Let $U$ be a scheme over $S$. Let $j = (s, t) : R \to U \times_S U$ be an étale equivalence relation on $U$ over $S$. Then the quotient $U/R$ is an algebraic space, and $U \to U/R$ is étale and surjective, in other words $(U, R, U \to U/R)$ is a presentation of $U/R$.

Proof. By Lemma 56.10.3 it suffices to prove that $U/R$ is an algebraic space. Let $U' \to U$ be a surjective, étale morphism. Then $\{U' \to U\}$ is in particular an fppf covering. Let $R'$ be the restriction of $R$ to $U'$, see Groupoids, Definition 38.3.3. According to Groupoids, Lemma 38.20.6 we see that $U/R \cong U'/R'$. By Lemma 56.10.1 $R'$ is an étale equivalence relation on $U'$. Thus we may replace $U$ by $U'$.

We apply the previous remark to $U' = \coprod U_i$, where $U = \bigcup U_i$ is an affine open covering of $S$. Hence we may and do assume that $U = \coprod U_i$ where each $U_i$ is an affine scheme.

Consider the restriction $R_i$ of $R$ to $U_i$. By Lemma 56.10.1 this is an étale equivalence relation. Set $F_i = U_i/R_i$ and $F = U/R$. It is clear that $\coprod F_i \to F$ is surjective. By Lemma 56.10.2 each $F_i \to F$ is representable, and an open immersion. By Lemma 56.10.4 applied to $(U_i, R_i)$ we see that $F_i$ is an algebraic space. Then by Lemma 56.10.3 we see that $U_i \to F_i$ is étale and surjective. From Lemma 56.8.3 it follows that $\coprod F_i$ is an algebraic space. Finally, we have verified all hypotheses of Lemma 56.8.4 and it follows that $F = U/R$ is an algebraic space. $\square$

    The code snippet corresponding to this tag is a part of the file spaces.tex and is located in lines 1571–1579 (see updates for more information).

    \begin{theorem}
    \label{theorem-presentation}
    Let $S$ be a scheme. Let $U$ be a scheme over $S$.
    Let $j = (s, t) : R \to U \times_S U$
    be an \'etale equivalence relation on $U$ over $S$.
    Then the quotient $U/R$ is an algebraic space,
    and $U \to U/R$ is \'etale and surjective, in other words
    $(U, R, U \to U/R)$ is a presentation of $U/R$.
    \end{theorem}
    
    \begin{proof}
    By Lemma \ref{lemma-when-it-works-it-works}
    it suffices to prove that $U/R$ is an algebraic space.
    Let $U' \to U$ be a surjective, \'etale morphism.
    Then $\{U' \to U\}$ is in particular an fppf covering.
    Let $R'$ be the restriction of $R$ to $U'$, see
    Groupoids, Definition \ref{groupoids-definition-restrict-relation}.
    According to
    Groupoids, Lemma \ref{groupoids-lemma-quotient-groupoid-restrict}
    we see that $U/R \cong U'/R'$.
    By Lemma \ref{lemma-pullback-etale-equivalence-relation} $R'$ is an
    \'etale equivalence relation on $U'$. Thus we may replace $U$ by $U'$.
    
    \medskip\noindent
    We apply the previous remark to $U' = \coprod U_i$, where
    $U = \bigcup U_i$ is an affine open covering of $S$. Hence we
    may and do assume that $U = \coprod U_i$ where
    each $U_i$ is an affine scheme.
    
    \medskip\noindent
    Consider the restriction $R_i$ of $R$ to $U_i$.
    By Lemma \ref{lemma-pullback-etale-equivalence-relation}
    this is an \'etale equivalence relation.
    Set $F_i = U_i/R_i$ and $F = U/R$.
    It is clear that $\coprod F_i \to F$ is surjective.
    By Lemma \ref{lemma-finding-opens} each $F_i \to F$
    is representable, and an open immersion.
    By Lemma \ref{lemma-presentation-quasi-compact}
    applied to $(U_i, R_i)$ we see that $F_i$ is an algebraic space.
    Then by Lemma \ref{lemma-when-it-works-it-works} we see that
    $U_i \to F_i$ is \'etale and surjective.
    From Lemma \ref{lemma-coproduct-algebraic-spaces}
    it follows that $\coprod F_i$ is an algebraic space.
    Finally, we have verified all
    hypotheses of Lemma \ref{lemma-glueing-algebraic-spaces}
    and it follows that $F = U/R$ is an algebraic space.
    \end{proof}

    Comments (1)

    Comment #216 by David Holmes on May 17, 2013 a 4:39 pm UTC

    Typo: in the first line of the proof, 'suffice' should read 'suffices'.

    There are also 6 comments on Section 56.10: Algebraic Spaces.

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