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Tag 03JH

Chapter 54: Properties of Algebraic Spaces > Section 54.12: The schematic locus

Lemma 54.12.1. 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.

Proof. Let $U \to X$ be an étale 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 54.4.4 we see that $$ \coprod X_i \longrightarrow X' $$ is a surjective map of sheaves on $(\textit{Sch}/S)_{fppf}$. 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 25.15.4 to see that $X'$ is a scheme. $\square$

    The code snippet corresponding to this tag is a part of the file spaces-properties.tex and is located in lines 1502–1507 (see updates for more information).

    \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}

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