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Tag 0B8A

Chapter 64: More on Morphisms of Spaces > Section 64.4: Monomorphisms

Lemma 64.4.1 (David Rydh). A flat monomorphism of algebraic spaces is representable by schemes.

Proof. Let $f : X \to Y$ be a flat morphism of algebraic spaces. To prove $f$ is representable, we have to show $X \times_Y V$ is a scheme for every scheme $V$ mapping to $Y$. Since being a scheme is local (Properties of Spaces, Lemma 54.12.1), we may assume $V$ is affine. Thus we may assume $Y = \mathop{\rm Spec}(B)$ is an affine scheme. Next, we can assume that $X$ is quasi-compact by replacing $X$ by a quasi-compact open. The space $X$ is separated as $X \to X \times_{\mathop{\rm Spec}(B)} X$ is an isomorphism. Applying Limits of Spaces, Lemma 58.17.3 we reduce to the case where $B$ is local, $X \to \mathop{\rm Spec}(B)$ is a flat monomorphism, and there exists a point $x \in X$ mapping to the closed point of $\mathop{\rm Spec}(B)$. Then $X \to \mathop{\rm Spec}(B)$ is surjective as generalizations lift along flat morphisms of separated algebraic spaces, see Decent Spaces, Lemma 56.7.3. Hence we see that $\{X \to \mathop{\rm Spec}(B)\}$ is an fpqc cover. Then $X \to \mathop{\rm Spec}(B)$ is a morphism which becomes an isomorphism after base change by $X \to \mathop{\rm Spec}(B)$. Hence it is an isomorphism by fpqc descent, see Descent on Spaces, Lemma 62.10.15. $\square$

    The code snippet corresponding to this tag is a part of the file spaces-more-morphisms.tex and is located in lines 286–289 (see updates for more information).

    \begin{lemma}[David Rydh]
    \label{lemma-flat-case}
    A flat monomorphism of algebraic spaces is representable by schemes.
    \end{lemma}
    
    \begin{proof}
    Let $f : X \to Y$ be a flat morphism of algebraic spaces.
    To prove $f$ is representable, we have to show
    $X \times_Y V$ is a scheme for every scheme $V$ mapping to $Y$.
    Since being a scheme is local (Properties of Spaces, 
    Lemma \ref{spaces-properties-lemma-subscheme}), we may
    assume $V$ is affine. Thus we may assume $Y = \Spec(B)$
    is an affine scheme. Next, we can assume that $X$ is quasi-compact
    by replacing $X$ by a quasi-compact open. The space $X$ is
    separated as $X \to X \times_{\Spec(B)} X$ is an isomorphism.
    Applying Limits of Spaces, Lemma \ref{spaces-limits-lemma-enough-local}
    we reduce to the case where $B$ is local, $X \to \Spec(B)$ is a
    flat monomorphism, and
    there exists a point $x \in X$ mapping to the closed point of $\Spec(B)$.
    Then $X \to \Spec(B)$ is surjective as generalizations
    lift along flat morphisms of separated algebraic spaces, see
    Decent Spaces, Lemma \ref{decent-spaces-lemma-generalizations-lift-flat}.
    Hence we see that $\{X \to \Spec(B)\}$ is an fpqc cover.
    Then $X \to \Spec(B)$ is a morphism which becomes an isomorphism
    after base change by $X \to \Spec(B)$. Hence it is an isomorphism by
    fpqc descent, see Descent on Spaces, Lemma
    \ref{spaces-descent-lemma-descending-property-isomorphism}.
    \end{proof}

    Comments (3)

    Comment #1563 by Matthew Emerton on July 11, 2015 a 4:33 am UTC

    Unless I'm misunderstanding, the ring $B$ morphs into the ring $A$ partway through the argument.

    Comment #1581 by Johan (site) on July 14, 2015 a 2:07 pm UTC

    Yep. Thanks. Fixed here.

    Comment #2457 by Matthieu Romagny on March 15, 2017 a 1:01 pm UTC

    First sentence of proof: flat morphism --> flat monomorphism.

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