# The Stacks Project

## Tag 0B8A

Lemma 67.4.1 (David Rydh). A flat monomorphism of algebraic spaces is representable by schemes.

Proof. Let $f : X \to Y$ be a flat monomorphism 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 57.12.1), 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 61.17.3 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 59.7.3. 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 65.10.15. $\square$

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 monomorphism 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}

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.

Comment #2496 by Johan (site) on April 13, 2017 a 11:17 pm UTC

Thanks, fixed here.

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