# The Stacks Project

## Tag 0B80

Lemma 58.17.3. Let $f: X \to S$ be a quasi-compact and quasi-separated morphism from an algebraic space to a scheme $S$. If for every $x \in |X|$ with image $s = f(x) \in S$ the algebraic space $X \times_S \mathop{\rm Spec}(\mathcal{O}_{S,s})$ is a scheme, then $X$ is a scheme.

Proof. Let $x \in |X|$. It suffices to find an open neighbourhood $U$ of $s = f(x)$ such that $X \times_S U$ is a scheme. As $X \times_S \mathop{\rm Spec}(\mathcal{O}_{S, s})$ is a scheme, then, since $\mathcal{O}_{S, s} = \mathop{\rm colim}\nolimits \mathcal{O}_S(U)$ where the colimit is over affine open neighbourhoods of $s$ in $S$ we see that $$X \times_S \mathop{\rm Spec}(\mathcal{O}_{S, s}) = \mathop{\rm lim}\nolimits X \times_S U$$ By Lemma 58.5.11 we see that $X \times_S U$ is a scheme for some $U$. $\square$

The code snippet corresponding to this tag is a part of the file spaces-limits.tex and is located in lines 3588–3594 (see updates for more information).

\begin{lemma}
\label{lemma-enough-local}
Let $f: X \to S$ be a quasi-compact and quasi-separated morphism from an
algebraic space to a scheme $S$. If for every $x \in |X|$ with image
$s = f(x) \in S$ the algebraic space $X \times_S \Spec(\mathcal{O}_{S,s})$
is a scheme, then $X$ is a scheme.
\end{lemma}

\begin{proof}
Let $x \in |X|$. It suffices to find an open neighbourhood $U$ of
$s = f(x)$ such that $X \times_S U$ is a scheme.
As $X \times_S \Spec(\mathcal{O}_{S, s})$ is a scheme, then, since
$\mathcal{O}_{S, s} = \colim \mathcal{O}_S(U)$ where the colimit is
over affine open neighbourhoods of $s$ in $S$ we see that
$$X \times_S \Spec(\mathcal{O}_{S, s}) = \lim X \times_S U$$
By Lemma \ref{lemma-limit-is-scheme} we see that $X \times_S U$
is a scheme for some $U$.
\end{proof}

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