# 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}

There are no comments yet for this tag.

## Add a comment on tag 0B80

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following box. So in case this where tag 0321 you just have to write 0321. Beware of the difference between the letter 'O' and the digit 0.

This captcha seems more appropriate than the usual illegible gibberish, right?