3
$\begingroup$

Let $S$ be a Noetherian scheme, let $Y$ be a scheme of finite type over $S$, and let $X$ be an algebraic space of finite type over $S$. Suppose that there is a morphism $f:Y \rightarrow X$ which is proper and birational. Must $X$ be a scheme?

See also When is an algebraic space a scheme?

Improve this question
$\endgroup$
3
  • 7
    $\begingroup$ No, that is not true. In fact, the opposite is true if you add a few hypotheses (e.g., if S is a finite type scheme over a field or over an excellent DVR, and if the algebraic space $X$ is separated). This is often called "Chow's Lemma for algebraic spaces". There is a reference in Knutson's "Algebraic Spaces" . . . $\endgroup$
    – Jason Starr
    Commented May 29, 2018 at 20:52
  • $\begingroup$ . . . I forget to look up the reference earlier. Here it is: Chow's Lemma, Theorem IV.3.1, p. 192, Donald Knutson, "Algebraic Spaces", LNM 203, Springer-Verlag, Berlin, 1971. $\endgroup$
    – Jason Starr
    Commented May 31, 2018 at 13:29
  • $\begingroup$ Thank you very much. Can you post it as an answer so I will accept it. $\endgroup$
    – Rami
    Commented Jun 1, 2018 at 21:42

1 Answer 1

Reset to default
2
$\begingroup$

I am just posting my comment as an answer. Chow's Lemma for algebraic spaces is Theorem IV.3.1, p. 192 of the following.

MR0302647 (46 #1791)
Knutson, Donald
Algebraic spaces.
Lecture Notes in Mathematics, Vol. 203.
Springer-Verlag, Berlin-New York, 1971. vi+261 pp.

For every separated, Noetherian scheme $S$, for every finitely presented, separated morphism of algebraic spaces, $X\to S$, there exists a morphism $f:Y\to X$ that is projective and birational such that $Y\to S$ is a quasi-projective morphism.

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.