0
$\begingroup$

Let $X$ be a scheme and $R$ be a proper equivalence relation on $X$.

What can be said about the geometric structure of the quotient $X/R$? Is it representable by a stack, for example?

$\endgroup$
5
  • $\begingroup$ What sort of stack are you after? $\endgroup$
    – David Roberts
    Nov 14, 2012 at 7:15
  • $\begingroup$ any, I only wanted to now what looks like such an object. $\endgroup$
    – prochet
    Nov 14, 2012 at 15:42
  • 1
    $\begingroup$ You should better make your question a bit more precise, because the possible answers depend on the sense you give to the quotient (is it a fibered category ? an fppf sheaf ?). $\endgroup$ Nov 15, 2012 at 8:13
  • $\begingroup$ I make the quotient as a sheaf. $\endgroup$
    – prochet
    Nov 15, 2012 at 16:46
  • 3
    $\begingroup$ If you consider the sheaf quotient, then being representable by an algebraic stack is nothing more than being representable by an algebraic space (a scheme-like thing, really). Anyway in order to have theorems you need to assume at least that the equivalence relation is flat. The best classical result close to what you are asking is Thm. 6.1 in Grothendieck's Bourbaki talk 212. $\endgroup$ Nov 16, 2012 at 10:24

0

Your Answer

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