5
$\begingroup$

In Deligne and Mumford's famous 1969 paper, The irreducibility of the space of curves of given genus, definition 4.6 (that of algebraic stacks) has the following footnote:

This definition is the "right" one only for quasi-separated stacks. It will however be sufficient for our purposes.

Note that their definition of an algebraic stack is:

  1. A stack on $Sch$ with the etale topology such that
  2. The diagonal is representable, and
  3. there is a representable etale surjection from a scheme.

and it immediately follows the definition of what it means for a stack to be quasi-separated, but they clearly do not mention quasi-separability here.

What do they mean by the footnote? What is the "right" definition for stacks which are not quasi-separated?

Improve this question
$\endgroup$

1 Answer 1

Reset to default
10
$\begingroup$

I think that what they might have in mind is that for non-quasi-separated Deligne-Mumford algebraic stacks one should not assume that the diagonal is represented by schemes, but by algebraic spaces. For quasi-separated Deligne-Mumford stacks this implies representability by schemes, but this is not true in general.

At least, this was my guess when I read Deligne-Mumford for the first time, as a graduate student. The first articles by Artin on algebraic spaces appeared in 1969 (a momentous year in algebraic geometry), and Deligne and Mumford must have been aware of his ideas.

Improve this answer
$\endgroup$
1
  • $\begingroup$ Hmm ok. So it was only a matter of what sort of space the stacks are presented by. Not as interesting as I'd hoped ;-) $\endgroup$
    – David Roberts ♦
    Commented Aug 18, 2011 at 10:54

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .