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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?

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up vote 10 down vote accepted

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.

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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 ;-) –  David Roberts Aug 18 '11 at 10:54
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