Question 1: The main reference on algebraic stacks (Laumon and Moret--Bailly) defines a separable algebraic stack as one having universally closed diagonal. For schemes separability is simply defined by the condition that the diagonal is a closed immersion. Why this difference?

Question 2: Presentations of algebraic stacks are defined (in LMB) as morphisms (with some properties) $X\to\mathscr{X}$ with $X$ an algebraic space. What does one lose by only considering schemes $X$ instead?

These questions are certainly well-known to experts in stack theory and I know there are a few around on MO, so I'm hopeful that I will be sufficiently enlightened.

  • $\begingroup$ My answers: #1: because if points have non-trivial automorphisms, the diagonal is never an immersion; #2: one does not lose anything. $\endgroup$
    – t3suji
    Dec 3, 2010 at 17:27
  • $\begingroup$ Indeed, for #1 this is simply Corollaire 8.1.1 in LMB. Thanks. But maybe there are some other reasons? And why is universally closed the "right" definition? $\endgroup$ Dec 3, 2010 at 17:59
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    $\begingroup$ Variants on t3suji's comment to #2: psychologically one loses something because $X \times_{\mathcal{X}} X$ is "just" an algebraic space in general even when $X$ is a scheme. (This is all tied up with having a theory for which Artin's method gives interesting existence results.) So to argue more efficiently when doing reduction steps via etale presentations, one usually reduces problems first from stacks to alg. spaces, and only then from alg. spaces to schemes. One should view algebraic spaces and not schemes as the building blocks for alg. stacks. In that sense the LMB defn is more natural. $\endgroup$
    – BCnrd
    Dec 3, 2010 at 17:59
  • $\begingroup$ Concerning #1: the real test of a good definition is that it allows for interesting results and examples, and can be verified in appropriate classes of cases. In that sense, this definition is a good one for stacks; it allows one to carry over many familiar properties of separatedness, for example. Presumably experience with how the concept is used and verified in practice will convince you. $\endgroup$
    – BCnrd
    Dec 3, 2010 at 18:03
  • $\begingroup$ Hmm, I will think seriously about this. Thanks, Brian. I had a suspicion that one does indeed lose something. Can you perhaps give me an explicit example (a reference perhaps) where it is actually essential to work with algebraic spaces instead of schemes. My intuition with algebraic spaces is a bit poor though so I always tend to think about schemes as presentations. $\endgroup$ Dec 3, 2010 at 19:02


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