Timeline for Exactly how is 'the diagonal is representable' used for algebraic stacks...
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Mar 4, 2011 at 15:11 | history | edited | David Carchedi |
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| Mar 4, 2011 at 14:38 | answer | added | David Carchedi | timeline score: 6 | |
| Mar 4, 2011 at 11:54 | comment | added | Mattia Talpo | One point (which doesn't answer your question, I know) is that representability of the diagonal follows from the existence of an atlas (i.e. a smooth surjective and representable morphism from an algebraic space $X\to \mathcal{X}$). See for example prop (4.3.2) of Champs Algébriques. | |
| Mar 4, 2011 at 5:58 | history | edited | David Roberts♦ | CC BY-SA 2.5 |
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| Mar 4, 2011 at 5:54 | comment | added | David Roberts♦ | @Scott - regarding 'the' - whoops! I didn't mean that. And yes, I think you've hit the nail on the head: I really want to know if one can avoid using representability of the diagonal explicitly in favour of properties arising from a presenting groupoid. | |
| Mar 4, 2011 at 5:41 | comment | added | S. Carnahan♦ | Since you can reconstruct the diagonal up to 2-isomorphism from a presenting groupoid, it seems tautological that there isn't any (isomorphism-invariant) property of the diagonal that can't be derived from properties of a presenting groupoid. Is that really the question you mean to ask? Also, I protest at your use of the word "the" in front of "presenting". | |
| Mar 4, 2011 at 3:00 | comment | added | David Roberts♦ | @Mike - but we can require representability of the atlas map separately, which is essential, and if pressed, require that any map from a scheme to the stack is representable. If it is this latter property which is used, then that is the sort of answer I am looking for. And any other reasons too, of course. | |
| Mar 4, 2011 at 2:14 | comment | added | Mike Skirvin | I've always thought that the condition that the diagonal is representable is useful because it is equivalent to requiring that any map from a scheme to your stack is representable. In particular, it implies that the map from an atlas of the stack to the stack is representable, which is certainly important. | |
| Mar 3, 2011 at 22:53 | history | asked | David Roberts♦ | CC BY-SA 2.5 |