Timeline for Why is this not an algebraic space?
Current License: CC BY-SA 2.5
8 events
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| Dec 24, 2009 at 18:02 | comment | added | Anton Geraschenko | @James Borger: Can you provide a reference for "the quotient has representable diagonal iff the relation is a scheme"? I recall working through a proof of this once (based on the proof of 5.7.2 in Raynaud-Gruson), but I remember that it required a quasi-finiteness hypothesis to apply EGA IV 18.12.12, and I didn't see a way to eliminate the hypothesis. If I have a chance, I'll try to reconstruct the argument and see where it came up. | |
| Dec 22, 2009 at 17:04 | comment | added | S. Carnahan♦ | I think I see my confusion. SGA1 Exp 1 1.4 seems to suggest that finite type is necessary for a morphism to be etale, but EGA4.4 17.3.1 replaces this with locally of finite presentation. | |
| Dec 22, 2009 at 12:26 | comment | added | JBorger | @C.S-P.: If you have an equivalence relation on a scheme, then the quotient should have representable diagonal if and only if the equivalence relation is a scheme. @S.C.: Yes, that's correct -- the projection isn't even quasi-compact | |
| Dec 22, 2009 at 7:29 | comment | added | S. Carnahan♦ | Perhaps I'm missing something, but I was under the impression that the projection from R to G_m is not of finite type. | |
| Dec 16, 2009 at 3:15 | comment | added | Chris Schommer-Pries | Thanks for these remarks! So the definition of algebraic space that I was taught only requires that the diagonal is representable, not necessarily quasi-compact. I'm not an algebraic geometer so I'm a little rusty. Does this example have a representable diagonal? Is there an easy criteria that would guarantee this? | |
| Dec 16, 2009 at 0:30 | history | edited | JBorger | CC BY-SA 2.5 |
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| Dec 15, 2009 at 23:02 | history | edited | JBorger | CC BY-SA 2.5 |
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| Dec 15, 2009 at 22:15 | history | answered | JBorger | CC BY-SA 2.5 |