Timeline for Clarifying an interpretation of algebraic spaces
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Apr 12, 2018 at 17:26 | vote | accept | Wenzhe | ||
| Feb 27, 2018 at 10:27 | history | edited | Niels | CC BY-SA 3.0 |
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| Feb 15, 2018 at 13:25 | comment | added | nfdc23 | @მამუკაჯიბლაძე: I'm not sure what you're asking for. There are quite a few "standard" references on algebraic spaces (e.g., Knutson's book, which has some quasi-compactness conditions on the diagonal, and the Raynaud-Gruson paper on flatness which removes that condition early in section 5.7 of Part I; and part of the Stacks Project), which all convey the importance of the representability condition on the diagonal. I also don't understand what you're aiming at with your (1) and (2) (e.g., Hom-functors among algebraic spaces are generally not representable), sorry. | |
| Feb 14, 2018 at 14:45 | comment | added | მამუკა ჯიბლაძე | @nfdc23 could you give a reference to read about this? It is very intriguing as it associates for me with (at least) two important things: "local homeomorphism" = "open with open diagonal" and a characterization of stacks as prestacks which are (1) "sheaves up to isomorphism" and (2) "have small homs", which means that hom presheaves are sheaves; so on the canonical (Giraud) site this means that both the isomorphism classes presheaf and the hom presheaves are representable. Do you know if these are related and/or relevant here? | |
| Feb 13, 2018 at 4:20 | comment | added | nfdc23 | @DenisNardin: The reason this answer is wrong is because it makes no contact with the condition of relative representability of the diagonal (which is where much of the geometric substance of the concept comes from). | |
| Feb 12, 2018 at 13:06 | comment | added | Denis Nardin | @მამუკაჯიბლაძე I think this is the moment where we need to pay attention on whether we're working in the big or in the small étale site (if I recall correctly in the small étale site all sheaves are indeed representable by algebraic spaces). But maybe I should let people who actually know this stuff answer... | |
| Feb 12, 2018 at 13:03 | comment | added | მამუკა ჯიბლაძე | @DenisNardin Sorry somehow I am confused but does not any étale sheaf have the same? | |
| Feb 12, 2018 at 12:37 | comment | added | Denis Nardin | @მამუკაჯიბლაძე I believe it means it receives a jointly surjective étale map from a disjoint union of representables. | |
| Feb 12, 2018 at 12:32 | history | edited | Niels | CC BY-SA 3.0 |
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| Feb 12, 2018 at 10:06 | comment | added | მამუკა ჯიბლაძე | Could you clarify a bit what do you mean by "locally covered" in the étale case? Since any sheaf in a subcanonical topology is a colimit of representables, and there are (I believe) étale sheaves that do not correspond to any algebraic space, one indeed needs the notion of "locally covered" more restrictive than just any colimit. In the Zariski case one can simply say you have an open cover by representables, but what is it for the étale site? | |
| Feb 12, 2018 at 8:49 | comment | added | nfdc23 | How is this encoding the relative representability of the diagonal? | |
| Feb 12, 2018 at 8:33 | history | answered | Niels | CC BY-SA 3.0 |