Timeline for An attempt to define partial properness and compactification for some maps between analytic spaces
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Aug 28, 2021 at 17:19 | comment | added | Longke Tang 唐珑珂 | @Z.M The locale in Clausen-Scholze is clearly not the classical one. | |
| Aug 28, 2021 at 12:24 | comment | added | Z. M | A weaker question is whether (the AnSpec of) a closed disc (with overconvergent power series ring) in a Riemann surface a steady subspace? If this is the case, then the locale obtained in Clausen-Scholze is probably not the classical one (of open subsets). | |
| Aug 28, 2021 at 12:17 | comment | added | Z. M | Sorry, I am still confused. Let me restrict to Riemann surfaces. How do you talk about openness so that closed discs are open? One of the confusions is that, for a compact Hausdorff space, there are two sites: the site of open subsets, and the site of compact subsets. The sheaf on the former is equivalent to an overconvergent sheaf on the later (Lurie HTT 7.3.4), while both have a corresponding locale if I understand correctly. | |
| Aug 27, 2021 at 21:29 | comment | added | Longke Tang 唐珑珂 | @Z.M Here the map considered is from a Riemann surface to a point. "Affine opens proper over Y" is just analytic spectra of overconvergent rings. I am viewing analytic spaces as locally analytically ringed locales to talk about openness. Actually the way to make a Riemann surface an analytic space is exactly to glue from analytic spectra of overconvergent rings on closed discs, so these serve as "affine opens" in analog with scheme theory. | |
| Aug 27, 2021 at 16:49 | comment | added | Z. M | Sorry, what do you mean by "coinciding with ... Riemann surfaces"? For example, what are "affine opens" in complex analytic geometry? I would guess, by mimicking Clausen's lecture, that for complex manifolds, they might be closed polydiscs (so not open) with the ring of overconvergent power series. This seems to coincide with Lurie's result that Clausen mentioned in the lectures if we need to compare it with classical "opens". | |
| Aug 27, 2021 at 9:57 | comment | added | Peter Scholze | Good question! In Etale cohomology of diamonds, a notion of "compactifiable" morphisms is constructed, asking that the map to this kind of "canonical compactification" is an open immersion. I think you can ask the same here, where "open immersion" should be understood as the existence of a left adjoint of $j^\ast$. It is probably true that these notions agree with the existing notions for adic spaces, and that this is good enough for an $f_!$ functor. But I would have to think more about this. | |
| S Aug 27, 2021 at 0:11 | review | First questions | |||
| Aug 27, 2021 at 2:18 | |||||
| S Aug 27, 2021 at 0:11 | history | asked | Longke Tang 唐珑珂 | CC BY-SA 4.0 |