Timeline for Is it true that $\operatorname{2-colim}_U \textsf{QCoh}(U) = \textsf{Vect}(K_X)$, as $U$ shrinks to the generic point?
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9 events
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| Jun 5, 2022 at 13:09 | comment | added | Maxime Ramzi | @Piotr : as crystalline pointed out (and as I maybe didn't make clear enough in my answer) I was taking a colimit in the category of presentable categories. Passing to compact objects recovers your claim about ordinary categories of finitely presented modules (and conversely) | |
| Jun 5, 2022 at 13:06 | comment | added | Maxime Ramzi | @crystalline : I agree with your comments, although there is a bit of higher categories involved, namely $(2,1)$-categories : I'm not considering a strict colimit of a strict functor :) | |
| Jun 5, 2022 at 10:57 | comment | added | crystalline | Look at Definition A.2 (colimit of presentables), A.3 (limit of presentables, but taken over right adjoints), and A.10 (colimit of small categories [of compact objects]). We have A.2 = A.3 and in the compactly generated case, A.2 = A.10; follow the references to Lurie's HTT for the proofs. This has nothing to do with ∞-categories, so you can ignore the "∞"s; or if you prefer, you may track down the corresponding statements in some books of Adamek and Rosicky on locally presentable 1-categories (note that "presentable" ∞-categories in [HTT] correspond to "locally presentable" 1-categories) | |
| Jun 5, 2022 at 10:41 | comment | added | Gabriel | Dear @crystalline, given that I know very little about $\infty$-categories, it would be really helpful (at least to me) if you could write an answer with at least some precise statements. | |
| Jun 5, 2022 at 10:35 | comment | added | Z. M | In case of the link the user "crystalline" being inaccessible in the future, it is Déglise, Frédéric (F-ENSLY-PM); Fasel, Jean (F-GREN-IF); Jin, Fangzhou (PRC-TONG-SM); Khan, Adeel A. (F-IHES), On the rational motivic homotopy category. J. Éc. polytech. Math. 8 (2021), 533–583. | |
| Jun 5, 2022 at 10:18 | comment | added | crystalline | This property is usually called continuity. This link jep.centre-mersenne.org/item/10.5802/jep.153.pdf Appendix A may be helpful (to both of you) to understand the relation between the "small" and "big" (presentable) category versions. Higher category magic is indeed not relevant, so no need to be scared. Do note that the point about reduction to affine opens addressed in the other answer (glossed over here because you only talk about the affine case) is important though. And of course, no noetherianness is necessary for either small or big version. | |
| Jun 5, 2022 at 9:36 | comment | added | Gabriel | Dear Piotr, perhaps what you say is related to @Z. M's comment on the other answer. If $R$ is the filtered colimit of $R_i$, then perhaps $\textsf{Mod}(R)$ is not the 2-colimit of $\textsf{Mod}(R_i)$ in the (2,1)-category of categories, but it is in the (2,1)-category of presentable categories. | |
| Jun 5, 2022 at 8:52 | history | edited | Piotr Achinger | CC BY-SA 4.0 |
added 1073 characters in body
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| Jun 5, 2022 at 8:44 | history | answered | Piotr Achinger | CC BY-SA 4.0 |