Timeline for Is there a ring stacky approach to $\ell$-adic or rigid cohomology?
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| S May 4, 2023 at 8:24 | history | bounty ended | Gabriel | ||
| S May 4, 2023 at 8:24 | history | notice removed | Gabriel | ||
| May 4, 2023 at 8:24 | vote | accept | Gabriel | ||
| May 2, 2023 at 12:17 | answer | added | Peter Scholze | timeline score: 18 | |
| May 2, 2023 at 9:11 | comment | added | Pavel Safronov | Just like the Betti stack, you can extend it to be constant in the "algebraic" direction to obtain a condensed stack. For constructible sheaves: you have to use the exit-path category in the Betti setting or the Galois category in the $\ell$-adic setting (see Chapter 13 of arxiv.org/abs/1807.03281). Note that these are not (condensed) $\infty$-groupoids, but rather (condensed) $\infty$-categories. | |
| May 2, 2023 at 8:30 | comment | added | Gabriel | Dear @PavelSafronov, do you know if it's possible to see this condensed shape as some sort of stack and/or if this also works for constructible sheaves? | |
| May 2, 2023 at 7:58 | comment | added | Pavel Safronov | For $\ell$-adic cohomology there is the condensed/pyknotic shape which is a condensed space whose category of local systems gives the category of lisse $\ell$-adic sheaves, see e.g. Appendix A in arxiv.org/abs/2012.02853. | |
| S May 2, 2023 at 7:35 | history | bounty started | Gabriel | ||
| S May 2, 2023 at 7:35 | history | notice added | Gabriel | Draw attention | |
| Apr 27, 2023 at 18:03 | comment | added | Gabriel | Dear @PiotrAchinger, if I understand correctly, given a scheme $X$ of finite-type over $\mathbb{C}$, its "Betti" space $X_B$ is just the constant sheaf (of anima) equal to the topological space $X_\text{an}$. (There's more about this in [Sim] and in the end of the first chapter in Scholze's notes about six-functor formalisms.) Do you have any idea for an étale counterpart? | |
| Apr 27, 2023 at 17:27 | comment | added | Piotr Achinger | I don’t understand the definition of $K_B$ in Porta–Sala but at first glance it looks like something which should have an etale homotopy counterpart. | |
| Apr 27, 2023 at 15:54 | comment | added | Gabriel | Since this post is already too big, let me explain the term ring stack in the title for those curious: in many cases it was remarked that it suffices to construct the stack $X_\text{stk}$ for $X=\mathbb{A}^1$. The resulting object is a "ring stack" and we have $X_\text{stk}=X\circ \mathbb{A}^1_\text{stk}$ as a functor. | |
| Apr 27, 2023 at 15:52 | history | asked | Gabriel | CC BY-SA 4.0 |