Timeline for Are condensed vector spaces over finite fields always solid?
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12 events
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| Jan 20, 2022 at 20:12 | comment | added | Z. M | Just to add that I was mistaken. The left adjoint from condensed sets to condensed abelian groups does exist and seems to coincide with the derived version. Indeed, it suffices to check the values at extremally disconnected sets. | |
| Jan 20, 2022 at 14:03 | vote | accept | Peter Kropholler | ||
| Jan 20, 2022 at 14:03 | answer | added | Peter Kropholler | timeline score: 1 | |
| Jan 17, 2022 at 12:21 | comment | added | Peter Scholze | The condensed $\mathbb F_p$-vector space $\mathbb F_p[S]$ sits in degree $0$ for any compact Hausdorff space $S$. It is naturally an increasing union of compact Hausdorff subsets $\mathbb F_p[S]_{\leq n}$ where this is the set of those sums $\sum_{s\in S} n_s [s]$ where one can choose the $n_s\in \mathbb Z$ with $\sum |n_s|\leq n$. See Proposition 2.1 and Exercise 2.3 in Analytic.pdf for the discussion for $\mathbb Z[S]$ (which carries over without much change to $\mathbb F_p[S]$). These things are not solid as soon as $S$ is infinite. | |
| Jan 15, 2022 at 14:05 | comment | added | Peter Kropholler | OK. I may have misunderstood: when I wrote F_p[S] I had in mind the result of applying the left adjoint to the forgetful functor from condensed F_p vector spaces to condensed sets to the condensed set S: I hope I am correct that such a left adjoint exists. | |
| Jan 15, 2022 at 13:46 | comment | added | Z. M | You would take $S$ to be a profinite set, otherwise $\mathbb F_p[S]$ lives in the derived category. | |
| Jan 15, 2022 at 12:34 | comment | added | Peter Kropholler | @Z.M Thanks for this advice, I had overlooked the discussion in math.uni-bonn.de/people/scholze/Analytic.pdf where Theorem 2.9 gives a very clear characterisation of the solid F_p vector spaces. I guess this means that typically, F_p[S] is not solid when S is a compact Hausdorff space. | |
| Jan 15, 2022 at 10:52 | comment | added | Maxime Ramzi | @Z.M : you should post this as an answer :) | |
| Jan 14, 2022 at 16:22 | comment | added | Z. M | If you look for a concrete example, see the example right after Exercise 2.3, and replacing $\mathbb Z$ by $\mathbb F_p$. | |
| Jan 14, 2022 at 16:19 | comment | added | Z. M | Does Theorem 2.9 of Lectures in Analytic Geometry answer your question? | |
| Jan 14, 2022 at 11:19 | history | edited | YCor |
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| Jan 14, 2022 at 10:42 | history | asked | Peter Kropholler | CC BY-SA 4.0 |