Timeline for Topology on $p$-adic period ring in an article by Fontaine
Current License: CC BY-SA 4.0
5 events
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| Jun 27, 2021 at 19:17 | comment | added | Peter Scholze | You're welcome! 1) Yes; just take their algebraic definition and do it internally in condensed rings. 2) Yes, everything translates directly. 3) I don't think anything has to be written, as nothing deep happens. I think you should be able to translate yourself, maybe referring to the lecture notes on my webpage for the basics. (But note that in my first p-adic Hodge theory paper, in hindsight I'm already discussing period rings from the condensed perspective, so maybe that's a little helpful.) | |
| Jun 27, 2021 at 7:23 | comment | added | DCM | 3. Is there a source for this more modern perspective, or is it all forthcoming? I would love to be able to ignore topology :) | |
| Jun 27, 2021 at 7:23 | comment | added | DCM | 2. In Fontaine’s theory you have the notion of a representation being cristalline / semi-stable / de Rham. Basically, if I recall correctly, a representation V of the Galois group G_k of a p-adic number field is semi-stable if the natural map (B_{st} \otimes_{Q_p} V)^{G_k} \to B_{st} \otimes_{Q_p} V is an isomorphism. If one just follows one’s nose, I would say that this should translate directly to the condensed world. Is this true? Namely, one embeds G_k as a condensed group, and one just does all the constructions there. | |
| Jun 27, 2021 at 7:22 | comment | added | DCM | Dear Peter, Thank you for this fascinating reply! Just three questions (maybe I should post these as separate question?) : 1. Can one construct all the period rings, such as B_{dr}, B_{st}, using the condensed perspective? | |
| Jun 26, 2021 at 19:58 | history | answered | Peter Scholze | CC BY-SA 4.0 |