Timeline for Difference between $K(1)$-local K theory and l-adic completion of etale $K$ theory
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Jul 31, 2022 at 18:03 | comment | added | Z. M | Just to add a remark that what I said was incorrect: even $\ell$-completely, $K$-theory does not seem to satisfy étale descent, but it seems to be controlled by the norm-residue isomorphism (I only heard these from talks so incompetent to say anything seriously). | |
| Jul 31, 2022 at 16:02 | comment | added | Maxime Ramzi | Basically if $\ell$ is invertible on $X$, it is invertible on $TC$, and therefore $L_{K(1)}TC$ should vanish. It follows that the $\ell$-adic completion of $K^{ét}$ should just be $L_{K(1)}K$ in degrees $\geq 0$ , if I'm not saying anything silly | |
| Jul 31, 2022 at 14:38 | comment | added | Maxime Ramzi | @Z.M : as I said, I don't know a lot about this - if you have an answer to add along the lines of your comment, I'm sure that would be very helpful | |
| Jul 31, 2022 at 9:29 | comment | added | Z. M | The OP seems to be interested in $\ell$-adic $K$-theory. If I remember correctly, it satisfies étale (hyper)descent, and would be much easier to analyze (e.g. via Gabber–Suslin rigidity)? | |
| Jul 31, 2022 at 9:00 | history | answered | Maxime Ramzi | CC BY-SA 4.0 |