Timeline for Lifting points via étale morphism of adic spaces
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5 events
| when toggle format | what | by | license | comment | |
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| Jun 17, 2016 at 16:15 | comment | added | js21 | The algebraic version of what you're looking for is Lemma 15.7.13 of stacks.math.columbia.edu/tag/07LW. | |
| Jun 17, 2016 at 14:23 | comment | added | nfdc23 | Have you worked out the analogue for classical rigid-analytic spaces (i.e., the case when $K$ is finite over the non-archimedean ground field)? If not, do that first! Hint: the key is to prove that the local ring on a rigid-analytic space is henselian (the proof of which rests on the analytic fact that a point isolated in its fiber admits an open neighborhood that is finite over an open neighborhood of its image; the details require some serious input about henselian local rings); even for complex-analytic spaces this is a serious theorem (see Houzel's Seminaire Cartan lectures). | |
| Jun 16, 2016 at 23:14 | comment | added | nfdc23 | In the setting of classical affinoid algebras that you are considering (you may as well take $X$ and $Y$ to be affinoid), etaleness is equivalent to the notion by that name in classical rigid-analytic geometry, and it is very far from etaleness in the ring/scheme-theoretic sense (e.g., consider an open immersion of affinoids, even just an inclusion of discs). So one cannot analyze this using just the theory of etale maps of schemes (though some of the ideas do adapt, working with rigid-analytic relative 1-forms and rigid-analytic flatness, etc.). | |
| Jun 16, 2016 at 22:27 | comment | added | Simone | math.stackexchange.com/q/1829116/77622 | |
| Jun 16, 2016 at 22:26 | history | asked | Simone | CC BY-SA 3.0 |