Timeline for Compact subgroups of general linear groups over affinoid algebras
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| Mar 3, 2017 at 16:37 | comment | added | Laurent Berger | As for your original question: what Chenevier proves in lemma 3.18 (page 22) of gaetan.chenevier.perso.math.cnrs.fr/articles/famgal.pdf should interest you. See also Bellovin's paper on p-adic Hodge theory in families, for instance the paragraph before prop 2.2.8 | |
| Mar 3, 2017 at 16:25 | comment | added | Laurent Berger | Ah, not quite. The question I was actually asked was the following. Let $D$ be the closed unit disk, and let $A^0 = Z_p\langle T \rangle$. Let $A$ be the ring of functions on $D \setminus \{ 0 \}$. Then is every compact subgroup of $GL_n(A)$ conjugate to a subgroup of $GL_n(A^0)$? The answer to this is no, and there is a reasonably simple counterexample. | |
| Mar 3, 2017 at 15:59 | comment | added | Laurent Berger | I think that I was asked this question a few years ago and that I came up with a simple counterexample. Let's see if I can manage to remember what it was... | |
| Mar 1, 2017 at 21:10 | comment | added | nfdc23 | Have you tried the case when $A$ is a non-smooth curve over $k$? Even for that I guess that one should find counterexamples. | |
| Mar 1, 2017 at 21:07 | comment | added | nfdc23 | It is harmless to pass to reduced $A$, but without requiring the reduced $A$ to be normal, probably it is false. (In the normal case $A^0$ is a noetherian normal ring, so the gulf between $A$ and $A^0$ is governed by finitely many dvr's, for generic points of $A^0/m_kA^0$.) I don't see a counterexample offhand (as it seems hard to control $A^0$ beyond the normal case), but it feels unlikely that for $A$ with arbitrary singularities there is just a single conjugacy class of maximal compact subgroups of ${\rm{GL}}_n(A)$. Hopefully someone else will have something more useful to say. | |
| Mar 1, 2017 at 16:13 | history | edited | user105552 | CC BY-SA 3.0 |
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| Mar 1, 2017 at 16:11 | comment | added | user105552 | I need to consider a general $k$-affinoid algebra $A = \mathbb{Q}_p \langle T_1, \dots, T_n \rangle / I$. | |
| Mar 1, 2017 at 14:57 | review | First posts | |||
| Mar 1, 2017 at 14:59 | |||||
| Mar 1, 2017 at 14:46 | history | asked | user105552 | CC BY-SA 3.0 |