Timeline for Integral smooth model of unramified reductive groups
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| Feb 11, 2019 at 0:18 | history | edited | YCor | CC BY-SA 4.0 |
edited tags, minor typos
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| Oct 4, 2017 at 0:41 | comment | added | nfdc23 | One can also use the SGA3-style methods to show that a reductive $O_K$-model exists if and only if the $K$-group is quasi-split and moreover splits over an unramified extension. These are "combinatorially classified" insofar as they are inner forms of the split form using some unramified Galois data (as one sees by staring at the proof that the quasi-split inner form is unique up to non-canonical isomorphism (see 7.2.12 in Conrad's article for a wider context), but without knowing what you intend to do with such a conclusion it is hard to tell how useful it could be. | |
| Oct 4, 2017 at 0:37 | comment | added | nfdc23 | There is no interesting theory of Neron models for connected reductive groups that aren't tori because by 10.1/8 of Neron models a smooth affine $K$-group admitting an lft Neron model cannot contain $\mathbf{G}_a$ as a $K$-subgroup (which forces the derived group to be $K$-anisotropic in the connected reductive case). As noted in Theorem 7.2.16 of Conrad's article, one can use the SGA3-style methods to show a reductive $O_K$-model is unique up to isomorphism if it exists, but there are lots of $K$-automorphisms that don't come from $O_K$-automorphisms of a chosen such $O_K$-model. | |
| Oct 3, 2017 at 20:57 | comment | added | Mayday | *c): And I just remember that Bruhat-Tits theory is good-behavior under \'etale descent (even though I don't know what this means explicitly), so according to a), we should have a construction of $\mathscr{G}_x$ independent of the Bruhat-Tits theory.\\ The reason of "avoid" is each time I want to explain the hyperspecial maximal compact subgroup of an unramified reductive group, I go through those I learned from Tits and Landvogt to define $\mathscr{G}_x$ and this is never a satisfactory answer. I expect there is an explanation of this just within the knowledge of reductive groups. | |
| Oct 3, 2017 at 20:55 | comment | added | Mayday | *b): In Chapter 2 of Landvogt's book in LNM, he defines, for a quasi-split reductive group $\mathrm{G}/K$, a connected smooth affine group scheme $\mathscr{G}_{\Omega}$ over $\mathscr{O}_K$ such that the $\mathscr{O}_K$ point of $\mathscr{G}$ is the stabilizer of the bounded subset $\Omega$ in an apartment of the building under the action of $\mathrm{G}$. The construction he used is that he constructs the integral model of the maximal torus and the root groups(regarding $\Omega$) first and then glue them with $\mathrm{G}$. The integral model of torus he used is simply the connected NR model. | |
| Oct 3, 2017 at 20:53 | comment | added | Mayday | @nfdc23, thanks for your answer and the reference for #1. For #2, the group scheme in my mind is the Bruhat-Tits group scheme with the $\mathscr{O}_K$ points the hyperspecial maximal compact subgroup of $\mathrm{G}$. The reason why I made a guess is: *a): If $\mathrm{G}/K$ is split, then $\mathscr{G}_x$ for a special point $x$ is the Chevalley group schemes of $\mathrm{G}$. cf., 3.4.2 of Tits' in Corvallis. And I know, the construction of the Chevalley group schemes comes from the root datum of $\mathrm{G}$. And in this case no Galois action. | |
| Oct 3, 2017 at 19:33 | comment | added | nfdc23 | For #1, the formation of the Neron-Raynaud model of tori commutes with unramified base change (10.1/3 in Neron Models), so to check if the canonical map $\mathscr{T} \to N(T)$ to the Neron-Raynaud model is an isomorphism onto the open relative identity component make an unramified base change to reduce to the case when $T$ is split. Then reduce to the case $T=\mathbf{G}_m$, which is easy to check directly from the construction of $N(\mathbf{G}_m)$ (10.1/5 in Neron Models). For #2, with paraphrase Bill Clinton, please define "the" in the first sentence (and "avoid" for what purpose?). | |
| Oct 3, 2017 at 19:03 | history | asked | Mayday | CC BY-SA 3.0 |