Timeline for Canonicality of group of integers for reductive groups over non-Archimedean local field
Current License: CC BY-SA 4.0
13 events
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| Sep 10, 2023 at 12:05 | vote | accept | user267839 | ||
| Sep 10, 2023 at 10:39 | comment | added | LSpice | Re, as @DavidLoedffler showed, my reference to well definedness of the image was wrong; the identification of $\mathcal G_K$ with $G$ allows inner automorphisms coming from $G(K)$, not just $\mathcal G(\mathcal O_K)$. | |
| Sep 10, 2023 at 9:28 | history | edited | user267839 | CC BY-SA 4.0 |
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| Sep 10, 2023 at 9:22 | comment | added | user267839 | which David Loeffler is refering to, namely the existence of such " nice enough" open compact $U \subset G(K)$? | |
| Sep 10, 2023 at 9:21 | comment | added | user267839 | @LSpice: yes thanks, in light of David Loeffler' s answer it seems that in order to assure the existence of such group one should pose for $G$ some additinal assumptions. Beeing 'split' seems to be very natural, I should add it. But could you give a reference or a brief sketch how this $O_K$-group scheme $\mathcal{G}$ is constructed? Is there a "standard" method? Do I understand it correctly, that this splitting assumption is one of possible assumptions which would garantee availability of the result from Bruhat- Tits theory to | |
| Sep 10, 2023 at 7:43 | answer | added | David Loeffler | timeline score: 2 | |
| Sep 10, 2023 at 0:01 | comment | added | LSpice | Is your group split? If so, then you can take the split reductive $\mathcal O_K$-group $\mathcal G$ of the same type, and identify $\mathcal G_K$ with $G$. This identification is not canonical, but the resulting image of $\mathcal G(\mathcal O_K)$ in $G(K)$ is. (This probably works just as well if $G$ is quasisplit, although I'd want to be a bit careful about Galois groups, and every reductive group over $K$ is quasisplit over an unramified extension of $K$ … but somewhere in there I lose the thread of well definedness!) | |
| Sep 9, 2023 at 23:35 | comment | added | user267839 | @DaveBenson: In the question I missed to mention that $G$ is considered as a variety over field $K$, sorry for confusion. So in language schemes it can be considered as a representable functor from finitely generated $K$- algebras to groups. So a priori it's not clear "what is" $G(\mathcal{O}_K)$, since $O_K$ is not a $K$ - algebra. Naively it only make sense when we fix/consider the closed embedding $ G \subset GL_n$ as sketched above. But my concern is if this $G(\mathcal{O}_K)$ really depends on choosen embedding $ G \subset GL_n$ or exist independently of such embedding abstractly | |
| Sep 9, 2023 at 23:26 | history | edited | user267839 | CC BY-SA 4.0 |
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| Sep 9, 2023 at 22:51 | history | edited | user267839 | CC BY-SA 4.0 |
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| Sep 9, 2023 at 22:32 | history | edited | user267839 | CC BY-SA 4.0 |
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| Sep 9, 2023 at 15:35 | history | edited | YCor | CC BY-SA 4.0 |
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| Sep 9, 2023 at 15:14 | history | asked | user267839 | CC BY-SA 4.0 |