Timeline for Bruhat decomposition over algebraically nonclosed fields
Current License: CC BY-SA 3.0
21 events
| when toggle format | what | by | license | comment | |
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| Oct 5, 2017 at 12:44 | answer | added | Vladimir_Z | timeline score: 1 | |
| Oct 2, 2017 at 12:29 | comment | added | LSpice | @nfdc23, I have deleted my comments, which I now think are irrelevant. I think that the point is the one you have pointed out: contrary to what is implicit in the statement, there is no such thing as "the corresponding class in the first [geometric] decomposition". Instead, I think that the point is that a given class $P w P$ with $w \in N_G(S)(k_{\text s})$ may be a union of 'Bruhat' cells (although I prefer to reserve Bruhat cells for Borel double cosets …), hence may be defined over $k$ without any of the individual cells being so defined (hence having a rational point). | |
| Oct 1, 2017 at 17:46 | history | edited | Mikhail Borovoi |
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| S Oct 1, 2017 at 17:14 | history | edited | Peter Heinig | CC BY-SA 3.0 |
Edited the Remark to adhere to modern notation in algebraic geometry (e.g., X_{k_s} denotes scalar extension of a k-scheme to a k_s-scheme). Stylistic improvements.
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| S Oct 1, 2017 at 17:14 | history | suggested | nfdc23 | CC BY-SA 3.0 |
Edited the Remark to adhere to modern notation in algebraic geometry (e.g., X_{k_s} denotes scalar extension of a k-scheme to a k_s-scheme).
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| Oct 1, 2017 at 16:38 | review | Suggested edits | |||
| S Oct 1, 2017 at 17:14 | |||||
| Oct 1, 2017 at 13:40 | history | edited | Vladimir_Z | CC BY-SA 3.0 |
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| S Oct 1, 2017 at 5:00 | history | suggested | nfdc23 | CC BY-SA 3.0 |
I have replace $H_k$ with $H(k)$ everywhere to get rid of the confusing notation about rational points.
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| Oct 1, 2017 at 3:25 | review | Suggested edits | |||
| S Oct 1, 2017 at 5:00 | |||||
| Oct 1, 2017 at 3:23 | comment | added | nfdc23 | @LSpice: Since $N_G(S)$ doesn't lie inside $N_G(T)$, if $w \in N_G(T)(k_s)$ and $P_{k_s}wP_{k_s}$ arises from a $k$-subgroup of $G$, it isn't clear if that "$P$-double coset" has any $k$-point. So in principle the possibility that is being said to be mentioned in some Remark of Borel-Tits isn't obviously ruled out. | |
| Oct 1, 2017 at 2:51 | comment | added | LSpice | Finally, you mention a remark in Borel–Tits. Where in the paper does this remark appear? I had a quick glance near (5.15), where the rational Bruhat decomposition is discussed, and didn't see it. It seems unlikely: for $PwP/P$ to be defined over $k$, it would have to be Galois-stable, which, by the geometric Bruhat decomposition, indicates that $w$ lies in $W(G, S)(k)$, hence, as @nfdc23 mentions, has a lift to $N_G(S)(k)$. | |
| Oct 1, 2017 at 2:48 | comment | added | LSpice | Also, what does your (2) mean? Do you mean to consider the same group $P$ as a parabolic inside several different groups $G$? I don't think that this can happen in any reasonable way. | |
| Oct 1, 2017 at 2:47 | comment | added | LSpice | I think that the issue @nfdc23 brings up has not been addressed by the edit. You're still writing $G_k$ sometimes and $N_G(S)(k)$ sometimes. Do these both mean the group of rational points? | |
| Sep 30, 2017 at 23:09 | comment | added | nfdc23 | I think that $N_G(S)(k)$ is reasonable notation. Since the finite etale $k$-group $W(G,S) = N_G(S)/Z_G(S)$ turns out to be constant with group of $k$-points always equal to $N_G(S)(k)/Z_G(S)(k)$ (a real miracle, in my opinion), one can also safely index those unions by the groups $W(G,T)(\overline{k})$ and $W(G,S)(k)$ respectively (where it is understood that $PwP$ means $P n_w P$ for a representative $n_w$ of $w$, the choice of which doesn't matter for the double coset). | |
| Sep 30, 2017 at 18:54 | history | edited | Vladimir_Z | CC BY-SA 3.0 |
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| Sep 30, 2017 at 18:47 | history | edited | Vladimir_Z | CC BY-SA 3.0 |
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| Sep 30, 2017 at 18:14 | comment | added | Vladimir_Z | Thank you for noticing confusion in my notations. I can understand the reason for putting $P(k)$ instead of $P_k$, but what you can advise me to change $N_G(S)_k$ for? $N(G,S)(k)$? | |
| Sep 30, 2017 at 18:04 | history | edited | Vladimir_Z | CC BY-SA 3.0 |
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| Sep 30, 2017 at 17:27 | comment | added | nfdc23 | Writing $W_k$ to denote the relative Weyl group (which is not the group of $k$-points of what you call $W$) and simultaneously writing $H_k$ to denote the group of $k$-points of a $k$-group ($P_k, G_k, N_G(S)_k$, etc.) is rather confusing notation. Can you please denote the group of $k$-points in another way, such as $H(k)$? While doing that, please also replace $G_k$ with $G(k)/P(k)$. The subsets $(PwP)(\overline{k})$ for $w\in W(G,S)$ are pairwise disjoint, so can you please also clarify your reason for interest in the possibility that some $PwP$ for $w\in W(G,T)$ is defined over $k$? | |
| Sep 30, 2017 at 16:17 | review | First posts | |||
| Sep 30, 2017 at 16:18 | |||||
| Sep 30, 2017 at 16:15 | history | asked | Vladimir_Z | CC BY-SA 3.0 |