Timeline for Chevalley Groups over an arbitrary ring.
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Apr 10 at 16:50 | comment | added | Μάρκος Καραμέρης | Is there a reference for the group injection $G(k)/Z(k)\to G/Z(k)$? How does this come from the central isogeny? I cannot find it in Carter's book. | |
| Dec 17, 2013 at 1:37 | vote | accept | M.B | ||
| Dec 17, 2013 at 1:37 | vote | accept | M.B | ||
| Dec 17, 2013 at 1:37 | |||||
| Nov 14, 2013 at 15:17 | comment | added | Marguax | The 1996 paper of Vavilov and Plotkin mentioned in my comment to Jim's answer addresses the role of "group generated by root groups" beyond the setting of fields. (It seems to be of more interest than I had expected, but nonetheless they also discuss the fact that with general rings these are smaller than the groups of ring-valued points of the Chevalley group schemes, and it is the latter groups of points which they call "Chevalley groups".) | |
| Nov 14, 2013 at 4:36 | comment | added | Marguax | In the end, the main issue is probably want you want to do with these things. There may be entrenched incompatible traditions in the group theory and group scheme communities for the meaning of "Chevalley group", and the arbiter of one's preference is what one wants to do with it. The algebraic geometer's viewpoint is very robust for varying rings and fields, and recovers the other viewpoint in a clean way. I am skeptical that "group generated by root groups" is a useful notion except over fields (where Bruhat decomposition is available), but I am open to hearing to the contrary. | |
| Nov 14, 2013 at 4:24 | comment | added | Marguax | @M.B.: There is a typo in 3.3(1): $G_{\pi,k}$ at the start should be $G_{\pi,K}$, with $K$ an algebraically closed field containing a given field $k$. In 3.3(5) he says (as I did) that for the "Chevalley group" $G_{\pi}$ as a $\mathbf{Z}$-group, the "group generated by $x_a(t)$'s" is the image in $G_{\pi}(k)$ of the $k$-points of the simply connected central cover of $G_{\pi}$. In section 4 he abandons pure group theory to make $G_{\pi}$ as a $\mathbf{Z}$-group scheme. Upshot: his abstract group $G_{\pi,k}$ is not $G_{\pi}(k)$. He calls $G_{\pi,k}$ a "Chevalley group", but this is dangerous! | |
| Nov 14, 2013 at 3:24 | history | edited | Marguax | CC BY-SA 3.0 |
added 2235 characters in body
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| Nov 14, 2013 at 3:06 | comment | added | M.B | Moreover I have seen the following paper by Borel on "Properties and linear representations of Chevalley groups". In this paper he also has defined the Chevalley groups over field with the way that I presented in my question. So I just want to know if there is any mistake if one define it over rings. Perhaps what you are trying to mention is that this construction is not general enough. Do I see it correctly? | |
| Nov 14, 2013 at 3:05 | comment | added | M.B | Dear Marguax. I am so thankful for answering to my question But unfortunately since I don't know much about the theory of algebraic groups I can not understand many points that you have made in your answer. It would be very good if you would please give me a reference with more details for these. | |
| Nov 14, 2013 at 0:51 | history | answered | Marguax | CC BY-SA 3.0 |