Timeline for Why central isogeny of reductive group over general field F map maximal F split torus onto a maximal split F torus
Current License: CC BY-SA 4.0
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| Apr 22, 2019 at 20:12 | comment | added | LSpice | @yshuaiqin, ah, now I see the relevance of pullbacks to your question! This answer explains why the image of a split torus is split. As you argue, the pre-image of a torus need not be a torus, but the reduced scheme underlying the pre-image is a torus, with the same image, and the rest of the argument goes through. | |
| Apr 22, 2019 at 19:05 | comment | added | yshuai Qin | @LSpice sorry I didn't write the question clear. The central isogeny is between two different reductive groups (could be not isomorphism to each other). So we don't assume the split ranks are same. | |
| Apr 22, 2019 at 12:35 | comment | added | LSpice | Just to emphasise, it's split because the isogeny induces a Galois-equivariant isomorphism of ratinalised character lattices. | |
| Apr 22, 2019 at 1:05 | history | edited | anon | CC BY-SA 4.0 |
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| Apr 22, 2019 at 0:56 | history | edited | anon | CC BY-SA 4.0 |
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| Apr 22, 2019 at 0:28 | history | answered | anon | CC BY-SA 4.0 |