Timeline for If $G$ is absolutely simple simply connected, why is G(F_v) quasisimple for almost every valuation v?
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| Jun 21, 2016 at 8:45 | comment | added | JGR | I did not ask about how to get geometrically connected fibers because I did find $IV_3$ 9.7.7. I found yesterday $IV_4$ 18.5.17 but I had already asked my question. Now that I see it, I think I would have probably spent a considerable amount of time ( if not infinite) finding and realizing that what I need was $IV_$$ 17.7.8. Thanks a lot, that was really helpful to me | |
| Jun 21, 2016 at 1:44 | comment | added | nfdc23 | By renaming $R[1/r]$ as $R$, you want that if $X$ is an $R$-scheme of finite type and $X_K$ is smooth then so is $X_{R[1/r']}$ for some nonzero $r' \in R$ (though fibers being geometrically connected for suitable $r'$ when $X_K$ is geometrically connected is also quite non-obvious, so I wonder why you don't ask about that too). And you want that $X(R) \rightarrow X(k)$ is surjective for smooth $X$ over a henselian local $R$ with residue field $k$. See IV$_4$ 17.7.8(ii) (and 17.7.11(iii)), IV$_3$ 9.7.7, IV$_4$, 18.5.17 (uses that smooth schemes are Zariski-locally etale over an affine space). | |
| Jun 18, 2016 at 8:56 | comment | added | JGR | Yes, the only part I still did not get is why can i choose $r'$ such that $\mathcal{G}_{R[1/r']}$ is smooth, could you point me to a reference as well as for the "surjectivity property" for smooth schemes over heneselian local rings? I know they are in EGA, but it is difficult to me to find these results in such a long book, plus most of the times the hypothesis for the theorems depend on general concepts that, although very likely they are trivially satisfied in my setting, I cannot recognize them. | |
| Jun 17, 2016 at 14:19 | comment | added | nfdc23 | Express $K[G]$ in terms of generators and relations over $K$, and likewise for the group law. This uses finitely many elements of $K$; by writing those as fractions we find a multiple $r \ne 0$ of all denominators occurring. Over $R[1/r]$ we thereby get an affine group scheme $\mathscr{G}$ of finite type with generic fiber $G$. More serious EGA input gives a multiple $r' \ne 0$ of $r$ so that $\mathscr{G}_{R[1/r']}$ is smooth with geometrically connected fibers over $R[1/r']$. "Yes" for the 2nd question. The 3rd is a property of smooth schemes over henselian local rings. | |
| Jun 17, 2016 at 7:21 | comment | added | JGR | Why is it true that i can choose r such that G is the generic fiber of a n R[1/r]-group scheme $\mathcal{G}$? The pullback over the v-adic completion just means considering it as an $\mathcal{O}_v$-scheme? Why is then true that $\mathcal{G}(\mathcal{O}_v)\to\mathcal{G}(\mathbb{F}_v)$ is surjective? I do not know to answer this questions when $k$ has positive characteristic. | |
| Jun 17, 2016 at 7:15 | comment | added | JGR | Thanks for your detailed answer. After a considerable amount of reading I have been able to follow the main ideas and concepts involved in it. However I am still confused on the part related to smoothness. | |
| Jun 13, 2016 at 15:27 | vote | accept | JGR | ||
| S Feb 20, 2016 at 5:09 | history | answered | nfdc23 | CC BY-SA 3.0 | |
| S Feb 20, 2016 at 5:09 | history | made wiki | Post Made Community Wiki by nfdc23 |