Timeline for Group schemes, adeles, double cosets, and étale cohomology
Current License: CC BY-SA 3.0
19 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| May 14, 2015 at 22:09 | vote | accept | Mikhail Borovoi | ||
| May 14, 2015 at 22:09 | comment | added | Mikhail Borovoi | @DanielLitt: Thank you! But this explanation should be included into your answer... | |
| May 14, 2015 at 19:32 | comment | added | Daniel Litt | @MikhailBorovoi: Descent data is given by an isomorphism between two trivialized $G$-torsors on $\text{Spec}(R[1/N]\otimes R_{\mathfrak{p}})$ for all $\mathfrak{p}$. If $\mathfrak{p}|N$, this is just $\text{Spec}(K_\mathfrak{p})$; an isomorphism between trivialized $G$-torsors on $\text{Spec}(K_\mathfrak{p})$ is just an element of $G(K_\mathfrak{p})$. If $\mathfrak{p}\nmid N$, we get the same thing over $\text{Spec}(R_\mathfrak{p})$; again an isomorphism is given by an element of $G(R_\mathfrak{p})$. | |
| May 14, 2015 at 18:53 | comment | added | Mikhail Borovoi | @DanielLitt: Please kindly also explain the sentence: "Now descent data is given by a choice of element of $G(R_{\mathfrak{p}})$ for each ${\mathfrak{p}}\in U$ and an element of $G(K_{\mathfrak{p}})$ for each ${\mathfrak{p}}|N$". | |
| May 14, 2015 at 18:47 | comment | added | Mikhail Borovoi | @DanielLitt: Thank you for the clarifying paragraph. Still I have questions. Please kindly explain the following sentence: "Now the descent data boils down to choosing elements of $G({\rm Frac}(K_{\mathfrak{p}}))$ for each $\mathfrak{p}|N$". | |
| May 14, 2015 at 16:49 | history | edited | Daniel Litt | CC BY-SA 3.0 |
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| May 14, 2015 at 16:49 | comment | added | Daniel Litt | @MikhailBorovoi: I've added a clarifying paragraph; let me know if you have any more questions. The structure of the argument above is: reverse engineer a description of descent data for the objects we're looking for (etale G-torsors which are trivial over $K$ and $R_\mathfrak{p}$ for each $\mathfrak{p}$) and then observe that elements of $c(G)$ give such descent data. | |
| May 14, 2015 at 14:21 | comment | added | Mikhail Borovoi | @DanielLitt Could you please describe the construction more clearly? We start from an element $g\in G(\mathbf{A}^f)$. How do we construct the corresponding torsor? From your exposition is seems that you start from a torsor.... | |
| May 14, 2015 at 7:56 | history | edited | Daniel Litt | CC BY-SA 3.0 |
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| May 14, 2015 at 7:55 | comment | added | Daniel Litt | @grghxy: Ah, perfect! | |
| May 14, 2015 at 7:54 | comment | added | grghxy | By excellence of $R$ and Artin approximation, a torsor that split over $R_{\mathfrak{p}}$ also splits over an etale neighborhood of $\mathfrak{p}$. In effect, it is a very strong condition for non-smooth $G$ that there be splitting over the completions of $R$! | |
| May 14, 2015 at 7:40 | comment | added | Daniel Litt | @grghxy: The reason I wrote it all out is that I wanted to check I wasn't going to use smoothness (assumed in that answer) and I'm a bit confused about why we land in $H^1_{et}$ rather than $H^1_{fpqc}$; as you'll note I had to assume weak approximation to get this. In the referenced answer this is not an issue b/c of smoothness. | |
| May 14, 2015 at 7:38 | comment | added | grghxy | Very true, but I had misread your answer to think you were saying something else. Now all is clear; doesn't the link you give to a related question also answer this one (assuming $G$ is flat and affine of finite type over the Dedekind base)? | |
| May 14, 2015 at 7:35 | comment | added | Daniel Litt | (Of course I agree that not all $G_K$-torsors will be trivial, e.g. if $G$ is $PGL_n$.) | |
| May 14, 2015 at 7:34 | history | edited | Daniel Litt | CC BY-SA 3.0 |
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| May 14, 2015 at 7:32 | comment | added | Daniel Litt | @grghxy: We're looking for a trivial $G_K$-torsor, since we're mapping into the kernel of a map to $H^1(K, G_K)\times \cdots$ | |
| May 14, 2015 at 7:30 | comment | added | grghxy | It isn't true that $G_K$-torsors must be trivial; that only holds under certain hypotheses on $G_K$ (and $K$). | |
| May 14, 2015 at 7:28 | history | answered | Daniel Litt | CC BY-SA 3.0 |