Timeline for Comparing algebraic group orbits over big and small algebraically closed fields
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| Mar 9, 2022 at 12:19 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
http -> https (the question was bumped anyway)
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| Jul 19, 2015 at 12:47 | comment | added | Jim Humphreys | @L Spice: Thanks for the edit, which I've made more concise. Older papers are now made openly accessible online by Elsevier's ScienceDirect, which is useful to note. | |
| Jul 19, 2015 at 12:43 | history | edited | Jim Humphreys | CC BY-SA 3.0 |
deleted 305 characters in body
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| S Jul 18, 2015 at 21:24 | history | suggested | LSpice | CC BY-SA 3.0 |
Inserted link to paper of Guralnick et al.
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| Jul 18, 2015 at 21:16 | review | Suggested edits | |||
| S Jul 18, 2015 at 21:24 | |||||
| Feb 22, 2013 at 22:03 | answer | added | anon | timeline score: 1 | |
| Dec 21, 2010 at 14:37 | vote | accept | Jim Humphreys | ||
| Dec 21, 2010 at 14:37 | comment | added | Jim Humphreys | Thanks for these interesting ways of working out the answer. It's hard to single out one "correct" one, but Torsten has the edge on brevity plus transparency in the classical setting of the question. Brian has provided the most fascinating extended discussion and best community-wiki answer one can imagine. George's careful, detailed answer improves on the published 1997 proof (like Torsten working within the classical setting). Thomas gives the most interesting meta-proof though it involves for a person of my background something of a black box. Again, thanks for all the insights. | |
| Dec 21, 2010 at 5:27 | comment | added | BCnrd | Jim, in the argument I give below in extensive detail (& generality), I directly prove $X(k)/G(k) \rightarrow X(K)/G(K)$ is bijective. But can be seen a-posteriori (& is "formal"; reductivity irrelevant). Indeed, if you know same size, just need injectivity. And that injectivity I prove early in my long answer, in a concrete manner (using nothing beyond Nullstellensatz). Here it is in other terms: if $x, x' \in X(k)$, form the "transporter variety" $T_{x,x'}$ inside of $G$. You want that this has a $k$-pt iff it has a $K$-point. Each says variety is non-empty, again by Nullstellensatz... | |
| Dec 20, 2010 at 23:22 | answer | added | George McNinch | timeline score: 6 | |
| Dec 20, 2010 at 15:57 | history | edited | Jim Humphreys | CC BY-SA 2.5 |
added 505 characters in body
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| Dec 20, 2010 at 6:24 | answer | added | Thomas Scanlon | timeline score: 11 | |
| Dec 19, 2010 at 20:24 | comment | added | BCnrd | An analogous situation where such a comparison occurs is this: if $X$ is finite type sept'd scheme over sep. closed field $k$ and if $K/k$ is sep. closed extn field, the natural pullback map of etale cohom. ${\rm{H}}^i(X,\mathbf{Q}_{\ell}) \rightarrow {\rm{H}}^i(X_K,\mathbf{Q}_{\ell})$ is an isom for any prime $\ell \ne {\rm{char}}(k)$ (and vast generalizations thereof). This is especially important when $k = \overline{\mathbf{Q}}$ and $K = \mathbf{C}$, since the former is where Galois gps act (when $X$ begins life over a number field) and the latter is topological (Artin comparison isom). | |
| Dec 19, 2010 at 18:58 | answer | added | BCnrd | timeline score: 18 | |
| Dec 19, 2010 at 17:12 | answer | added | Torsten Ekedahl | timeline score: 10 | |
| Dec 19, 2010 at 15:46 | history | asked | Jim Humphreys | CC BY-SA 2.5 |