Timeline for Algebraic groups acting on affine varieties with finite-dim orbits in the coordinate ring
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 9, 2022 at 21:52 | comment | added | Joshua Ruiter | I wanted to know if there were situations where this could be deduced perhaps just working with $K[X]$ or the fact that $\operatorname{SL}_3(\mathbb{Z})$ is generated by elements of finite order. | |
| Apr 9, 2022 at 21:46 | vote | accept | Joshua Ruiter | ||
| Apr 9, 2022 at 21:43 | comment | added | Joshua Ruiter | No, this is not what I meant to ask. I am studying the situation in which $G$ is an elementary subgroup of a Chevalley group, e.g. $\operatorname{SL}_n(\mathbb{Z})$. I am interested in finding an example of such $G$ acting on an affine variety in a locally finite way, but that we have deduced local-finiteness in some other way than already knowing it is just the restriction of an algebraic action of $\operatorname{SL}_n(k)$. | |
| Apr 8, 2022 at 9:27 | comment | added | Friedrich Knop | Your question was "Are there examples ... where this finite-dimensionality property ... holds for some other reason than the action being algebraic?" and the answer in "no" in the sense of my answer. Could it be that you meant to ask "Given a subgroup $G$ of an algebraic group $\overline G$ and $\overline G$ acts algebraically on $X$ does the induced $G$-action on $X$ have special properties?"?. | |
| Apr 7, 2022 at 14:51 | comment | added | Joshua Ruiter | This argument is exactly my motivation for the question. My goal is to find an example where this implication tells you something meaningful about an action which you did not already know was algebraic. | |
| Apr 7, 2022 at 9:03 | history | answered | Friedrich Knop | CC BY-SA 4.0 |