Timeline for Closure of an orbit under the action of an algebraic group
Current License: CC BY-SA 3.0
11 events
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| Aug 12, 2012 at 0:17 | comment | added | user22479 | @Jesko: If $G$ is an affine group scheme over a ring $R$ and $M$ is any $R$-module then a "representation" of $G$ on $M$ is an assignment to each $R$-algebra $S$ of an $S$-linear action of $G(S)$ on $M_S$ functorially in $S$. By a Yoneda argument, this data quantified over "all" $S$ is encoded in a single such action with $S=R[G]$. This applies to $G$-actions on affine $R$-schemes via the coordinate ring, so $M=R[G]$ is a representation of $G$ and thus of any closed $R$-subgroup of $G$. For $G=GL_1$, a representation on $M$ is a $Z$-grading! The proof early in SGA3 is a clever computation. | |
| Jul 27, 2012 at 12:22 | history | bounty ended | Jesko Hüttenhain | ||
| Jul 23, 2012 at 12:22 | comment | added | Jesko Hüttenhain | @quasi-coherent: Actually, I thought about formulating the question in a similar manner, and my problem was the following: Why does the action of our central $\mathrm{GL}_1$ on $Z$ define a grading? I can call an $f\in k[Z]$ "homogeneous of degree $d$" if $f(\lambda z)=\lambda^d f(z)$ for all $z\in Z$ and $\lambda\in\mathrm{GL}_1$, but how do I know that this yields a decomposition into direct summands? | |
| Jul 21, 2012 at 22:37 | comment | added | user22479 | @Jesko: Rather than speak in terms of cones and homogeneous polynomials, it seems more intrinsic (albeit equivalent) to say that we are given the action of an abstract connected reductive $G$ on a smooth affine $X$, choosing a central ${\rm{GL}}_1$ in $G$, and contemplating the singular locus of the closure $Z$ of the orbit of a point with (connected?) reductive stabilizer as well as the graded ring $k[Z]$ (grading defined by the ${\rm{GL}}_1$-action on $Z$). I know this doesn't constitute any genuine progress, so feel free to ignore it (but from this viewpoint it looks kind of hopeless). | |
| Jul 20, 2012 at 11:27 | history | bounty started | Jesko Hüttenhain | ||
| Jul 19, 2012 at 5:51 | history | edited | Jesko Hüttenhain | CC BY-SA 3.0 |
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| Jul 18, 2012 at 15:16 | comment | added | Marc Palm | The closure of the conjugacy classes are describable by this theory. This is at least the canonical example with which I would start with. The closure of an orbit contains a finite number of smaller dimensional orbits. Only the diagonazable elements (in some algebraic extension) have closed orbits. Unipotent matrices have never closed orbits and so forth. | |
| Jul 18, 2012 at 12:45 | comment | added | Jesko Hüttenhain | At the risk of sounding dumb, how does the rational canonical form relate to my question? | |
| Jul 18, 2012 at 11:31 | history | edited | Jesko Hüttenhain | CC BY-SA 3.0 |
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| Jul 18, 2012 at 9:16 | comment | added | Marc Palm | The similarity classes in $M_{n \times n}(k)$ (conjugation action by $Gl(n,k)$) are well understood by the rational canonical form, but you probably know this? If not, probably you want to have a look at Laumon "Drinfeld modules .. " Part 1 Chapter 4 Section 3. | |
| Jul 18, 2012 at 9:10 | history | asked | Jesko Hüttenhain | CC BY-SA 3.0 |