Timeline for Representation of a group scheme
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 3, 2019 at 21:32 | vote | accept | CommunityBot | ||
| Nov 13, 2017 at 8:49 | comment | added | Geordie Williamson | For further discussion see Aside 9.4 in Chapter 9, in Milne, "Basic Theory of Affine Group Schemes": www.jmilne.org/math/CourseNotes/AGS.pdf | |
| Nov 13, 2017 at 2:02 | comment | added | Jim Humphreys | I strongly agree with nfdc23's comment and especially the last part. I'm unaware of any setting in which one has to consider "rational representations" over a base that isn't at least a commutative ring such as a field. For example, since Jantzen's book is about representation theory, he has no need to work in greater generality over arbitrary schemes. A field is usually complicated enough. | |
| Nov 12, 2017 at 22:29 | comment | added | nfdc23 | I believe it remains an unsolved problem to determine if all smooth affine group schemes over the ring $k[\epsilon]$ of dual numbers over a field arise as a closed subgroup scheme of some ${\rm{GL}}_n$. So that device which is so useful over fields is not available more generally. Anyway, one can always contemplate $S$-group homomorphisms $\mathcal{G} \to {\rm{GL}}_n$ (even if no interesting ones exist), but calling these "rational representations" sounds weird when the base isn't a field. | |
| Nov 12, 2017 at 21:48 | answer | added | Jim Humphreys | timeline score: 6 | |
| Nov 12, 2017 at 21:25 | history | edited | user100841 | CC BY-SA 3.0 |
added 17 characters in body
|
| Nov 12, 2017 at 21:22 | comment | added | Jason Starr | What if $S$ is $\text{Spec}\ \mathbb{C}$ and $\mathcal{G}$ is an Abelian variety? Do you want to add a hypothesis that $\mathcal{G}$ is a closed subgroup scheme of $\textbf{GL}_{n,S}$, or at least that it is affine over $S?$ | |
| Nov 12, 2017 at 21:21 | history | asked | user100841 | CC BY-SA 3.0 |