Timeline for Connectedness of Centralizers in $GL_n$
Current License: CC BY-SA 3.0
9 events
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| Oct 27, 2011 at 21:34 | comment | added | Jim Humphreys | @Michael: I'm just following by habit the older practice of working with linear algebraic groups over algebraically closed fields. But the language of schemes and closed points is certainly more flexible. | |
| Oct 26, 2011 at 21:52 | comment | added | Michael Thaddeus | Why the hypothesis that the field is algebraically closed in this theorem? If not, can't I just pass to an algebraic closure? At least if by "element" I mean a k-valued point... | |
| May 17, 2011 at 15:14 | comment | added | Jim Humphreys | @HNuer: The argument for general linear groups or centralizers to be connected just requires showing that the underlying affine variety is irreducible. (The algebra of regular functions on a principal open set is a nice subalgebra of the field of fractions of a polynomial algebra.) @darij: I didn't answer directly your question about Milne's early definition of "connected" (assuming characteristic 0), which is based on a lot of structure theory one wants to develop. There's a long history of comparing abstract group notions with algebraic group notions (Chevalley, Borel-Tits, ...) | |
| May 17, 2011 at 10:34 | comment | added | HNuer | Sorry, now I understand, this is a linear subspace, and thus isomorphic to some affine space. Since affine spaces are irreducible any non-empty open set in them is irreducible (in particular principle open affines) and thus connected. Thanks for the suggestion. | |
| May 17, 2011 at 10:08 | comment | added | HNuer | Jim, is it somehow obvious that principle open sets in affine varieties are connected? | |
| May 16, 2011 at 20:59 | comment | added | Jim Humphreys | Milne's discussion of connectedness is much more elaborate than this (see his Chapter 13), while his overall framework for "algebraic groups" is much broader than the older Borel-Chevalley theory of linear algebraic groups. So it's important to be careful with the terminology when comparing his treatment with earlier sources. | |
| May 16, 2011 at 17:26 | comment | added | darij grinberg | Interesting; Milne defines "connected" as "no quotient is a nontrivial finite group". Is this equivalent? | |
| May 16, 2011 at 17:13 | history | edited | Jim Humphreys | CC BY-SA 3.0 |
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| May 16, 2011 at 15:47 | history | answered | Jim Humphreys | CC BY-SA 3.0 |