Timeline for Does every reductive group scheme admit a maximal torus?
Current License: CC BY-SA 3.0
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| May 6, 2014 at 16:43 | comment | added | S. Carnahan♦ | Ok, sorry. I was confusing monomial transformations (from the char. lattice) with linear transformations. | |
| May 6, 2014 at 16:34 | comment | added | David E Speyer | Or, less formally, by assumption we have a torus $T_x$ in each fiber $GL(E_x)$. The elements in those torii all share a common eigenspace decomposition of $E_x$. | |
| May 6, 2014 at 16:29 | comment | added | David E Speyer | Given a scheme $X$, a rank $r$ vector bundle $E$ on $X$, and a torus $T \subset GL(E)$, we can consider the closed subscheme of $\mathbb{P}(E)$ given by $\{ (x,v) \in \mathbb{P}(E) : t \cdot v = v \ \forall_{t \in T_x} \}$. (Equality as points in projective space, so proportionality as elements of the vector bundle.) If I am not confused, in each fiber $\mathbb{P}(E_x)$ this should be a reduced zero dimensional scheme of length $r$. (So, if we work over the complex numbers, $r$ points over each closed point.) | |
| May 6, 2014 at 16:22 | comment | added | S. Carnahan♦ | What do you mean when you say that the eigenspaces of $T_x$ form two points in $\mathbb{P}(E)$? I am having trouble seeing a canonical decomposition of $E_x$ into eigenspaces. | |
| May 6, 2014 at 15:59 | history | edited | David E Speyer | CC BY-SA 3.0 |
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| May 6, 2014 at 13:59 | history | edited | David E Speyer | CC BY-SA 3.0 |
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| May 6, 2014 at 13:45 | history | answered | David E Speyer | CC BY-SA 3.0 |