4
votes
Accepted
Definition of a character of a linear algebraic group
Yes, please don't feel bad about this confusion. It is for semi-historical reasons as KReiser rightly pointed out above, and can be quite confusing.
Namely, let us fix a field $K$ and an (affine) ...
- 51.9k
2
votes
Accepted
If a hom. $\phi:G\to H$ of diagonalisable linear algebraic groups is injective, then the induced hom. $\phi^*:X^*(H)\to X^*(G)$ is surjective
Unless I'm missing something here, the exercise is wrong as stated.
Let $k$ of characteristic $p>0$ and let $G=H=\Bbb G_m$. Let $\phi:G \to G$ be the map that raises everything to the $p$-th power....
- 16.9k
2
votes
Accepted
Weil restriction of a base change
No, they need not be isomorphic as schemes.
It might be helpful to see an example. Let $\mathbb{C}/\mathbb{R}$ be an extension of fields.
Consider, some algebraic group over $\mathbb{R}$. I will take ...
- 2,275
1
vote
Let $G$ be a linear algebraic group and $C_G(x)$ be the centralizer of $x$. Show that $C_G(x)$ is a closed subgroup for all $x\in G$.
Let $g_nx = x g_n$ for all $g_n \in G$. If $g_n \rightarrow g$ then $g_nx \rightarrow gx$ and $x g_n \rightarrow xg$ and since $g_n x = x g_n$, we have that $xg = gx$. Hence $g \in C_G$. Hence $C_G(x)$...
- 2,750
1
vote
CW complex structure on the Flag variety
I would suggest this writeup, although it’s for Grassmannians:
https://www.mathematicalgemstones.com/uncategorized/the-cw-complex-structure-of-the-grassmannian/
- 5,507
1
vote
Accepted
Correspondence between ideals of Lie algebra and connected normal subgroups of connected algebraic group
Let $G$ be a smooth connected (affine) algebraic group over some ground field $k$ and write $\mathfrak{g}$ for its Lie algebra.
As was already pointed out in Armado's answer, we have the adjoint ...
- 413
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