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Presumably this is treated in detail in chapter 7 of the book The Classical Groups and K-Theory, by A.J.Hahn and O.T.O'Meara. On page 424 it says in theorem 7.2.23 that the elementary subgroup has i …

Let $k$ be a commutative ring and let $G$ be a flat affine algebraic group scheme over $k$. Let $G$ act by algebra automorphisms on the commutative $k$-algebra $A$. So $G(R)$ acts by $R$-algebra auto …

In positive characteristic the only connected groups of finite homological dimension are the tori. We need the following result from Jantzen, Representations of algebraic groups. [J, I 5.13], [J, I …

No. Let $G=SL_n$, acting on its defining representation $V$, with $n\geq2$. Let $R=\mathbb{Z}[X_1,\dots,X_n]$ be the obvious $\mathbb{Z}$-form of the ring of polynomial functions on $V$. Let $p$ be a …

You want to show that $T_eH=\{D\in T_eG\mid\delta_DI\subset I\}$. This is something general about smooth subvarieties of a variety. The left hand side maps to the right hand side by functoriality of t …

To get some clarity it may help to consider the example where $G=\{\pmatrix{1&t\cr 0&e^u}\mid t, u\in \mathbb R\}$. The adjoint action of its Lie algebra is not right for a unipotent group.

Let me summarize. We take a basis $x$, $y$ of $E$ and the characteristic is $p$. There are two cases where there is a surjective map $E\otimes S^r(E)\to S^{r-1}E$. The first case is when $r=p-1$. The …