Spahn relation of certain classes of group schemes (Rev #2, changes)

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Decomposition of k-groups

One way the characterize certain classes of $k$ -groups is via exact sequences

0\to G^{ex}\to G\to G_{ex}\to 0

$k$ -group	$G^{ex}$	$G_{ex}$
~~formal~~ formal k-group? ~~$k$~~ ~~-group~~	~~connected~~connected?	~~étale~~étale		p.34
~~finite k-group~~finite k-group?	~~infinitesimal~~infinitesimal?	étale	splits if $k$ is perfect	p.35
~~affine k-group~~affine k-group?	~~multiplicative~~multiplicative	~~unipotent~~smooth?	~~splits if~~ $G/G_{red}$ ~~$k$~~ is infinitesimal ~~is perfect~~	~~p.39~~
connected formal $k$ -group	$G_{red}$ is smooth	$G/G_{red}$ is infinitesimal		p. 43

(p. 39)

If $k$ is perfect any finite affine $k$ -group $G$ is in a unique way the product of four subgroups

G=a\times b\times c\times d

where

$D$	$D(G)$
affine commutative $k$ -group	$\hat D(G)$ is affine commutative formal $k$ -group	p.27
finite commutative $k$ -group	finite commutative $k$ -group	p.27
constant $k$ -group	diagonalizable $k$ -group	p.36
étale k-group	multiplicative k-group	p.37
multiplicative k-group	$\hat D(G)$ is étale formal $k$ -group	p.37
unipotent k-group	$\hat D(G)$ connected formal group	p.38
$Fim_k$	$Feu_k$

Let $K\in M_k$ be a field, let $k_s$ be the separable clusure of $k$ , let $\overline k$ denote the algebraic closure of $k$ .


$G$	$G\otimes_k K$	$G\otimes_k k_s$	$G\otimes_k \overline k$
multiplicative	diagonalizable	diagonalizable	diagonalizable

Revision on May 31, 2012 at 16:48:28 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.