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relations of certain classes of group schemes

We describe the relation of certain classes of group schemes.

The page numbering refers to the text:

• Michel Demazure, lectures on p-divisible groups web

Summary (Constructing examples of group schemes from basic examples)

(…)

Decomposition of k-groups

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

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

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

Definition

(p. 39)

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

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

where

1. $a\in {\mathrm{Fem}}_k$ is a formal étale multiplicative $k$ group.

2. $b\in {\mathrm{Feu}}_k$ is a formal étale unipotent $k$ group.

3. $c\in {\mathrm{Fim}}_k$ is a formal infinitesimal multiplicative $k$ group.

4. $d\in {\mathrm{Fem}}_k$ is a infinitesimal unipotent $k$ group.

Duality of k-groups

$D$	$D(G)$
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
${\mathrm{Fim}}_k$	${\mathrm{Feu}}_k$

Skalar extension and skalar restriction

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	p.38
étale		constant	constant	p.17

Examples of $k$-groups

	unipotent	multiplicative	étale	connected	infinitesimal	diagonalizable	p-divisible
unipotent
multiplicative
étale
connected
infinitesimal
diagonalizable
p-divisible							$({\mathbb{Q}}_p/{\mathbb{Z}}_p{)}_k$ and A(p)