# nLab 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

(…)

## 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 k-groupconnected?étale?p.34
finite k-group?infinitesimal?étalesplits if $k$ is perfectp.35
affine k-group?multiplicativeunipotentsplits if $k$ is perfectp.39
connected formal $k$-group$G_{red}$ is smooth?$G/G_{red}$ is infinitesimalp.43
###### Definition

(p. 39)

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

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

where

1. $a\in Fem_k$ is a formal étale multiplicative $k$ group.

2. $b\in Feu_k$ is a formal étale unipotent $k$ group.

3. $c\in Fim_k$ is a formal infinitesimal multiplicative $k$ group.

4. $d\in Fem_k$ is a infinitesimal unipotent $k$ group.

## Duality of k-groups

$D$$D(G)$
affine commutative $k$-group$\hat D(G)$ is affine commutative formal $k$-groupp.27
finite commutative $k$-groupfinite commutative $k$-groupp.27
constant k-group?diagonalizable $k$-groupp.36
étale k-groupmultiplicative k-groupp.37
multiplicative k-group$\hat D(G)$ is étale formal $k$-groupp.37
unipotent k-group$\hat D(G)$ connected formal groupp.38
$Fim_k$$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$
multiplicativediagonalizablediagonalizablediagonalizablep.38
étaleconstantconstantp.17

## Examples of $k$-groups

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

Last revised on August 1, 2018 at 18:28:14. See the history of this page for a list of all contributions to it.