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

Summary (Constructing examples of group schemes from basic examples)

(…)

Decomposition of k-groups

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

0G exGG ex00\to G^{ex}\to G\to G_{ex}\to 0
kk-groupG exG^{ex}G exG_{ex}
formal k-groupconnected?étale?p.34
finite k-group?infinitesimal?étalesplits if kk is perfectp.35
affine k-group?multiplicativesmooth?G/G redG/G_{red} is infinitesimalp.43
Definition

(p. 39)

If kk is perfect any finite affine kk-group GG is in a unique way the product of four subgroups

G=a×b×c×dG=a\times b\times c\times d

where

  1. aFem ka\in Fem_k is a formal étale multiplicative kk group.

  2. bFeu kb\in Feu_k is a formal étale unipotent kk group.

  3. cFim kc\in Fim_k is a formal infinitesimal multiplicative kk group.

  4. dFem kd\in Fem_k is a infinitesimal unipotent kk group.

Duality of k-groups

DDD(G)D(G)
affine commutative kk-groupD^(G)\hat D(G) is affine commutative formal kk-groupp.27
finite commutative kk-groupfinite commutative kk-groupp.27
constant k-group?diagonalizable kk-groupp.36
étale k-groupmultiplicative k-groupp.37
multiplicative k-groupD^(G)\hat D(G) is étale formal kk-groupp.37
unipotent k-groupD^(G)\hat D(G) connected formal groupp.38
Fim kFim_kFeu kFeu_k

Skalar extension and skalar restriction

Let KM kK\in M_k be a field, let k sk_s be the separable clusure of kk, let k¯\overline k denote the algebraic closure of kk.

GGG kKG\otimes_k KG kk sG\otimes_k k_sG kk¯G\otimes_k \overline k
multiplicativediagonalizablediagonalizablediagonalizablep.38
étaleconstantconstantp.17

Examples of kk-groups

unipotentmultiplicativeétaleconnectedinfinitesimaldiagonalizablep-divisible
unipotent
multiplicative
étale
connected
infinitesimal
diagonalizable
p-divisible( p/ p) k(\mathbb{Q}_p/\mathbb{Z}_p)_k and A(p)
Revised on July 20, 2012 21:49:51 by Stephan Alexander Spahn (79.219.126.169)