We describe the relation of certain classes of group schemes.
The page numbering refers to the text:
(…)
One way the characterize certain classes of $k$-groups is via exact sequences
$k$-group | $G^{ex}$ | $G_{ex}$ | ||
---|---|---|---|---|
formal k-group | connected? | étale? | p.34 | |
finite k-group? | infinitesimal? | étale | splits if $k$ is perfect | p.35 |
affine k-group? | multiplicative | smooth? | $G/G_{red}$ 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
where
$a\in Fem_k$ is a formal étale multiplicative $k$ group.
$b\in Feu_k$ is a formal étale unipotent $k$ group.
$c\in Fim_k$ is a formal infinitesimal multiplicative $k$ group.
$d\in Fem_k$ is a infinitesimal unipotent $k$ group.
$D$ | $D(G)$ | |
---|---|---|
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 | p.38 |
étale | constant | constant | p.17 |
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) |