We describe the relation of certain classes of group schemes. The page numbering refers to the text: ## 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 group scheme|formal k-group]] | [[connected group scheme|connected]] | [[étale group scheme|étale]] | | p.34 | | [[finite k-group]] | [[infinitesimal group scheme|infinitesimal]] | étale | splits if $k$ is perfect |p.35 | | [[affine group scheme|affine k-group]] | [[multiplicative group scheme|multiplicative]] | [[unipotent group scheme|unipotent | splits if $k$ is perfect | p.39 | | connected formal $k$-group | $G_{red}$ is [[smooth group scheme|smooth]] | $G/G_{red}$ is infinitesimal | | p. 43 | +-- {: .num_defn} ###### 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. 1. $b\in Feu_k$ is a formal étale unipotent $k$ group. 1. $c\in Fim_k$ is a formal infinitesimal multiplicative $k$ group. 1. $d\in Fem_k$ is a infinitesimal unipotent $k$ group. =-- ## Duality of k-groups | $D$ | $D(G)$ | | |--|--|--| | affine commutative $k$-group | $\hat D(G)$ is affine commutative formal $k$-group | p.27 | | finite commutative $k$-group | finite commutative $k$-group | p.27 | | [[constant group scheme|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| |$Fim_k$|$Feu_k$| | ## 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 group scheme|multiplicative]] | | | | | | | | | [[étale scheme|étale]] | | | | | | | | | connected | | | | | | | | | infinitesimal | | | | | | | | | [[diagonalizable group scheme|diagonalizable]] | | | | | | | | |p-divisible | | | | | | |$(\mathbb{Q}_p/\mathbb{Z}_p)_k$ and [[the p-divisible group A(p)|A(p)]] |