# Contents * table of contents {:toc} ## constant (group) scheme $Sch_k$ is [[copower|copowered (= tensored)]] over $Set$. We define the _constant $k$-scheme_ on a set $E$ by $$E_k:=E\otimes Sp_k k=\coprod_{e\in E}Sp_k k$$ For a scheme $X$ we compute $M_k(E_k,E)=Set(Sp_k k,X)^E=X(k)^E=Set(E,X(k))$ and see that there is an adjunction $$((-)_k\dashv (-)(k)):Sch_k\to Set$$ ## étale (group) scheme An _étale $k$-scheme_ is defined to be a directed colimit of $k$-spectra $Sp_k k^\prime$ of finite separable field-extensions $k^\prime$ of $k$. ## affine (group) scheme An affine $k$-scheme $G:=Spec_k A$ is a representable object in $k.Fun$. We obtain a group law $G\times G\to G$ if ## formal (group) scheme ## local (=connected) group scheme ## multiplicative group scheme +-- {: .num_defn} ###### Definition and Remmark A group scheme is called *multiplicative group scheme* if the following equivalent conditions are satisfied: 1. $G\otimes_k k_s$ is [[diagonalizable group scheme|diagonalizable]]. 1. $G\otimes_k K$ is diagonalizable for a field $K\in M_k$. 1. $G$ is the Cartier dual of an étale $k$-group. 1. $\hat D(G)$ is an étale $k$-formal group. 1. $Gr_k(G,\alpha_k)=0$ 1. (If $p\neq 0)$, $V_G$ is an epimorphism 1. (If $p\neq 0)$, $V_G$ is an isomorphism =-- +-- {: .num_remark} ###### Remark Let $G_const$ dnote a [[constant group scheme]], let $E$ be an [[étale group scheme]]. Then we have the following [[Cartier duality|cartier duals]]: 1. $D(G_const)$ is [[diagonalizable group scheme|diagonalizable]]. 1. $D(E)$ is [[multiplicative group scheme|multiplicative]] =-- ## diagonalizable group scheme ## unipotent group scheme ## smooth formal group scheme ## $p$-divisible group scheme