# Contents * table of contents {:toc} ## Summary (random tour through the examples) Let $k$ be some base field. We start with the **constant group scheme** $E_k$ defined by some classical group $E$ which gives in every component just the group $E$. Next we visit the notion of **étale group scheme**. This is not itself constant but becomes so by scalar extension to the separable closure $k_sep$ of $k$. So far these examples ''do nothing'' with the (multiplicative and additive) structure of the ring in which we evaluate our group scheme. But the next instances do this: We define the **additive-** and the **multiplicative group scheme** by $\alpha_k: R\mapsto R^+$ and $\mu_k:R\mapsto R^\times$ sending a $k$-ring to to its underlying additive- and multiplicative group, respectively. These have the ''function rings'' $O_k(\alpha_k)=k[T]$ and $O_(\mu_k)=K[T,T^{-1}]$ and since $(O_k\dashv Spec_k):k.Ring\stackrel{Spec_k}{\to} k.Aff$ we note that our basic building blocks $\alpha_k$ and $\mu_k$ are in fact representable $k$-functors aka. **affine group schemes**. We observe that we have $k.Gr(\mu_k,\alpha_k)=0$ and call in generalization of this property any group scheme $G$ satisfying $k.Gr(G,\alpha_k)=0$ **multiplicative group scheme**. (We could have also the idea to call $G$ satisfying $k.Gr(\mu_k,G)=0$ ''additive'' but I didn't see this.) By some computation of the hom spaces $k.Gr(G,\mu_k)$ involving co- and birings we see that these are again always values of a representable $k$-functor $D(G)(-):=(-).Gr(G\otimes_k (-),\mu_{(-)})$; this functor we call the **Cartier dual of $G$**. If for example $G$ is a **finite group scheme** $D(G)$ also is, and moreover $D$ is a contravariant autoequivalence (''duality'') of $k.fin.comm.Grp$; in general it is also a **[[nLab:duality]]** in some specific sense. By taking the Cartier dual $D(E_k)$ of a constant group scheme we obtain the notion of a **diagonlizable group scheme**. To justify this naming we compute some value $D(E_k)(R)=hom_{Grp}(E_k\otimes_k R, \mu_R)\simeq hom_{Grp}(E_k,R^\times)\simeq hom_Alg(k[E_k],R)$ where $k[E_k]$ denotes the [[group algebra]] of $E_k$ and the last isomorphism is due to the universal property of group rings; we observe that the last equality tells us that $G=Spec\,k[E_k]$ and recall that a $\zeta\in E_k\subset k[E_k]$ is called a [[nLab:character]] of $G$ (and one calls a group generated by these ''diagonalizable''). ## constant (group) scheme Recall that $Spec_k k=*$ is the terminal object of $k.Sch$. $k.Sch$ is [[copower|copowered (= tensored)]] over $Set$. We define the _constant $k$-scheme_ on a set $E$ by $$E_k:=E\otimes *=\coprod_{e\in E}*$$ For a scheme $X$ we compute $M_k(E_k,E)=Set(*,X)^E=X(k)^E=Set(E,X(k))$ and see that there is an adjunction $$((-)_k\dashv (-)(k)):k.Sch\to Set$$ This is just the [[nLab:global section|constant-sheaf-global-section adjunction]]. ## étale (group) scheme An _étale $k$-scheme_ is defined to be a directed colimit of $k$-spectra $Spec_k k^\prime$ of finite separable field-extensions $k^\prime$ of $k$. For an étal group scheme $X=colim_{k^\prime \in T} Spec_k k^\prime$ we have $$X\otimes_k k_sep\simeq(colim_{k^\prime \in T}Spec_k k^\prime)\otimes_k k_sep\simeq colim_{k^\prime \in T} Spec_{k_sep} k^\prime\simeq colim_{k^\prime \in T} *$$ ## affine (group) scheme (see also [[coalgebras, corings and birings in the theory of group shemes]]) An affine $k$-scheme $G:=Spec_k A$ is a representable object in $k.Fun$. We obtain a group law $G\times G\to G$ induced by $A$ if $A$ satisfies the dual axioms of a [[nLab:group object]]. We denote the structure maps called comultiplication, counit, and converse by $\Delta:A\to A\otimes A$ $\epsilon: A\to *$ $\sigma:A\to A$ ### Examples The additive group $\alpha_k$ The multiplicative group $\mu_k$ The kernels of group homomorphisms. In particular the kernel $ker\, (-)^n:\mu_k\to \mu_k$. ### Mapping spaces ## 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