# 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 $k$. The importance of étale affine is that the category of them is equivalent to that of [[nLab:Galois module|Galois modules]] by $E\mapsto E \otimes_k k_sep=\cup_{K/k \,sep\,fin} E(K)$ 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''). Revisiting the condition $k.Gr(H,\alpha_k)=0$ by which we defined multiplicative group schemes and considering a group scheme $G$ satisfying this condition for all **sub group-schemes** $H$ of $G$ we arrive at the notion of **unipotent group scheme**. By the structure theorem of **decomposition of affine groups** we can proof that $G$ is unipotent iff the **completion of group schemes** (which gives us-by the usual technic of [[nLab:completion]]- a **formal (group) scheme** $\hat X$ if $X$ is a group scheme) of the Cartier dual of $G$, i.e. $\hat D(G)$ is a **connected formal group scheme** also called **local group scheme** since a local group scheme $Q=Spec_k A$ is defined to be the [[nLab:spectrum]] of a local ring; this requirement in turn is equivalent to $Q(K)=hom(A,K)=\{0\}$ hence the first name ''connected''. There is also a connection between connected and étale schemes: For any formal group scheme there is an essentially unique [[nLab:exact sequence]] $0\to G^\circ\to G\to \pi_0(G)\to 0$ where $G^\circ$ is connected and $\pi_0(G)$ is étale. Such decomposition in exact sequences we obtain in further cases: $0\to G^{ex}\to G\to G_{ex}\to 0$ where |$k$-group | $G^{ex}$ | $G_{ex}$ | | | |-------|-------|-------|-------|--------| | [[nLab:formal scheme|formal]] | [[nLab:connected object|connected]] | [[étale group scheme|étale]] | | p.34 | | finite | [[nLab:infinitesimal object|infinitesimal]] | étale | splits if $k$ is perfect |p.35 | | [[nLab:ring spectrum|affine]] | multiplicative | [[unipotent group scheme|unipotent | splits if $k$ is perfect | p.39 | | connected | $G_{red}$ is [[nLab:smooth scheme|smooth]] | $G/G_{red}$ is infinitesimal |p.43 | | where a **smooth (group) scheme** is defined to be the spectrum of a finite dimensional (over k) [[nLab:power series]] algebra, a (group) scheme is called **finite (group) scheme** if we restrict in all necessary definitions to $k$-ring which are finite dimensional $k$-[[nLab:vector space|vector spaces]], and a (group) scheme is called **infinitesimal (group) scheme** if it is finite and local. If moreover $k$ is a perfect field 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 $a\in Fem_k$ 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, and $d\in Fem_k$ is a infinitesimal unipotent $k$ group. ## 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