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Summary (random tour through the examples)

Start Let with the$k$ 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 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 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 copowered (= tensored)? over $Set$. We define the constant $k$-scheme on a set $E$ by

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

This is just the 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

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 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

diagonalizable group scheme

unipotent group scheme

smooth formal group scheme

$p$-divisible group scheme