Spahn examples of (group) schemes (Rev #4)

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

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

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} *

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.

A group scheme is called multiplicative group scheme if the following equivalent conditions are satisfied:

Let $G_const$ dnote a constant group scheme, let $E$ be an étale group scheme. Then we have the following cartier duals:

Revision on July 19, 2012 at 22:42:08 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.