Spahn examples of (group) schemes (Rev #1, changes)

Showing changes from revision #0 to #1: Added | ~~Removed~~ | ~~Chan~~ged

constant (group) scheme

$Sch_k$ is 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

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 $k$ .

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

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 18:27:29 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.