Spahn examples of (group) schemes (Rev #2, changes)

Showing changes from revision #1 to #2: Added | ~~Removed~~ | ~~Chan~~ged

constant (group) scheme
étale (group) scheme
affine (group) scheme
formal (group) scheme
local (=connected) group scheme
multiplicative group scheme
diagonalizable group scheme
unipotent group scheme
smooth formal group scheme
$p$ -divisible group scheme

constant (group) scheme

~~$Sch_k$~~ Recall that is $Spec_k k=*$ ~~copowered (= tensored)?~~ is the terminal object of ~~over~~ $k.Sch$ ~~$Set$~~ .~~. We define the~~ ~~constant $k$ -scheme~~ ~~on a set~~ ~~$E$~~ by

~~$E_k:=E\otimes Sp_k k=\coprod_{e\in E}Sp_k k$~~

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

~~For a scheme $X$ we compute $M_k(E_k,E)=Set(Sp_k k,X)^E=X(k)^E=Set(E,X(k))$ and see that there is an adjunction~~

E_k:=E\otimes *=\coprod_{e\in E}*

~~$((-)_k\dashv (-)(k)):Sch_k\to Set$~~

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 constant-sheaf-global-section adjunction.

étale (group) scheme

An étale $k$ -scheme is defined to be a directed colimit of $k$ -spectra ${Sp Spec}_{k} k^{'}$ ~~Sp_k~~ 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

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 induced by $A$ if $A$ satisfies the dual axioms of a group object.

formal (group) scheme

local (=connected) group scheme

multiplicative group scheme

Definition and Remmark

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

$G\otimes_k k_s$ is diagonalizable.
$G\otimes_k K$ is diagonalizable for a field $K\in M_k$ .
$G$ is the Cartier dual of an étale $k$ -group.
$\hat D(G)$ is an étale $k$ -formal group.
$Gr_k(G,\alpha_k)=0$
(If $p\neq 0)$ , $V_G$ is an epimorphism
(If $p\neq 0)$ , $V_G$ is an isomorphism

Remark

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

$D(G_const)$ is diagonalizable.
$D(E)$ is multiplicative

diagonalizable group scheme

unipotent group scheme

smooth formal group scheme

$p$ -divisible group scheme

Revision on July 19, 2012 at 20:23:17 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.