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

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constant (group) scheme

Recall that Spec kk=*Spec_k k=* is the terminal object of k.Schk.Sch.

k.Schk.Sch is copowered (= tensored)? over SetSet. We define the constant kk-scheme on a set EE by

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

For a scheme XX we compute M k(E k,E)=Set(*,X) E=X(k) E=Set(E,X(k))M_k(E_k,E)=Set(*,X)^E=X(k)^E=Set(E,X(k)) and see that there is an adjunction

(() k()(k)):k.SchSet((-)_k\dashv (-)(k)):k.Sch\to Set

This is just the constant-sheaf-global-section adjunction.

étale (group) scheme

An étale kk-scheme is defined to be a directed colimit of kk-spectra Spec kk Spec_k k^\prime of finite separable field-extensions k k^\prime of kk.

For an étal group scheme X=colim k TSpec kk X=colim_{k^\prime \in T} Spec_k k^\prime we have

X kk sep(colim k TSpec kk ) kk sepcolim k TSpec k sepk colim k T*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 kk-scheme G:=Spec kAG:=Spec_k A is a representable object in k.Funk.Fun.

We obtain a group law G×GGG\times G\to G induced by AA if AA 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:

  1. G kk sG\otimes_k k_s is diagonalizable.

  2. G kKG\otimes_k K is diagonalizable for a field KM kK\in M_k.

  3. GG is the Cartier dual of an étale kk-group.

  4. D^(G)\hat D(G) is an étale kk-formal group.

  5. Gr k(G,α k)=0Gr_k(G,\alpha_k)=0

  6. (If p0)p\neq 0), V GV_G is an epimorphism

  7. (If p0)p\neq 0), V GV_G is an isomorphism

Remark

Let G constG_const dnote a constant group scheme, let EE be an étale group scheme. Then we have the following cartier duals:

  1. D(G const)D(G_const) is diagonalizable.

  2. D(E)D(E) is multiplicative

diagonalizable group scheme

unipotent group scheme

smooth formal group scheme

pp-divisible group scheme

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.