Spahn examples of (group) schemes (Rev #7, changes)

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Summary (random tour through the examples)

Start with the constant group scheme E kE_k defined by some classical group EE which gives in every component just the group EE. 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 sepk_sep of kk. 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 α k:RR +\alpha_k: R\mapsto R^+ and μ k:RR ×\mu_k:R\mapsto R^\times sending a kk-ring to to its underlying additive- and multiplicative group, respectively. These have the ‘’function rings’‘ O k(α k)=k[T]O_k(\alpha_k)=k[T] and O (μ k)=K[T,T 1]O_(\mu_k)=K[T,T^{-1}] and since (O kSpec k):k.RingSpec kk.Aff(O_k\dashv Spec_k):k.Ring\stackrel{Spec_k}{\to} k.Aff we note that our basic building blocks α k\alpha_k and μ k\mu_k are in fact representable kk-functors aka. affine group schemes. We observe that we have k.Gr(μ k,α k)=0k.Gr(\mu_k,\alpha_k)=0 and call in generalization of this property any group scheme GG satisfying k.Gr(G,α k)=0k.Gr(G,\alpha_k)=0 multiplicative group scheme. (We could have also the idea to call GG satisfying k.Gr(μ k,G)=0k.Gr(\mu_k,G)=0 ‘’additive’‘ but I didn’t see this.)

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

(see also coalgebras, corings and birings in the theory of group shemes)

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. We denote the structure maps called comultiplication, counit, and converse by

Δ:AAA\Delta:A\to A\otimes A

ϵ:A*\epsilon: A\to *

σ:AA\sigma:A\to A

Examples

The additive group α k\alpha_k

The multiplicative group μ k\mu_k

The kernels of group homomorphisms. In particular the kernel ker() n:μ kμ kker\, (-)^n:\mu_k\to \mu_k.

Mapping spaces

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 20, 2012 at 18:03:48 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.