Spahn category theoretic aspects of the theory of group schemes (Rev #1)

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(Group) functors and affine (group) schemes

Let kk be a ring. Let k.Ringk.Ring denote the category of kk-rings. Let k.Funk.Fun denote the category of (contravariant) functors X:k:RingSetX:k:Ring\to Set. Let k.Affk.Aff denote the category of representable kk-functors; we call this category thecategory of affine kk-schemes and an object of this category we write as

Spec kA:{k.RingSet Rhom(A,R)Spec_k A:\begin{cases} k.Ring\to Set \\ R\mapsto hom(A,R) \end{cases}

We obtain in this way a functor

Spec k:k.Ringk.FunSpec_k:k.Ring\to k.Fun

This functor has a left adjoint

(O kSpec k):k.RingSpec kk.Fun(O_k\dashv Spec_k):k.Ring\stackrel{Spec_k}{\to} k.Fun

assigning to a kk-functor its ring of functions. This adjunction restricts to an adjoint equivalence

(O kSpec k):k.RingSpec kk.Aff(O_k\dashv Spec_k):k.Ring\stackrel{Spec_k}{\to} k.Aff

Constant (group) scheme

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

E k:=ESp kk= eESp kkE_k:=E\otimes Sp_k k=\coprod_{e\in E}Sp_k k

For a scheme XX we compute M k(E k,E)=Set(Sp kk,X) E=X(k) E=Set(E,X(k))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()(k)):Sch kSet((-)_k\dashv (-)(k)):Sch_k\to Set

If EE is a group E kE_k is a group scheme.

Étale (group) scheme

(see also Grothendieck's Galois theory)

An étale group scheme over a field kk is defined to be a directed colimit

colim (kk )TSepSpeck colim_{(k\hookrightarrow k^\prime)\in T\subseteq Sep} Spec\, k^\prime

where TT denotes some set of finite separable field extensions of kk.

Cartier dual of a finite flat commutative group scheme

Let GG be a commutative kk-group functor (in cases of interest this is a finite flat commutative group scheme). Then the Cartier dual D(G)D(G) of GG is defined by

D(G)(R):=Gr R(G kR,μ R)D(G)(R):=Gr_R(G\otimes_k R,\mu_R)

where μ k\mu_k denotes the group scheme assigning to a ring its multiplicative group R ×R^\times consisting of the invertible elements of RR.

This definition deserves the name duality since we have

hom(G,D(H))=hom(H,D(G))=hom(G×H,μ k)hom(G,D(H))=hom(H,D(G))=hom(G\times H,\mu_k)

p-torsion

Let s:RSs:R\to S be a morphism of rings. Then we have an adjunction

(s *s *):S.Mods *R.Mod(s^*\dashv s_*):S.Mod\stackrel{s_*}{\to} R.Mod

from the category of SS-modules to that of RR-modules where

s *:AA sSs^*:A\mapsto A\otimes_s S

is called scalar extension and s *s_* is called scalar restriction.

If XX denotes some scheme over a kk-ring for kk being a field of characteristic pp, we define its pp-torsion component-wise by X (p)(R):=X(s *R)X^{(p)}(R):=X(s_* R).

p-divisible groups

Witt rings and Dieudonné modules

Revision on July 19, 2012 at 13:38:30 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.