Spahn
category theoretic aspects of the theory of group schemes

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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 the category 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

and it restricts moreover to an adjoint equivalence

(O kSpec k):k.BiRingSpec kk.Aff.comm.Gr(O_k\dashv Spec_k):k.Bi Ring\stackrel{Spec_k}{\to} k.Aff.comm.Gr

between the categories of kk-birings and the category of commutative affine kk-group schemes. To see this be aware that a kk-biring is a commutative ring object in k.Ring.comm opk.Aff.Sch{k.Ring.comm}^{op}\simeq k.Aff.Sch (where the latter denotes the category of affine schemes).

(Group) schemes

A kk-functor is called a kk-scheme if it is a sheaf for the Zariski Grothendieck topology on k.Ring opk.Ring^{op}.

We will consider the moral of this op-ing below.

To give more details, recall that the closed sets of the Zariski topology on the spectrum SpecASpec A of a kk-ring AA is defined by

V(I):={PSpecA|IP}V(I):=\{P\in Spec \, A|I\subseteq P\}

We can characterize the the elements of V(a)V(a) also by

e a(P)=0iffPV(a)e_a(P)=0\, iff\, P\in V(a)

where

e a:{Spec(A)Quot(A/P) PamodP1e_a:\begin{cases} Spec (A)\to Quot(A/P) \\ P\mapsto \frac{a\,mod\,P}{1} \end{cases}

where Quot(A/P)Quot(A/P) denotes the quotient field (aka. field of fractions) of the integral domain A/PA/P.

This construction generalizes to kk-functors by defining an open subfunctor of a kk-functor XX by

V(I):R{xX(R)|Ix}V(I):R\mapsto\{x\in X(R)| I\subseteq x\}

where IO(X)I\subseteq O(X). By the above alternative characterization, the assigned set consists precisely of those xx for which f(x)=0f(x)=0 for all fIf\in I.

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.

We denote the category of kk-schemes by k.Schk.Sch.

A kk-group scheme is a group object in k.Schk.Sch. The category of kk-group schemes we denote by k.Grpk.Grp.

Formal (group) scheme

Let kk be a field. A finite kk-ring is defined to be a kk-algebra which is a finite dimensional kk-vector space. The category of finite kk-rings we denote by fin.k.Ringfin.k.Ring. A finite kk-functor is defined to be a covariant functor X:fin.k.RingSetX:fin.k.Ring\to Set. The category of finite kk-functors we denote by fin.k.Funfin.k.Fun. A finite kk-scheme is defined to be kk-scheme which is a finite kk-functor. The category of finite kk-schemes we denote by fin.k.Schfin.k.Sch. Analogously we define the category of finite group schemes.

A formal group scheme is defined to be a codirected colimit of finite kk-schemes.

Recall that we have a covariant embedding

Spec k:k.Ring opk.FunSpec_k:k.Ring^{op}\to k.Fun

but we equivalently an embedding

Spec *:k.coRingk.FunSpec^*:k.co Ring\to k.Fun

where by k.coRingk.co Ring we denote the category of kk-corings. A coring is a comonoid in the category of affine schemes (the latter is the opposite category of k.Ringk.Ring). If we restrict to finite kk-rings by linear algebra we have a bijection A *R=hom(A,k)hom(A,R)A^*\otimes R=hom(A,k)\simeq hom(A,R) and can write

Spec kC:R{uA *R|Δ Ru=uu,ϵ Ru=1}Spec_k C: R\mapsto\{u\in A^*\otimes R|\Delta_R u=u\otimes u, \epsilon_R u=1\}

where CC is a kk-coring, RR a finite kk-ring, Δ R\Delta_R the skalar-extended comultiplication, ϵ R\epsilon_R the skalar-extended counit.

Examples of (group) schemes

examples of (group) schemes

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

Last revised on August 22, 2012 at 13:04:31. See the history of this page for a list of all contributions to it.