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group scheme

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Definition

A group scheme is a group object in the category of schemes (or in a category of some schemes as for instance that of schemes over a fixed base scheme); in particular a group scheme is a group functor. As explained at group object there are two equivalent ways of realizing this:

One way is to define it as a functor G:CRingSetG:C Ring\to Set equipped with a transformation m:G×GGm:G\times G\to G satisfying the properties spelled out at group object.

The other way is to define it as a functor SchGrpSch\to Grp from the category of schemes to that of (discrete) groups whose composition with the forgetful functor GrpSetGrp\to Set is representable.

Grothendieck emphasized the study of schemes over a fixed base scheme. Following this idea in the functor of points formalism, a group scheme over a scheme XX is a functor

G:(Sch/X) opGrp G: (Sch /X)^{op} \to Grp

A morphism of group schemes f:GHf:G\to H is a morphism of schemes that is a group homomorphism on any choice of values of points. This is more easily stated by saying that a morphism of group schemes must be a natural transformation between the functor of points; i.e. ff is required to be a natural transformation of functors with values in the category GrpGrp of groups (instead of with values in Set); an equivalent way to state this is that ff needs to satisfy fm=m(f×f)f m=m(f\times f) if m:G×GGm:G\times G\to G denotes the group law on GG.

This construction generalizes to ind-schemes (as for example formal schemes) to that of a formal group scheme.

Overview

Let kk be some base field. We 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. The importance of étale affine is that the category of them is equivalent to that of Galois modules by EE kk sep= K/ksepfinE(K)E\mapsto E \otimes_k k_sep=\cup_{K/k \,sep\,fin} E(K)

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.) By some computation of the hom spaces k.Gr(G,μ k)k.Gr(G,\mu_k) involving co- and birings we see that these are again always values of a representable kk-functor D(G)():=().Gr(G k(),μ ())D(G)(-):=(-).Gr(G\otimes_k (-),\mu_{(-)}); this functor we call the Cartier dual of GG. If for example GG is a finite group scheme D(G)D(G) also is, and moreover DD is a contravariant autoequivalence (”duality”) of k.fin.comm.Grpk.fin.comm.Grp; in general it is also a duality in some specific sense. By taking the Cartier dual D(E k)D(E_k) of a constant group scheme we obtain the notion of a diagonlizable group scheme. To justify this naming we compute some value D(E k)(R)=hom Grp(E k kR,μ R)hom Grp(E k,R ×)hom Alg(k[E k],R)D(E_k)(R)=hom_{Grp}(E_k\otimes_k R, \mu_R)\simeq hom_{Grp}(E_k,R^\times)\simeq hom_Alg(k[E_k],R) where k[E k]k[E_k] denotes the group algebra of E kE_k and the last isomorphism is due to the universal property of group rings; we observe that the last equality tells us that G=Speck[E k]G=Spec\,k[E_k] and recall that a ζE kk[E k]\zeta\in E_k\subset k[E_k] is called a character of GG (and one calls a group generated by these ”diagonalizable”). Revisiting the condition k.Gr(H,α k)=0k.Gr(H,\alpha_k)=0 by which we defined multiplicative group schemes and considering a group scheme GG satisfying this condition for all sub group-schemes HH of GG we arrive at the notion of unipotent group scheme. By the structure theorem of decomposition of affine groups we can proof that GG is unipotent iff the completion of group schemes (which gives us-by the usual technic of completion- a formal (group) scheme X^\hat X if XX is a group scheme) of the Cartier dual of GG, i.e. D^(G)\hat D(G) is a connected formal group scheme also called local group scheme since a local group scheme Q=Spec kAQ=Spec_k A is defined to be the spectrum of a local ring; this requirement in turn is equivalent to Q(K)=hom(A,K)={0}Q(K)=hom(A,K)=\{0\} hence the first name ”connected”. There is also a connection between connected and étale schemes: For any formal group scheme there is an essentially unique exact sequence

(1)0G Gπ 0(G)00\to G^\circ\to G\to \pi_0(G)\to 0

where G G^\circ is connected and π 0(G)\pi_0(G) is étale. Such decomposition in exact sequences we obtain in further cases: 0G exGG ex00\to G^{ex}\to G\to G_{ex}\to 0 where

kk-groupG exG^{ex}G exG_{ex}
formalconnectedétale?p.34
finiteinfinitesimalétalesplits if kk is perfectp.35
affinemultiplicativesmoothG/G redG/G_{red} is infinitesimalp.43

where a smooth (group) scheme is defined to be the spectrum of a finite dimensional (over k) power series algebra, a (group) scheme is called finite (group) scheme if we restrict in all necessary definitions to kk-ring which are finite dimensional kk-vector spaces, and a (group) scheme is called infinitesimal (group) scheme if it is finite and local. If moreover kk is a perfect field any finite affine kk-group GG is in a unique way the product of four subgroups G=a×b×c×dG=a\times b\times c\times d where aFem ka\in Fem_k is a formal étale multiplicative kk group, bFeu kb\in Feu_k is a formal étale unipotent kk group, cFim kc\in Fim_k is a formal infinitesimal multiplicative kk group, and dFem kd\in Fem_k is a infinitesimal unipotent kk group.

If we now shift our focus to colimits- or more generally to codirected systems of finite group schemes, in particular the notion of p-divisible group is an extensively studied case because the pp-divisible group G(p)G(p) of a group scheme encodes information on the p-torsion of the group scheme GG. To appreciate the definition of G(p)G(p) we first recall that for any group scheme GG we have the relative Frobenius morphism F G:GG (p)F_G:G\to G^{(p)} to distinguish it from the absolute Frobenius morphism F G abs:GGF^{abs}_G:G\to G which is induced by the Frobenius morphism of the underlying ring kk. The passage to the relative Frobenius is necessary since in general it is not true that the absolute Frobenius respects the base scheme. Now we define G[p n]:=kerF G nG[p^n]:=ker\; F^n_G where the kernel is taken of the Frobenius iterated nn-times and the codirected system

G[p]pG[p 2]ppG[p n]pG[p n+1]pG[p]\stackrel{p}{\to}G[p^2]\stackrel{p}{\to}\dots \stackrel{p}{\to}G[p^{n}]\stackrel{p}{\to}G[p^{n+1}]\stackrel{p}{\to}\dots

is then called the pp-divisible group of GG. As cardinality (in group theory also called rank) of this objects we have card(G[p j])=p jhcard(G[p^j])=p^{j\cdot h} for some hh\in \mathbb{N}; this hh is called the height of GG. Moreover we have (p1) the G[p i]G[p^i] are finite group schemes (we assumed this by definition), (p2) the sequences of the form 0kerp jι jkerp j+kp jkerp k00\to ker\, p^j\xhookrightarrow{\iota_j} ker p^{j+k}\stackrel{p^j}{\to}ker p^k\to 0 are exact, (p3) G= jkerp jid GG=\cup_j ker\, p^j\cdot id_G and one can show that if we start with any codirected system (G i) i(G_i)_{i\in \mathbb{N}} satisfying (p1)(p2) we have that colim iG icolim_i G_i satisfies (p3) and ker(F G n)G nker( F^n_G)\simeq G_n - in other words the properties (p1)(p2) give an equivalent alternative definition of pp-divisible groups (and (p3) leads some authors to ”identify” GG and G(p)G(p)). Basic examples of pp-divisible groups are ( p/ p) k h(\mathbb{Q}_p/\mathbb{Z}_p)^h_k which is (up to isomorphism) the unique example of a constant pp-divisible group of height hh and A(p)A(p) where AA is a commutative variety with a group law (aka. algebraic group). A(p)A(p) is called the Barsotti-Tate group of an abelian variety; if the dimension of AA is gg the height of A(p)A(p) is 2g2g. Now, what about decomposition of pp-divisible groups? We have even one more equivalent ”exactness” characterization of pp-divisible formal groups by: GG is pp-divisible iff in the connected-étale decomposition given by the exact sequence displayed in (1) we have ,(p1 p1^\prime), π 0(G)(k¯)( p/ p) r\pi_0(G)(\overline k)\simeq (\mathbb{Q}_p/\mathbb{Z}_p)^r for some rr\in \mathbb{N} and ,(p2 p2^\prime), G G^\circ is of finite type (= the spectrum of a Noetherian ring), smooth, and the kernel of its Verschiebung morphism (this is the left adjoint the Frobenius morphism) is finite. Of course this characterization of pp-divisiblity by exact sequences gives rise to propositions on dimensions and subgroups of pp-divisible groups.

(…)

In cases where kk is a field of prime characteristic pp, there is some special kk-functor which is a group functor and even a ring functor (a kk-functor equipped with a ring structure) - namely the functor W:k.Ring.commλ.RingSetW:k.Ring. comm\to \lambda.Ring\hookrightarrow Set whose image is the category Λ\Lambda of lambda-rings; the objects W(R)W(R) of Λ\Lambda are also called Witt vectors since they are infinite sequences of elements of RR (this justifies at least ”vectors”). WW possesses a left adjoint (VdasvW)(V\dasv W) forgetting the lambda-structure and the couniversal property? associated to this adjunction states that for a kk-ring RR we have that W(R)W(R) is the couniversal object such that all so called Witt polynomials w n(x 0,x n):=x 0 p n+px 1 p n1+p 2x 2 p n2++p nx nw_n(x_0,\dots x_n):=x_0^{p^n}+p\cdot x_1^{p^{n-1}}+p^2 \cdot x_2^{p^{n-2}}+\dots+p^n\cdot x_n are ring homomorphisms. For this special kk-group WW we revisit some construction we have done above for general kk-groups: we firstly make the eponymous remark that the Verschiebung morphism V W(R):(a 1,a 2,,a n,)(0,a 1,a 2,,a n,)V_W(R):(a_1,a_2,\dots,a_n,\dots)\mapsto (0,a_1,a_2,\dots,a_n,\dots) is given by shifting (German: Verschiebung) one component to the right. By abstract nonsense we have also Frobenius. An important proposition concerning the ring of Witt vectors is that for a perfect field kk, W(k)W(k) is a discrete valuation ring. The next construction we visit with W(R)W(R) is Cartier duality of finite Witt groups (here we forget that W(R)W(R) is even a ring): For this note that the ring of finite Witt vectors W fin(R)W_fin(R) is an ideal in W(R)W(R) and we have Frobenius and Verschiebung also in this truncated case; more precisely we have for each nn a Frobenius F W n:W nW nF_{W_n}:W_n\to W_n where W n(R)W_n(R) denotes the ring of Witt vectors of length nn. With this notation we find ker(F W n m)D(ker(F W n n)ker(F^m_{W_n})\simeq D(ker(F^n_{W_n}).

Since W(k)W(k) is a ring we can ask of its modules in general; however there is in particular one W(k)W(k)-module of interest which is called the Dieudonné module M(G)M(G) of GG. It can be defined in two equivalent ways: 1. as a W(k)W(k)-module MM equipped with two endomorphisms of FF and VV satisfying the ”Witt-Frobenius identities

(WF1): FV=VF=pFV=VF=p

(WF2): Fw=w (p)FFw=w^{(p)} F

(WF3): wV=Vw (p)w V=V w^{(p)}

or 2. as a left module over the Dieudonné ring which is the (noncommutative ring) generated by W(k)W(k) and two variables FF and VV satisfying (WF1)(WF2)(WF3) in which case every element of D kD_k can uniquely be written as a finite sum

i>0a iV i+a 0+ i>0a iF i\sum_{i\gt 0} a_{-i} V^i + a_0 + \sum_{i\gt 0} a_i F^i

(…)

Examples

  • For a field kk the terminal kk-scheme Sp kkSp_k k is a group scheme in a unique way.

  • An affine group scheme. Affine group varieties are called linear algebraic groups.

  • Complete group varieties are called abelian varieties.

  • Given any group GG, one can form the constant group scheme? G XG_X over XX.

  • etale group scheme? is the spectrum of a commutative Hopf algebra. In this case the multiplication- resp. inversion- reps. unit map are given by comultiplication? reps. antipodism? resp. counit in the Hopf algebra.

  • The functor μ:=𝔾 m\mu:=\mathbb{G}_m is a group scheme given by 𝔾 m(S)=Γ(S,𝒪 S) ×\mathbb{G}_m(S)=\Gamma(S, \mathcal{O}_S)^\times. A scheme is sent to the invertible elements of its global functions. This group scheme is called the multiplicative group scheme. In context of p-divisible groups the kernels of the kk-group scheme endomorphisms of 𝔾 m\mathbb{G}_m defined by () n:xx n(-)^n:x\mapsto x^n for an integer nn are of particular interest. These kernels give the group schemes of the nn-th root of unity.

  • diagonalizable group scheme. Note that the multiplicative group scheme is diagonalizable.

  • multiplicative group scheme also called group scheme of multiplicative type. Every diagonalizable group scheme is in particular of multiplicative type.

  • The additive group scheme assigns to a ring its additive group. Also here the kernels of the powering-by-n map are of interest. These kernels give the group schemes of the nn-th nilpotent element?.

  • Group schemes can be constructed by restriction of scalars.

  • The functor α:=𝔾 a\alpha:=\mathbb{G}_a is a group scheme given by 𝔾 a(S)=Γ(S,𝒪 S)\mathbb{G}_a(S)=\Gamma(S, \mathcal{O}_S) the additive group of the ring of global functions. This group scheme is called the additive group scheme.

  • connected group scheme? (is synonymous to local group scheme?)

  • unipotent group scheme (these are Cartier duals of local group schemes)

  • the kernel of any group scheme morphism is a group scheme.

  • Every algebraic group is in particular a group scheme.

Properties

Cartier duality

(main article: Cartier duality)

Suppose now that GG is a finite flat commutative group scheme (over XX). The Cartier dual of GG is given by the functor G D(S)=Hom(GS,𝔾 mS)G^D(S)=Hom (G\otimes S, \mathbb{G}_m \otimes S). The Hom is taken in the category of group schemes over SS.

For example, α p Dα p\alpha_p^D\simeq \alpha_p.

Dieudonné module

(main article: Dieudonné module)

There are certain correspondences (Theorem AcuTheorem Fftc) between certain categories of group schemes and certain categories of Dieudonné modules.

Definition

A Dieudonné module is a module over the Dieudonné ring D kD_k of a field kk of prime characteristic pp.

Definition

The Dieudonné ring of kk is the ring generated by two objects F,VF,V subject to the relations

FV=VF=pFV=VF=p
Fw=w σFFw=w^\sigma F
wV=Vw σw V=V w^\sigma

where

σ:{W(k)W(k) (w 1,w 2,)(w 1 p,w 2 p,)\sigma:\begin{cases} W(k)\to W(k) \\ (w_1,w_2,\dots)\mapsto (w_1^p,w_2^p,\dots) \end{cases}

denotes the endomorphism of the Witt ring W(k)W(k) of kk given by raising each component of the Witt vectors to the pp-th power; this means that σ\sigma is component-wise given by the Frobenius endomorphism of the file kk.

The Dieudonné ring is a \mathbb{Z}-graded ring where the degree nn-part is the 11-dimensional free module generated by V nV^{-n} if n<0n\lt 0 and by F nF^n if n>0n\gt 0

Theorem

(III.5, Acu kTor VD kModAcu_k\simeq Tor_V D_kMod)

(see also Dieudonné module for more details concerning this theorem)

Let kk be a perfect field of prime characteristic pp. Since kk is perfect Frobenius is an automorphism.

On the left we have the category of affine commutative unipotent group schemes. On the right we have the category of all D_k-modules of VV-torsion. The (contravariant) equivalence is given by

M:{Acu k Tor VD kMod G colim nAcu k(G,W nk)M:\begin{cases} Acu_k&\to& Tor_V D_kMod \\ G&\mapsto&colim_n Acu_k(G,W_{nk}) \end{cases}

where we recall that how the colimit of the hom space can be multiplied by the generators of the Dieudonné ring.

Theorem

(III.6, Feu kTor VD kModFeu_k\simeq Tor_V D_kMod)

Theorem

(III.6, Fiu kTor FD kModFiu_k\simeq Tor_F D_kMod)

Theorem

(III.8, Torf p(finW(k)Mod,F,V)Torf_p\simeq (fin W(k) Mod,F,V))

Theorem

(III.9, FftcD^ kMod fin.len.quotFftc\simeq \hat D_k Mod_{fin.len.quot})

References

  • M. Artin, J. E. Bertin, M. Demazure, P. Gabriel, A. Grothendieck, M. Raynaud, J.-P. Serre, Schemas en groupes, i.e. SGA III-1, III-2, III-3

  • Michel Demazure, P. Gabriel, Groupes algebriques, tome 1 (later volumes never appeared), Mason and Cie, Paris 1970

  • Michel Demazure, lectures on p-divisible groups web

  • W. Waterhouse, Introduction to affine group schemes, GTM 66, Springer 1979.

  • D. Mumford, Abelian varieties, 1970, 1985.

  • J. C. Jantzen, Representations of algebraic groups, Acad. Press 1987 (Pure and Appl. Math. vol 131); 2nd edition AMS Math. Surveys and Monog. 107 (2003; reprinted 2007)

Revised on November 11, 2013 23:09:39 by Urs Schreiber (145.116.131.252)