Spahn examples of (group) schemes

Contents

Summary (random tour through the examples)

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}
formalconnectedétalep.34
finiteinfinitesimalétalesplits if kk is perfectp.35
affinemultiplicativesmooth?G/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

(…)

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

Last revised on July 21, 2012 at 20:27:22. See the history of this page for a list of all contributions to it.