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Demazure, lectures on p-divisible groups, II.10, smooth formal groups

This entry is about a section of the text

Definition

A not necesarily commutative connected formal group G=SpfAG=\Sp f A is called smooth formal k-group? if AA is a power-series algebra k[[X 1,...X n]]k[ [X_1,...X_n] ] in nn variables

The coproduct Δ:AA^A\Delta:A\to A\hat \otimes A is given by a set af formal power series Φ(X,Y)=(Φ i(X 1,...,X n,Y 1,...,Y n)),i 1,...,n\Phi(X,Y)=(\Phi_i(X_1,..., X_n, Y_1,..., Y_n)), i_1,...,n satisfying the axioms (Ass),(Un) and (Com). Such a set {Φ i}\{\Phi_i\} is called a Dieudonné group law.

Theorem

Let G=SpfAG=Sp f A be a (not necessarily commutative) connected formal group of finite type. 1.If p=0p=0 then GG is smooth.

  1. If p¬=0p\not =0 then the following conditions are equivalent

  2. GG is smooth

  3. A kk p1A\otimes_k k^{p-1} is reduced.

  4. F GF_G is an epimorphism.

The previous theorem can be strengthened:

Theorem

(Cartier)

Let p=0p=0, let G=Sp *CG=Sp^* C be a connected (not necessarily commutative) formal kk-group (realized as the formal spectrum of a k-coring? CC).

  1. CC is the universal enveloping algebra of the Lie algebra \mathcal{g} of GG.

  2. The category of connected formal kk-groups is equivalent to the category of all Lie algebras over kk.

  3. If \mathcal{g} is finite dimensional then GG is smooth.

  4. If GG is commutative \mathcal{g} is abelian.

  5. G(α ) (I)G\simeq (\alpha^\circ)^{(I)}; by duality any unipotent (commutative) kk-group is a power of the additive group.

Theorem

(Dieudonné-Cartier-Gabriel) Let p>0p\gt 0, let kk be a perfect field of characteristic pp let G=Sp *CG=Sp^* C be a connected (not necessarily commutative) connected formal kk-group of finite type?, let HH be a subgroup of GG, let G/H:=SpfAG/H:=Spf A (this quit ion has not been defined in these lectures).

Then AA is of the form

k[[X 1,,X n]][Y 1,,Y d]/(Y 1 p r 1,,Y d p r d)k [ [ X_1,\dots,X_n] ][Y_1,\dots,Y_d]/(Y_1^{p^{r_1}},\dots,Y_d^{p^{r_d}})

This applies for instance to A=O^ G,eA=\hat O_{G,e}, for an algebraic group GG.

Corollary

Let p0p\ge 0, let GG be a connected formal group? (=local formal group) of finite type?. Then

  1. If kk is prefect, there exists a unique exact sequence of connected groups
0G redGG/G red00\to G_{red}\to G\to G/G_{red}\to 0

with G redG_{red} smooth and G/G redG/G_{red} infinitesimal?.

  1. For large rr, the group G/kerF G rG/ker F^r_G is smooth.
Corollary

Let GG be a connected formal group of finite type, let n:=dimGn:=dim G. Then rk(coherF G i)rk(co her F^i_G) is bounded and

rk(kerF G i)=p nirk(cokerF G i)rk(ker F^i_G)=p^{n i}\cdot rk(coker F^i_G)
Corollary
  1. Let 0G GG n00\to G^\prime\to G\to G^n\to 0 be an exact sequence of connected formal groups. Then dim(G)=dim(G )+dim(G n)dim(G)=dim(G^\prime)+dim(G^n).

  2. If f:G Gf:G^\prime \to G is a morphism of connected formal groups, with GG smooth and dimG=dimG dim G=dim G^\prime, then ff is an epimorphism iff kerfker f is finite.

Revised on May 27, 2012 13:36:32 by Stephan Alexander Spahn (79.227.168.80)