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

This entry is about a section of the text

• Michel Demazure, lectures on p-divisible groups, web

Definition

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

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

Theorem

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

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

2. $G$ is smooth

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

4. $F_G$ is an epimorphism.

The previous theorem can be strengthened:

Theorem

(Cartier)

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

1. $C$ is the universal enveloping algebra of the Lie algebra $\mathcal{g}$ of $G$.

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

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

4. If $G$ is commutative $\mathcal{g}$ is abelian.

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

Theorem

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

Then $A$ is of the form

$$k [ [ X_1,\dots,X_n] ][Y_1,\dots,Y_d]/(Y_1^{p^{r_1}},\dots,Y_d^{p^{r_d}})$$

This applies for instance to $A=\hat O_{G,e}$, for an algebraic group $G$.

Corollary

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

1. If $k$ is prefect, there exists a unique exact sequence of connected groups

$$0\to G_{red}\to G\to G/G_{red}\to 0$$

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

1. For large $r$, the group $G/ker F^r_G$ is smooth.

Corollary

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

$$rk(ker F^i_G)=p^{n i}\cdot rk(coker F^i_G)$$

Corollary

1. Let $0\to G^\prime\to G\to G^n\to 0$ be an exact sequence of connected formal groups. Then $dim(G)=dim(G^\prime)+dim(G^n)$.

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