nLab 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 $G=SpfA$ is called smooth formal k-group? if $A$ is a power-series algebra $k\left[\left[{X}_{1},...{X}_{n}\right]\right]$ in $n$ variables

The coproduct $\Delta :A\to A\stackrel{^}{\otimes }A$ is given by a set af formal power series $\Phi \left(X,Y\right)=\left({\Phi }_{i}\left({X}_{1},...,{X}_{n},{Y}_{1},...,{Y}_{n}\right)\right),{i}_{1},...,n$ satisfying the axioms (Ass),(Un) and (Com). Such a set $\left\{{\Phi }_{i}\right\}$ is called a Dieudonné group law.

Theorem

Let $G=\mathrm{Sp}fA$ be a (not necessarily commutative) connected formal group of finite type. 1.If $p=0$ then $G$ is smooth.

1. If $p¬=0$ 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 $G={\mathrm{Sp}}^{*}C$ 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 $ℊ$ of $G$.

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

3. If $ℊ$ is finite dimensional then $G$ is smooth.

4. If $G$ is commutative $ℊ$ is abelian.

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

Theorem

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

Then $A$ is of the form

$k\left[\left[{X}_{1},\dots ,{X}_{n}\right]\right]\left[{Y}_{1},\dots ,{Y}_{d}\right]/\left({Y}_{1}^{{p}^{{r}_{1}}},\dots ,{Y}_{d}^{{p}^{{r}_{d}}}\right)$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 $A={\stackrel{^}{O}}_{G,e}$, 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}_{\mathrm{red}}\to G\to G/{G}_{\mathrm{red}}\to 0$0\to G_{red}\to G\to G/G_{red}\to 0

with ${G}_{\mathrm{red}}$ smooth and $G/{G}_{\mathrm{red}}$ infinitesimal?.

1. For large $r$, the group $G/\mathrm{ker}{F}_{G}^{r}$ is smooth.
Corollary

Let $G$ be a connected formal group of finite type, let $n:=\mathrm{dim}G$. Then $\mathrm{rk}\left(\mathrm{co}\mathrm{her}{F}_{G}^{i}\right)$ is bounded and

$\mathrm{rk}\left(\mathrm{ker}{F}_{G}^{i}\right)={p}^{ni}\cdot \mathrm{rk}\left(\mathrm{coker}{F}_{G}^{i}\right)$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 $\mathrm{dim}\left(G\right)=\mathrm{dim}\left({G}^{\prime }\right)+\mathrm{dim}\left({G}^{n}\right)$.

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

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