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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.
Let $G=Sp f A$ be a (not necessarily commutative) connected formal group of finite type. 1.If $p=0$ then $G$ is smooth.
If $p\not =0$ then the following conditions are equivalent
$G$ is smooth
$A\otimes_k k^{p-1}$ is reduced.
$F_G$ is an epimorphism.
The previous theorem can be strengthened:
(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? $C$).
$C$ is the universal enveloping algebra of the Lie algebra $\mathcal{g}$ of $G$.
The category of connected formal $k$-groups is equivalent to the category of all Lie algebras over $k$.
If $\mathcal{g}$ is finite dimensional then $G$ is smooth.
If $G$ is commutative $\mathcal{g}$ is abelian.
$G\simeq (\alpha^\circ)^{(I)}$; by duality any unipotent (commutative) $k$-group is a power of the additive group.
(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
This applies for instance to $A=\hat O_{G,e}$, for an algebraic group $G$.
Let $p\ge 0$, let $G$ be a connected formal group? (=local formal group) of finite type?. Then
with $G_{red}$ smooth and $G/G_{red}$ infinitesimal?.
Let $G$ be a connected formal group of finite type, let $n:=dim G$. Then $rk(co her F^i_G)$ is bounded and
Let $0\to G^\prime\to G\to G^n\to 0$ be an exact sequence of connected formal groups. Then $dim(G)=dim(G^\prime)+dim(G^n)$.
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.