# Formal group

An algebraic analogue of the concept of a local Lie group (cf. Lie group, local). The theory of formal groups has numerous applications in algebraic geometry, class field theory and cobordism theory.

A formal group over a field $k$ is a group object in the category of connected affine formal schemes over $k$ ( see , , , ). Let $H _{A}$ be the functor that associates with an algebra $B$ the set of algebra homomorphism $A \rightarrow B$ from some Noetherian commutative local $k$ - algebra $A$ with maximal ideal $m$ and field of residues $k$ , complete in the $m$ - adic topology, such that the homomorphisms map $m$ into the set $\mathop{\rm nil}\nolimits (B)$ of nilpotent elements of $B$ . Then a connected affine formal scheme is a covariant functor $H$ from the category of finite-dimensional commutative $k$ - algebras $B$ into the category of sets that is isomorphic to an $H _{A}$ . That $H$ is a group object means that there is a group structure given on all the sets $H (B)$ such that for every $k$ - algebra homomorphism $B _{1} \rightarrow B _{2}$ the corresponding mapping $H (B _{1} ) \rightarrow H (B _{2} )$ is a group homomorphism. If all the groups $H (B)$ are commutative, then the formal group $H$ is said to be commutative. Every connected group scheme $G$ over $k$ defines a formal group $G: \ B \rightarrow G (B)$ . Here one can take as $A$ the completion of the local ring of $G$ at the unit element.

If $A$ is the ring $k [[X _{1} \dots X _{2} ]]$ of formal power series in $n$ variables over $k$ , then $H$ is called an $n$ - dimensional formal Lie group. For a connected algebraic group $G$ over $k$ , $\widehat{G}$ is a formal Lie group. A formal Lie group $H$ is isomorphic, as a functor in the category of sets, to the functor $D ^{n} : \ B \rightarrow \mathop{\rm nil}\nolimits (B) ^{n}$ that associates with an algebra $B$ the $n$ - fold Cartesian product of its nil radical $\mathop{\rm nil}\nolimits (B)$ with itself. The group structure on the sets $H (B) = \mathop{\rm nil}\nolimits (B) ^{n}$ is given by a formal group law — a collection of $n$ formal power series in $2n$ variables $X _{1} \dots X _{n} ,\ Y _{1} \dots Y _{n}$ : $$F _{1} (X _{1} \dots X _{n} ,\ Y _{1} \dots Y _{n} ) \dots$$ $$F _{n} (X _{1} \dots X _{n} ,\ Y _{1} \dots Y _{n} ),$$ satisfying the following conditions:$$F _{i} (X,\ 0) = X _{i} , F _{i} (0,\ Y) = Y _{i} ,$$ $$F _{i} (X _{1} \dots X _{n} ,\ F _{1} (Y,\ Z) \dots F _{n} (Y,\ Z)) =$$ $$= F _{i} (F _{1} (X,\ Y) \dots F _{n} (X,\ Y),\ Z _{1} \dots Z _{n} ) .$$ Here $X = (X _{1} \dots X _{n} )$ , $Y = (Y _{1} \dots Y _{n} ),$ $Z = (Z _{1} \dots Z _{n} )$ , $0 = (0 \dots 0)$ . This group law on the sets $H (B) = \mathop{\rm nil}\nolimits (B) ^{n}$ is given by the formulas$$(x _{1} \dots x _{n} ) \circ (y _{1} \dots y _{n} ) = (z _{1} \dots z _{n} ),$$ where $z _{i} = F _{i} (x _{1} \dots x _{n} ,\ y _{1} \dots y _{n} )$ ; because $x$ and $y$ are nilpotent, all except a finite number of terms of the series are zero. Every formal group law gives group structures on $\mathop{\rm nil}\nolimits (B) ^{n}$ by means of

and converts the functor $D ^{n}$ into a formal Lie group. The concept of a formal group law, and thus of a formal Lie group, can be generalized to the case of arbitrary commutative ground rings (see , ). Sometimes by a formal group one means just a formal Lie group or even a formal group law.

Just as for local Lie groups (cf. Lie group, local) one can define the Lie algebra of a formal Lie group. Over fields $k$ of characteristic 0 the correspondence between a formal Lie group and its Lie algebra defines an equivalence of the respective categories. In characteristic $p > 0$ the situation is more complicated. Thus, over an algebraically closed field (for $p > 0$ ) there is a countable number of pairwise non-isomorphic one-dimensional commutative formal Lie groups , while all one-dimensional Lie algebras are isomorphic . Over perfect fields of finite characteristic, commutative formal Lie groups are classified by means of Dieudonné modules (see , ).

The theory of formal groups over fields can be generalized to the case of arbitrary formal ground schemes .

How to Cite This Entry:
Formal group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Formal_group&oldid=44249
This article was adapted from an original article by Yu.G. Zarkhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article