# Formal group

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

If is the ring of formal power series in variables over , then is called an -dimensional formal Lie group. For a connected algebraic group over , is a formal Lie group. A formal Lie group is isomorphic, as a functor in the category of sets, to the functor that associates with an algebra the -fold Cartesian product of its nil radical with itself. The group structure on the sets is given by a formal group law — a collection of formal power series in variables :  satisfying the following conditions:   Here ,  , . This group law on the sets is given by the formulas where ; because and are nilpotent, all except a finite number of terms of the series are zero. Every formal group law gives group structures on by means of

and converts the functor 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 of characteristic 0 the correspondence between a formal Lie group and its Lie algebra defines an equivalence of the respective categories. In characteristic the situation is more complicated. Thus, over an algebraically closed field (for ) 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=17665
This article was adapted from an original article by Yu.G. Zarkhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article