Formal Group

A formal group is a group object in the category of formal schemes.

• If is a functor from Artinian algebras to groups which is left exact, then it representable (G is the functor of points of a formal group. (left exactness of a functor is equivalent to commuting with finite projective limits).
• If is a group scheme then, the formal completion of G at the identity has the structure of a formal group.
• A smooth group scheme is isomorphic to . Some people call a formal group scheme smooth if the converse holds.
• formal smoothness asserts the existence of lifts of deformations and can apply to formal schemes that are larger than points. A smooth formal group scheme is a special case of a formal group scheme.
• Given a smooth formal group, one can construct a formal group law and a field by choosing a uniformizing set of sections.
• The (non-strict) isomorphisms between formal group laws induced by change of parameters make up the elements of the group of coordinate changes on the formal group.

Formal groups and formal group laws can also be defined over arbitrary schemes, rather than just over commutative rings or fields, and families can be classified by maps from the base to a parametrizing object.

The moduli space of formal group laws is a disjoint union of infinite dimensional affine spaces, whose components are parametrized by dimension, and whose points are parametrized by admissible coefficients of the power series F. The corresponding moduli stack of smooth formal groups is a quotient of this space by a canonical action of the infinite dimensional groupoid of coordinate changes.

Over an algebraically closed field, the substack of one dimensional formal groups is either a point (in characteristic zero) or an infinite chain of stacky points parametrizing heights. In characteristic zero, the closure of each point contains all points of greater height. This difference gives formal groups a rich geometric theory in positive and mixed characteristic, with connections to the Steenrod algebra, p-divisible groups, Dieudonné theory, and Galois representations. For example, the Serre-Tate theorem implies that the deformations of a group scheme are strongly controlled by those of its formal group, especially in the case of supersingular abelian varieties. For supersingular elliptic curves, this control is complete, and this is quite different from the characteristic zero situation where the formal group has no deformations.

A formal group is sometimes defined as a cocommutative Hopf algebra (usually with some extra conditions added, such as being pointed or connected). This is more or less dual to the notion above. In the smooth case, choosing coordinates is equivalent to taking a distinguished basis of the formal group ring.

Some authors use the term formal group to mean formal group law.