Formal group - Knowlpedia

Definition

A formal group over a base $S$ is a group object in the category of formal schemes over $S$ . It consists of a formal scheme $\mathcal G$ with multiplication, identity, and inverse morphisms

m:\mathcal G\times_S\mathcal G\to\mathcal G,\qquad e:S\to\mathcal G,\qquad i:\mathcal G\to\mathcal G,

satisfying the associative, unit, and inverse diagrams. It is commutative when $m=m\circ\tau$ , where $\tau$ swaps the two factors.

Finite-dimensional formal Lie groups

For the characteristic-zero Lie correspondence used here, $S=\operatorname{Spec}k$ for a field $k$ , and a finite-dimensional formally smooth formal group means one whose pointed underlying formal scheme is isomorphic to a formal disc

(\mathcal G,e)\cong \left(\operatorname{Spf}k[[X_1,\ldots,X_n]],0\right)

for some finite $n$ . The isomorphism is not part of the coordinate-free formal group; choosing one supplies coordinates.

Broader usages include commutative formal groups over general bases and formal groups not represented by a finite-dimensional formal disc. Those objects are not silently included in the finite-dimensional equivalence.

Coordinates

After a coordinate choice, the pullback of multiplication is determined by

m^*(X_i)=F_i(X,Y),

and the group diagrams become the identities for a formal group law. Different coordinate choices give isomorphic laws, so the formal group is the intrinsic object and the law is a presentation.

Tangent structure

The tangent space at the identity carries a canonical Lie bracket, producing the tangent Lie algebra $\operatorname{Lie}(\mathcal G)$ . In characteristic zero and within the finite-dimensional formal-disc category, this construction is an equivalence of categories, not merely an assignment of an invariant.

References

Michiel Hazewinkel, Formal Groups and Applications, AMS Chelsea Publishing, 2012. AMS book record. Relevant: Chapters 2 and 7, multidimensional formal groups and bialgebras.
Jean-Pierre Serre, Lie Algebras and Lie Groups, second edition, Springer, 1992. Publisher record. Relevant: Part I, formal groups and Lie algebras.