Morphism of formal group laws

Definition

Let $F$ be an $n$ -dimensional formal group law over $R$ , and let $G$ be $m$ -dimensional. A morphism of formal group laws

f:F\longrightarrow G

is a tuple $f(X)\in(X)R[[X_1,\ldots,X_n]]^m$ satisfying

f(F(X,Y))=G(f(X),f(Y)).

Composition is formal substitution, and the identity morphism is the coordinate tuple $X$ .

Isomorphisms and strict isomorphisms

When $m=n$ , the morphism $f$ is an isomorphism precisely when its linear coefficient matrix $Df(0)$ lies in $\operatorname{GL}_n(R)$ . The formal inverse function theorem then supplies a unique compositional inverse, and the homomorphism identity shows that inverse also respects the laws.

An isomorphism is strict when

Df(0)=I_n.

Thus every strict isomorphism preserves the chosen tangent coordinates to first order, while a general isomorphism may also change the tangent basis.

Coordinate changes

An invertible pointed tuple $f$ transports a law $F$ to a law $G$ by requiring

f(F(X,Y))=G(f(X),f(Y)).

This is a change of coordinates on the same coordinate-free formal group. Accordingly, properties invariant under formal-group-law isomorphism do not depend on the selected parameters.

Tangent functor

Taking linear terms sends $f$ to

Df(0):T_0F\longrightarrow T_0G.

It is a Lie algebra homomorphism for the tangent brackets. Over a characteristic-zero field, every finite-dimensional Lie algebra homomorphism integrates to one and only one formal-group-law morphism.

References

Michiel Hazewinkel, Formal Groups and Applications, AMS Chelsea Publishing, 2012. AMS book record. Relevant: Chapters 1, 2, and 4, homomorphisms and strict isomorphisms.
Jean-Pierre Serre, Lie Algebras and Lie Groups, second edition, Springer, 1992. Publisher record. Relevant: Part I, morphisms in formal Lie theory.