Formal group laws as coordinates on formal groups

Statement

Fix a commutative base field $k$ and $n\geq0$ . Giving a group-object structure on the pointed formal disc

\widehat{\mathbb A}^{\,n}_k =\operatorname{Spf}k[[X_1,\ldots,X_n]]

is equivalent to giving an $n$ -dimensional formal group law over $k$ . Under this equivalence, formal group homomorphisms are exactly morphisms of formal group laws in the chosen coordinates.

The coordinate calculation

The product of two formal $n$ -discs has coordinate ring $k[[X,Y]]$ . Because morphisms of formal spectra reverse arrows, multiplication

m:\widehat{\mathbb A}^{\,n}_k\times \widehat{\mathbb A}^{\,n}_k\longrightarrow \widehat{\mathbb A}^{\,n}_k

is determined by a continuous homomorphism

m^*:k[[X]]\longrightarrow k[[X,Y]],\qquad X_i\longmapsto F_i(X,Y).

The unit, associativity, and inverse diagrams dualize exactly to the unit, associativity, and inverse identities for $F$ .

A pointed formal map $f$ between two such group objects is represented by a tuple of zero-constant-term series. The group-homomorphism square dualizes to

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

Thus both objects and morphisms agree, not only their isomorphism classes.

Coordinate Hopf-algebra formula

For a group law $F=(F_1,\ldots,F_n)$ , the coordinate ring $A=k[[x_1,\ldots,x_n]]$ has comultiplication

\Delta(x_i)=F_i(x_1,\ldots,x_n,y_1,\ldots,y_n) \in A\widehat\otimes_kA\cong k[[x_1,\ldots,x_n,y_1,\ldots,y_n]].

The counit sends $x_i$ to $0$ , and the antipode sends $x_i$ to the $i$ -th component of the inverse power series. These maps form the coordinate Hopf algebra of the formal group.

Chosen coordinates versus intrinsic objects

For a formal group whose underlying pointed formal scheme is merely isomorphic to a formal disc, one must first choose such an isomorphism. A different choice transports $F$ by an invertible pointed substitution. The coordinate-free group is independent of that choice.

This distinction matters categorically: formal group laws are presentations by parameters, while the formal group is the represented geometric object.

References

Michiel Hazewinkel, Formal Groups and Applications, AMS Chelsea Publishing, 2012. AMS book record. Relevant: Chapters 2 and 7, power-series and bialgebra descriptions.
The Stacks Project Authors, “Formal schemes à la EGA.” Section 87.2, Tag 0AHY. Relevant: the anti-equivalence between affine formal schemes and admissible topological rings.