Additive and multiplicative formal group laws

Example

Over a commutative ring $R$ , the additive formal group law and multiplicative formal group law are

F_a(X,Y)=X+Y, \qquad F_m(X,Y)=X+Y+XY.

Both are one-dimensional commutative formal group laws. Their inverse series are

i_a(X)=-X, \qquad i_m(X)=-\frac{X}{1+X}=-X+X^2-X^3+\cdots.

Why the multiplicative formula appears

Use the coordinate $X=u-1$ near the identity $u=1$ of the multiplicative group. Since

(1+X)(1+Y)-1=X+Y+XY,

ordinary multiplication becomes $F_m$ in this identity-centered coordinate. Thus $F_a$ is the law near $0$ on the additive group, while $F_m$ is the law near $1$ on the multiplicative group.

Logarithms over characteristic zero

Over a $\mathbb Q$ -algebra,

\log_{F_a}(X)=X,\qquad \log_{F_m}(X)=\log(1+X) =\sum_{n\geq1}(-1)^{n+1}\frac{X^n}{n}.

The identity

\log(1+F_m(X,Y)) =\log(1+X)+\log(1+Y)

exhibits the multiplicative law as strictly isomorphic to the additive law after rational denominators are allowed.

Positive-characteristic distinction

In characteristic $p$ ,

[p]_{F_a}(X)=0, \qquad [p]_{F_m}(X)=(1+X)^p-1=X^p.

Therefore these laws can have the same one-dimensional abelian tangent Lie algebra without being isomorphic. This is the basic example behind the positive-characteristic warning.

References

Michiel Hazewinkel, Formal Groups and Applications, AMS Chelsea Publishing, 2012. AMS book record. Relevant: Chapter 1, basic one-dimensional examples.
J. F. Adams, Stable Homotopy and Generalised Homology, University of Chicago Press, 1974. Relevant: Part II, additive and multiplicative formal group laws.