Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 22 / 3 / i / Solution

In this one-dimensional setting, a commutative formal group law over a commutative ring $R$ is a formal power series $F(X,Y)\in R[[X,Y]]$ withIn particular $F(X,Y)=X+Y+$ terms of total degree at least two. There is a unique formal inverse $i_F(T)=-T+O(T^2)$; coefficient recursion solves $F(T,i_F(T))=0$.

A morphism from formal group law $F$ to $G$ is $f(T)\in TR[[T]]$ satisfyingThe invertible morphism criterion for formal group laws isNecessity follows by differentiating $g\circ f=T$ at zero for an inverse $g$. Conversely, write $f(T)=a_{1}T+a_2{T}^2+\cdots$ with $a_1$ a unit. In constructing $g(T)=b_{1}T+b_2{T}^2+\cdots$, the coefficient of $T$ fixes $b_1=a_1^{-1}$; at degree $n$, the equation $f(g(T))=T$ has the form $a_1{b}_n+$ an already known expression $=0$. This determines every $b_n$ over $R$. The same construction gives an inverse on the other side, and uniqueness makes the two inverses agree. Finally apply $g$ to the morphism identity with $X=g(U),Y=g(V)$ to obtainThus the inverse is itself a morphism of formal group laws, not merely an inverse formal power series. Over a general ring, nonzero derivative is insufficient: it must be a unit.