Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 136 / 5

A one-dimensional commutative formal group law over a ring $R$ is a series $F(X,Y)\in R[[X,Y]]$ satisfyingThe identity axiom gives $F(X,Y)=X+Y+$ terms of total degree at least two. SeekAfter $a_2,\dots ,a_{n-1}$ have been chosen, the coefficient of $X^n$ in $F(X,i(X))$ is $a_n$ plus a known expression in the earlier coefficients. There is therefore a unique choice of $a_n$ making it zero. Recursion constructs the formal inverse $i(X)$ with $F(X,i(X))=0$ and $i(X)\equiv -X(\mathrm{mod}{X}^2)$.

The formal additive group and formal multiplicative group have lawsIf $R$ is an algebra over a field whose scalar field is $\mathbb{Q}$, the serieshas linear coefficient one and satisfiesIt is therefore the exponential isomorphism between the additive and multiplicative formal groups, with inverse $\log (1+X)$.

Repeated formal addition givesfor positive $n$, and the same formulas extend to all $n\in \mathbb{Z}$ using the formal inverse.

Now let $R$ be a field of characteristic $p>0$ and let $h:{\hat{\mathbb{G}}}_a\to {\hat{\mathbb{G}}}_m$ be a homomorphism. Compatibility with multiplication by $p$ givesThe left side is $h(0)=0$, while the Frobenius identity gives the right side asThe formal power series ring over a field is an integral domain, so $h(X{)}^p=0$ implies $h=0$. Hence there are no nonzero homomorphisms from the formal additive group to the formal multiplicative group.