Repeated formal addition gives
$$
[n]_{\widehat{\mathbb G}_a}(X)=nX,
\qquad
[n]_{\widehat{\mathbb G}_m}(X)=(1+X)^n-1
$$
for 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 $h:\widehat{\mathbb G}_a\to\widehat{\mathbb G}_m$ be a homomorphism. Compatibility with multiplication by $p$ gives
$$
h([p]_{\widehat{\mathbb G}_a}(X))
=[p]_{\widehat{\mathbb G}_m}(h(X)).
$$
The left side is $h(0)=0$, while the Frobenius identity gives the right side as
$$
(1+h(X))^p-1=h(X)^p.
$$
The <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.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-136/5/c/solution