Retraction given by a principal ideal

Question

Let $R$ be a commutative ring. Let $ f : R \to A$ be a smooth morphism of relative dimension 1. Suppose $f$ admits a retraction $r : A \to R$ i.e $r f = Id_R$. Is it true that the kernel of $r$ is a principal ideal ?

Answer 1

The kernel is locally principal, but need not be principal.

Let $R$ be an algebraically closed field, let $A$ be the ring of regular functions of a smooth affine curve $C$, then any closed point of $C$ defines a section (=rectraction), but the corresponding maximal ideal is not principal in general because otherwise $A$ would be principal. Concrete example: $C=$ a projective smooth curve of positive genus minus one point. Then no maximal ideal of $A$ is principal.