  Let $B$ be a Noetherian normal local ring, and $G\subset\Aut(B)$ a cyclic
group of local automorphisms of prime order. Let $A$ be the ring of
$G$-invariants of $B$, assume that $A$ is Noetherian. We study the invariant
morphism; in particular, we prove that $B$ is a monogenous $A$-algebra if and
only if the augmentation ideal of $B$ is principal. If in particular $B$ is
regular, we prove that $A$ is regular if the augmentation ideal of $B$ is
principal.
