If $f: Y\to Z$ is a finite, surjective morphism of normal integral schemes (of finite type over a field) and $y$ is a prime divisor of $Y$, is then also $z= f(y)$ of codimension 1? We have an inclusion $\mathcal{O}_{Z,z} \to \mathcal{O}_{Y,y}$ and the latter is a DVR, i.e. it has dimension 1. Does the integrality of both local rings imply that $\operatorname{codim}z =1$?
More generally, is there a relationship between $\dim y, \dim f(z)$ and $\dim f^{-1}(z'), \dim z'$ for $z'\in Z$ arbitrary?