Let $\phi : X \to Y$ be a dominant morphism of irreducible varieties over an algebraically closed field $\mathsf{k}$, and for the sake of simplicity, let us assume that $X$ and $Y$ are affine. Then the comorphism $\phi^\sharp : \mathsf{k}[Y] \to \mathsf{k}[X]$ is an injection. This defines a $\mathsf{k}[Y]$-module structure on $\mathsf{k}[X]$.
My question concerns whether we can characterise in terms of $\phi$ the necessary conditions required to make $\mathsf{k}[X]$ free as a $\mathsf{k}[Y]$-module?