To a regular(or polynomial) map $f: X \to Y$ between affine varieties we associate its pullback $f^\ast: K[Y] \to K[X]$ and it holds that f is an isomorphism iff $f^\ast$ is an isomorphism.
Now if $\mathcal{O}_X,p$ denotes the local ring of X at p and $\phi: \mathcal{O}_{X,p} \to \mathcal{O}_{Y,q}$ is an ring homomorphism/isomorphism, to what "kind" of morphism between the varieties X and Y does it correspond? Are these the so called rational maps? Or do rational maps correspond to K-morphisms of the function fields $K(X)$ and $K(Y)$?