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 $K(Y)$?
Best Answer
Maps of the form $\phi$ correspond to rational maps from $Y$ to $X$ which are regular at $q$. If $\varphi$ is an isomorphism, then it induces an isomorphism on fraction fields (I am assuming that "variety" means irreducible here since you are talking about function fields), i.e., an isomorphism from $K(Y)$ to $K(X)$.
In general, (not necessarily iso-)morphisms of function fields $K(Y)$ to $K(X)$ correspond to dominant rational maps $f: X \rightarrow Y$, i.e., those with Zariski-dense image. For a map $\phi$ -- or any homomorphism of integral domains -- to extend to a homomorphism on the fraction fields, it is necessary and sufficient that it be injective.