# Ring homomorphism and affine scheme

How to describe all ring homomorphisms $f: A \rightarrow B$, such that corresponding affine scheme morphism $f: Spec \, B \rightarrow Spec \, A$ is open immersion?

• $\def\Spec{\operatorname{Spec}}$Don't we have $f\colon\Spec B \to \Spec A$? – martini Oct 30 '12 at 9:54
• Of course! I'm sorry. I've edited. – user46336 Oct 30 '12 at 10:06
• I asked the same question here: mathoverflow.net/questions/20782/… – Manny Reyes Oct 30 '12 at 10:54

• $A$ is an integrally closed domain,
• $B$ is contained in $\mathrm{Frac}(A)$ and finitely presented over $A$ (as $A$-algebra),
• $f$ is quasi-finite (i.e. for all prime ideals $p$ of $A$, $B/pB$ is artinian),
then $f$ is an open immersion. This is a form of Zariski's Main Theorem.