4
$\begingroup$

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?

  • 4
    I asked the same question here: http://mathoverflow.net/questions/2$0$782/ring-theoretic-characterization-of-open-affines2012-10-30

1 Answers 1

2

The answers in the link given by Manny are great. Let me just add another sufficient condition which may be simpler to check than, e.g. flatness. If

  • $A$ 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.