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?
Ring homomorphism and affine scheme
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algebraic-geometry
commutative-algebra
ring-theory
schemes
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4I asked the same question here: http://mathoverflow.net/questions/2$0$782/ring-theoretic-characterization-of-open-affines – 2012-10-30
1 Answers
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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.