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Suppose $f: A \rightarrow B$ is a local homomorphism of local rings. Assume that $A$ and $B$ are noetherian, regular and $\mathrm{Spec} B \rightarrow \mathrm{Spec} A$ is quasi-finite. Is is necessary that $f$ is flat?

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    $\begingroup$ Obviously not. But this looks like homework, and anyways is not appropriate for mathoverflow. Try math.stackexchange.com instead? Or consider a closed immersion. $\endgroup$
    – Will Sawin
    Jun 4, 2012 at 0:09
  • $\begingroup$ You need a dimension condition to hold. $\endgroup$
    – Mahdi Majidi-Zolbanin
    Jun 4, 2012 at 5:51

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