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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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Obviously not. But this looks like homework, and anyways is not appropriate for mathoverflow. Try instead? Or consider a closed immersion. –  Will Sawin Jun 4 '12 at 0:09
You need a dimension condition to hold. –  Mahdi Majidi-Zolbanin Jun 4 '12 at 5:51

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