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Let $A$ be a Dedekind domain and let $B:=A[X]/(f(X))$, where $f(X)\in A[X]$ is some monic polynomial such that $B$ is a domain. If I take the canonical map $A\longrightarrow B$, then it nduces a continuos map $\phi\colon \mathsf{Spec} \: B\longrightarrow \mathsf{Spec} \: A$. Now let us consider a maximal ideal $\mathfrak{p}\subset A$ and and let $\mathfrak{q}\in \phi^{-1}(\mathfrak{p})$. My question is:

Q. Are there necessary and sufficient conditions on $f(X)$ such that the localization $B_\mathfrak{q}$ is a regular ring?

I just posted, without results, the question on MSE, but maybe this is the most suitable place.

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  • $\begingroup$ MSE link $\endgroup$
    – Steve D
    Commented Aug 29, 2017 at 13:26
  • $\begingroup$ @AlexM. I see that my comment came across a bit more negative than I intended. An earlier version stated: this is true when $f$ is étale at $\mathfrak q$, i.e. if $f'(\alpha) \not \in \mathfrak q$ (where $\alpha$ is a root of $f$). If that fails, I don't think there is an easy way to tell. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Aug 29, 2017 at 19:42

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