The Stacks project

Example 58.31.4 (Standard tamely ramified morphism). Let $A$ be a Noetherian ring. Let $f \in A$ be a nonzerodivisor such that $A/fA$ is reduced. This implies that $A_\mathfrak p$ is a discrete valuation ring with uniformizer $f$ for any minimal prime $\mathfrak p$ over $f$. Let $e \geq 1$ be an integer which is invertible in $A$. Set

\[ C = A[x]/(x^ e - f) \]

Then $\mathop{\mathrm{Spec}}(C) \to \mathop{\mathrm{Spec}}(A)$ is a finite locally free morphism which is ├ętale over the spectrum of $A_ f$. The finite ├ętale morphism

\[ \mathop{\mathrm{Spec}}(C_ f) \longrightarrow \mathop{\mathrm{Spec}}(A_ f) \]

is tamely ramified over $\mathop{\mathrm{Spec}}(A)$ in codimension $1$. The tameness follows immediately from the characterization of tamely ramified extensions in More on Algebra, Lemma 15.114.7.

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