The Stacks project

Example 58.31.3 (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.


Comments (0)

There are also:

  • 4 comment(s) on Section 58.31: Tame ramification

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0EYF. Beware of the difference between the letter 'O' and the digit '0'.