The Stacks project

Example 58.13.3. Let $X$ be a normal integral Noetherian scheme with function field $K$. Purity of branch locus (see below) tells us that if $X$ is regular, then it suffices in Lemma 58.13.2 to consider the inertia groups $I = \pi _1(\mathop{\mathrm{Spec}}(K_ x^{sh}))$ for points $x$ of codimension $1$ in $X$. In general this is not enough however. Namely, let $Y = \mathbf{A}_ k^ n = \mathop{\mathrm{Spec}}(k[t_1, \ldots , t_ n])$ where $k$ is a field not of characteristic $2$. Let $G = \{ \pm 1\} $ be the group of order $2$ acting on $Y$ by multiplication on the coordinates. Set

\[ X = \mathop{\mathrm{Spec}}(k[t_ it_ j, i, j \in \{ 1, \ldots , n\} ]) \]

The embedding $k[t_ it_ j] \subset k[t_1, \ldots , t_ n]$ defines a degree $2$ morphism $Y \to X$ which is unramified everywhere except over the maximal ideal $\mathfrak m = (t_ it_ j)$ which is a point of codimension $n$ in $X$.

Comments (0)

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Your email address will not be published.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 0BTE. Beware of the difference between the letter 'O' and the digit '0'.