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 $X$.

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View Example 58.13.3 as pdf