The Stacks project

Lemma 58.31.1. Let $X' \to X$ be a morphism of locally Noetherian schemes. Let $U \subset X$ be a dense open. Assume

  1. $U' = f^{-1}(U)$ is dense open in $X'$,

  2. for every prime divisor $Z \subset X$ with $Z \cap U = \emptyset $ the local ring $\mathcal{O}_{X, \xi }$ of $X$ at the generic point $\xi $ of $Z$ is a discrete valuation ring,

  3. for every prime divisor $Z' \subset X'$ with $Z' \cap U' = \emptyset $ the local ring $\mathcal{O}_{X', \xi '}$ of $X'$ at the generic point $\xi '$ of $Z'$ is a discrete valuation ring,

  4. if $\xi ' \in X'$ is as in (3), then $\xi = f(\xi ')$ is as in (2).

Then if $f : Y \to U$ is finite ├ętale and $Y$ is unramified, resp. tamely ramified over $X$ in codimension $1$, then $Y' = Y \times _ X X' \to U'$ is finite ├ętale and $Y'$ is unramified, resp. tamely ramified over $X'$ in codimension $1$.

Proof. The only interesting fact in this lemma is the commutative algebra result given in More on Algebra, Lemma 15.114.9. $\square$


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