The Stacks project

Lemma 58.31.5. In the situation of Lemma 58.31.4 the normalization of $X$ in $Y$ is a finite locally free morphism $\pi : Y' \to X$ such that

  1. the restriction of $Y'$ to $X \setminus D$ is isomorphic to $Y$,

  2. $D' = \pi ^{-1}(D)_{red}$ is an effective Cartier divisor on $Y'$, and

  3. $D'$ is a regular scheme.

Moreover, étale locally on $X$ the morphism $Y' \to X$ is a finite disjoint union of morphisms

\[ \mathop{\mathrm{Spec}}(A[x]/(x^ e - f)) \to \mathop{\mathrm{Spec}}(A) \]

where $A$ is a Noetherian ring, $f \in A$ is a nonzerodivisor with $A/fA$ regular, and $e \geq 1$ is invertible in $A$.

Proof. This is just an addendum to Lemma 58.31.4 and in fact the truth of this lemma follows almost immediately if you've read the proof of that lemma. But we can also deduce the lemma from the result of Lemma 58.31.4. Namely, taking the normalization of $X$ in $Y$ commutes with étale base change, see More on Morphisms, Lemma 37.18.2. Hence we see that we may prove the statements on the local structure of $Y' \to X$ étale locally on $X$. Thus, by Lemma 58.31.4 we may assume that $X = \mathop{\mathrm{Spec}}(A)$ where $A$ is a Noetherian ring, that we have a nonzerodivisor $f\in A$ such that $A/fA$ is regular, and that $Y$ is a finite disjoint union of spectra of rings $A_ f[x]/(x^ e - f)$ where $e$ is invertible in $A$. We omit the verification that the integral closure of $A$ in $A_ f[x]/(x^ e - f)$ is equal to $A' = A[x]/(x^ e - f)$. (To see this argue that the localizations of $A'$ at primes lying over $(f)$ are regular.) We omit the details. $\square$


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