The Stacks project

Proof. Let $\eta _1, \ldots , \eta _ r$ be the generic points of the irreducible components of $X$ passing through $x$. The base change of the normalization to $\mathop{\mathrm{Spec}}(\mathcal{O}_{X, x})$ is the spectrum of the integral closure of $\mathcal{O}_{X, x}$ in $\prod \kappa (\eta _ i)$. This follows from our construction of the normalization of $X$ in Morphisms, Definition 29.54.1 and Morphisms, Lemma 29.53.1; you can also use the description of the normalization in Morphisms, Lemma 29.54.3. Thus we reduce to the following algebra problem. Let $A$ be a Noetherian local ring; recall that this implies the henselization $A^ h$ and strict henselization $A^{sh}$ are Noetherian too (More on Algebra, Lemma 15.45.3). Let $\mathfrak p_1, \ldots , \mathfrak p_ r$ be its minimal primes. Let $A'$ be the integral closure of $A$ in $\prod \kappa (\mathfrak p_ i)$. Problem: show that $A' \otimes _ A A^ h$, resp. $A' \otimes _ A A^{sh}$ is constructed from the Noetherian local ring $A^ h$, resp. $A^{sh}$ in the same manner.

Since $A^ h$, resp. $A^{sh}$ are colimits of étale $A$-algebras, we see that the minimal primes of $A$ and $A^{sh}$ are exactly the primes of $A^ h$, resp. $A^{sh}$ lying over the minimal primes of $A$ (by going down, see Algebra, Lemmas 10.39.19 and 10.30.7). Thus More on Algebra, Lemma 15.45.13 tells us that $A^ h \otimes _ A \prod \kappa (\mathfrak p_ i)$, resp. $A^{sh} \otimes _ A \prod \kappa (\mathfrak p_ i)$ is the product of the residue fields at the minimal primes of $A^ h$, resp. $A^{sh}$. We know that taking the integral closure in an overring commutes with étale base change, see Algebra, Lemma 10.147.2. Writing $A^ h$ and $A^{sh}$ as a limit of étale $A$-algebras we see that the same thing is true for the base change to $A^ h$ and $A^{sh}$ (you can also use the more general Algebra, Lemma 10.147.5). $\square$