Lemma 37.19.4 (Normalization and henselization). Let X be a locally Noetherian scheme. Let \nu : X^\nu \to X be the normalization morphism. Then for any point x \in X the base change
X^\nu \times _ X \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x}^ h) \to \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x}^ h), \quad \text{resp.}\quad X^\nu \times _ X \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x}^{sh}) \to \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x}^{sh})
is the normalization of \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x}^ h), resp. \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x}^{sh}).
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
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