The Stacks project

Lemma 37.17.4 (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 29.52.1 and Morphisms, Lemma 29.51.1; you can also use the description of the normalization in Morphisms, Lemma 29.52.3. 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 15.44.3). 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 10.38.19 and 10.29.7). Thus More on Algebra, Lemma 15.44.13 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 10.144.2. 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 10.144.5). $\square$


Comments (2)

Comment #4646 by on

Does this Lemma hold for the completion of the strict Henselisation of the local ring?

Let be the completion of the strict henselization. Then is the fiber in bijection with .

Comment #4792 by on

In general (for strictly henselian Noetherian rings) completion does not commute with normalization. For excellent rings it will be true (hint: use something like Lemma 15.51.6). Sorry, but your second question about the fibre did not make sense to me.


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