Lemma 54.13.3. Let $(A, \mathfrak m, \kappa )$ be a local Noetherian domain. Assume there exists an alteration $f : X \to \mathop{\mathrm{Spec}}(A)$ with $X$ regular. Then

1. there exists a nonzero $f \in A$ such that $A_ f$ is regular,

2. the integral closure $B$ of $A$ in its fraction field is finite over $A$,

3. the $\mathfrak m$-adic completion of $B$ is a normal ring, i.e., the completions of $B$ at its maximal ideals are normal domains, and

4. the generic formal fibre of $A$ is regular.

Proof. Parts (1) and (2) follow from Lemma 54.13.1. We have to redo part of the proof of that lemma in order to set up notation for the proof of (3). Set $C = \Gamma (X, \mathcal{O}_ X)$. By Cohomology of Schemes, Lemma 30.19.2 we see that $C$ is a finite $A$-module. As $X$ is normal (Properties, Lemma 28.9.4) we see that $C$ is normal domain (Properties, Lemma 28.7.9). Thus $B \subset C$ and we conclude that $B$ is finite over $A$ as $A$ is Noetherian. By Lemma 54.13.2 in order to prove (3) it suffices to show that the $\mathfrak m$-adic completion $C^\wedge$ is normal.

By Algebra, Lemma 10.97.8 the completion $C^\wedge$ is the product of the completions of $C$ at the prime ideals of $C$ lying over $\mathfrak m$. There are finitely many of these and these are the maximal ideals $\mathfrak m_1, \ldots , \mathfrak m_ r$ of $C$. (The corresponding result for $B$ explains the final statement of the lemma.) Thus replacing $A$ by $C_{\mathfrak m_ i}$ and $X$ by $X_ i = X \times _{\mathop{\mathrm{Spec}}(C)} \mathop{\mathrm{Spec}}(C_{\mathfrak m_ i})$ we reduce to the case discussed in the next paragraph. (Note that $\Gamma (X_ i, \mathcal{O}) = C_{\mathfrak m_ i}$ by Cohomology of Schemes, Lemma 30.5.2.)

Here $A$ is a Noetherian local normal domain and $f : X \to \mathop{\mathrm{Spec}}(A)$ is a regular alteration with $\Gamma (X, \mathcal{O}_ X) = A$. We have to show that the completion $A^\wedge$ of $A$ is a normal domain. By Lemma 54.11.2 $Y = X \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A^\wedge )$ is regular. Since $\Gamma (Y, \mathcal{O}_ Y) = A^\wedge$ by Cohomology of Schemes, Lemma 30.5.2, we conclude that $A^\wedge$ is normal as before. Namely, $Y$ is normal by Properties, Lemma 28.9.4. It is connected because $\Gamma (Y, \mathcal{O}_ Y) = A^\wedge$ is local. Hence $Y$ is normal and integral (as connected and normal implies integral for Noetherian schemes). Thus $\Gamma (Y, \mathcal{O}_ Y) = A^\wedge$ is a normal domain by Properties, Lemma 28.7.9. This proves (3).

Proof of (4). Let $\eta \in \mathop{\mathrm{Spec}}(A)$ denote the generic point and denote by a subscript $\eta$ the base change to $\eta$. Since $f$ is an alteration, the scheme $X_\eta$ is finite and faithfully flat over $\eta$. Since $Y = X \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A^\wedge )$ is regular by Lemma 54.11.2 we see that $Y_\eta$ is regular (as a limit of opens in $Y$). Then $Y_\eta \to \mathop{\mathrm{Spec}}(A^\wedge \otimes _ A \kappa (\eta ))$ is finite faithfully flat onto the generic formal fibre. We conclude by Algebra, Lemma 10.164.4. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0BGI. Beware of the difference between the letter 'O' and the digit '0'.