The Stacks project

Lemma 15.52.4. Let $(A, \mathfrak m)$ be a Noetherian local ring. The following are equivalent

  1. $A$ is Nagata, and

  2. the formal fibres of $A$ are geometrically reduced.

Proof. Assume (2). By Algebra, Lemma 10.162.14 we have to show that if $A \to B$ is finite, $B$ is a domain, and $\mathfrak m' \subset B$ is a maximal ideal, then $B_{\mathfrak m'}$ is analytically unramified. Combining Lemmas 15.51.9 and 15.51.4 and Proposition 15.51.5 we see that the formal fibres of $B_{\mathfrak m'}$ are geometrically reduced. In particular $B_{\mathfrak m'}^\wedge \otimes _ B L$ is reduced where $L$ is the fraction field of $B$. It follows that $B_{\mathfrak m'}^\wedge $ is reduced, i.e., $B_{\mathfrak m'}$ is analytically unramified.

Assume (1). Let $\mathfrak q \subset A$ be a prime ideal and let $K/\kappa (\mathfrak q)$ be a finite extension. We have to show that $A^\wedge \otimes _ A K$ is reduced. Let $A/\mathfrak q \subset B \subset K$ be a local subring finite over $A$ whose fraction field is $K$. To construct $B$ choose $x_1, \ldots , x_ n \in K$ which generate $K$ over $\kappa (\mathfrak q)$ and which satisfy monic polynomials $P_ i(T) = T^{d_ i} + a_{i, 1} T^{d_ i - 1} + \ldots + a_{i, d_ i} = 0$ with $a_{i, j} \in \mathfrak m$. Then let $B$ be the $A$-subalgebra of $K$ generated by $x_1, \ldots , x_ n$. (For more details see the proof of Algebra, Lemma 10.162.14.) Then

\[ A^\wedge \otimes _ A K = (A^\wedge \otimes _ A B)_\mathfrak q = B^\wedge _\mathfrak q \]

Since $B^\wedge $ is reduced by Algebra, Lemma 10.162.14 the proof is complete. $\square$

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