Lemma 15.51.6. Let $A$ be a $P$-ring where $P$ satisfies (B) and (D). Let $I \subset A$ be an ideal and let $A^\wedge$ be the completion of $A$ with respect to $I$. Then the fibres of $A \to A^\wedge$ have $P$.

Proof. The ring map $A \to A^\wedge$ is flat by Algebra, Lemma 10.97.2. The ring $A^\wedge$ is Noetherian by Algebra, Lemma 10.97.6. Thus it suffices to check the third condition of Lemma 15.51.2. Let $\mathfrak m' \subset A^\wedge$ be a maximal ideal lying over $\mathfrak m \subset A$. By Algebra, Lemma 10.96.6 we have $IA^\wedge \subset \mathfrak m'$. Since $A^\wedge /IA^\wedge = A/I$ we see that $I \subset \mathfrak m$, $\mathfrak m/I = \mathfrak m'/IA^\wedge$, and $A/\mathfrak m = A^\wedge /\mathfrak m'$. Since $A^\wedge /\mathfrak m'$ is a field, we conclude that $\mathfrak m$ is a maximal ideal as well. Then $A_\mathfrak m \to A^\wedge _{\mathfrak m'}$ is a flat local ring homomorphism of Noetherian local rings which identifies residue fields and such that $\mathfrak m A^\wedge _{\mathfrak m'} = \mathfrak m'A^\wedge _{\mathfrak m'}$. Thus it induces an isomorphism on complete local rings, see Lemma 15.43.9. Let $(A_\mathfrak m)^\wedge$ be the completion of $A_\mathfrak m$ with respect to its maximal ideal. The ring map

$(A^\wedge )_{\mathfrak m'} \to ((A^\wedge )_{\mathfrak m'})^\wedge = (A_\mathfrak m)^\wedge$

is faithfully flat (Algebra, Lemma 10.97.3). Thus we can apply (D) to the ring maps

$A_\mathfrak m \to (A^\wedge )_{\mathfrak m'} \to (A_\mathfrak m)^\wedge$

to conclude because the fibres of $A_\mathfrak m \to (A_\mathfrak m)^\wedge$ have $P$ as $A$ is a $P$-ring. $\square$

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