Lemma 15.51.1. Let $R$ be a Noetherian ring. Let $P$ be a property as above. Then $R$ is a $P$-ring if and only if for every pair of primes $\mathfrak q \subset \mathfrak p \subset R$ the $\kappa (\mathfrak q)$-algebra

has property $P$.

Lemma 15.51.1. Let $R$ be a Noetherian ring. Let $P$ be a property as above. Then $R$ is a $P$-ring if and only if for every pair of primes $\mathfrak q \subset \mathfrak p \subset R$ the $\kappa (\mathfrak q)$-algebra

\[ (R/\mathfrak q)_\mathfrak p^\wedge \otimes _{R/\mathfrak q} \kappa (\mathfrak q) \]

has property $P$.

**Proof.**
This follows from the fact that

\[ R_\mathfrak p^\wedge \otimes _ R \kappa (\mathfrak q) = (R/\mathfrak q)_\mathfrak p^\wedge \otimes _{R/\mathfrak q} \kappa (\mathfrak q) \]

as algebras over $\kappa (\mathfrak q)$. $\square$

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