The Stacks project

Lemma 15.50.2. Let $R$ be a Noetherian ring. Then $R$ is a G-ring if and only if for every pair of primes $\mathfrak q \subset \mathfrak p \subset R$ the algebra

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

is geometrically regular over $\kappa (\mathfrak q)$.

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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