Lemma 15.50.4. Let $R$ be a Noetherian ring. Then $R$ is a G-ring if and only if for every finite free ring map $R \to S$ the formal fibres of $S$ are regular rings.

**Proof.**
Assume that for any finite free ring map $R \to S$ the ring $S$ has regular formal fibres. Let $\mathfrak q \subset \mathfrak p \subset R$ be primes and let $\kappa (\mathfrak q) \subset L$ be a finite purely inseparable extension. To show that $R$ is a G-ring it suffices to show that

is a regular ring. Choose a finite free extension $R \to R'$ such that $\mathfrak q' = \mathfrak qR'$ is a prime and such that $\kappa (\mathfrak q')$ is isomorphic to $L$ over $\kappa (\mathfrak q)$, see Algebra, Lemma 10.159.3. By Algebra, Lemma 10.97.8 we have

where $\mathfrak p_ i'$ are the primes of $R'$ lying over $\mathfrak p$. Thus we have

Our assumption is that the rings on the right are regular, hence the ring on the left is regular too. Thus $R$ is a G-ring. The converse follows from Lemma 15.50.3. $\square$

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