The Stacks project

Proof. Let $A$ be a Noetherian complete local ring. By Lemma 15.50.2 it suffices to check that $B = A/\mathfrak q$ has geometrically regular formal fibres over the minimal prime $(0)$ of $B$. Thus we may assume that $A$ is a domain and it suffices to check the condition for the formal fibres over the minimal prime $(0)$ of $A$. Let $K$ be the fraction field of $A$.

We can choose a subring $A_0 \subset A$ which is a regular complete local ring such that $A$ is finite over $A_0$, see Algebra, Lemma 10.160.11. Moreover, we may assume that $A_0$ is a power series ring over a field or a Cohen ring. By Lemma 15.50.3 we see that it suffices to prove the result for $A_0$.

Assume that $A$ is a power series ring over a field or a Cohen ring. Since $A$ is regular the localizations $A_\mathfrak p$ are regular (see Algebra, Definition 10.110.7 and the discussion preceding it). Hence the completions $A_\mathfrak p^\wedge $ are regular, see Lemma 15.43.4. Hence the fibre $A_{\mathfrak p}^\wedge \otimes _ A K$ is, as a localization of $A_\mathfrak p^\wedge $, also regular. Thus we are done if the characteristic of $K$ is $0$. The positive characteristic case is the case $A = k[[x_1, \ldots , x_ d]]$ which is a special case of Lemma 15.50.5. $\square$

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