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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


Comments (2)

Comment #10047 by on

Lemma 045K can be cited at the end of the sentence "Thus we are done if the characteristic of is ".

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