The Stacks project

Lemma 15.43.6. Let $A$ be a Noetherian local ring.

  1. If $A^\wedge $ is reduced, then so is $A$.

  2. In general $A$ reduced does not imply $A^\wedge $ is reduced.

  3. If $A$ is Nagata, then $A$ is reduced if and only if $A^\wedge $ is reduced.

Proof. As $A \to A^\wedge $ is faithfully flat we have (1) by Algebra, Lemma 10.164.2. For (2) see Algebra, Example 10.119.5 (there are also examples in characteristic zero, see Algebra, Remark 10.119.6). For (3) see Algebra, Lemmas 10.162.13 and 10.162.10. $\square$

Comments (2)

Comment #4260 by DS on

Can I suggest adding to the lemma: (4) If \hat{A} is normal, then so is A (with a reference to 033G as the proof)?

Comment #4430 by on

OK, I've added the statement as a separate lemma. See here.

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