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

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

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

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

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

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

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

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 10.163.2. 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 10.161.13 and 10.161.10.
$\square$

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## Comments (2)

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