Lemma 35.15.4. The property $\mathcal{P}(S) =$“$S$ is regular” is local in the smooth topology.

**Proof.**
This is clear from Lemma 35.15.3 above since a locally Noetherian scheme is regular if and only if it is locally Noetherian and $(R_ k)$ for all $k \geq 0$.
$\square$

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