The Stacks project

Remark 59.33.4. Let $S$ be a scheme. Let $s \in S$. If $S$ is locally Noetherian then $\mathcal{O}_{S, s}^ h$ is also Noetherian and it has the same completion:

\[ \widehat{\mathcal{O}_{S, s}} \cong \widehat{\mathcal{O}_{S, s}^ h}. \]

In particular, $\mathcal{O}_{S, s} \subset \mathcal{O}_{S, s}^ h \subset \widehat{\mathcal{O}_{S, s}}$. The henselization of $\mathcal{O}_{S, s}$ is in general much smaller than its completion and inherits many of its properties. For example, if $\mathcal{O}_{S, s}$ is reduced, then so is $\mathcal{O}_{S, s}^ h$, but this is not true for the completion in general. Insert future references here.

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