The Stacks project

Definition 59.33.2. Let $S$ be a scheme. Let $\overline{s}$ be a geometric point of $S$ lying over the point $s \in S$.

1. The étale local ring of $S$ at $\overline{s}$ is the stalk of the structure sheaf $\mathcal{O}_ S$ on $S_{\acute{e}tale}$ at $\overline{s}$. We sometimes call this the strict henselization of $\mathcal{O}_{S, s}$ relative to the geometric point $\overline{s}$. Notation used: $\mathcal{O}_{S, \overline{s}}^{sh}$.

2. The henselization of $\mathcal{O}_{S, s}$ is the henselization of the local ring of $S$ at $s$. See Algebra, Definition 10.155.3, and Theorem 59.32.8. Notation: $\mathcal{O}_{S, s}^ h$.

3. The strict henselization of $S$ at $\overline{s}$ is the scheme $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{s}}^{sh})$.

4. The henselization of $S$ at $s$ is the scheme $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}^ h)$.