Lemma 10.153.4. Let $(R, \mathfrak m, \kappa )$ be a henselian local ring.

If $R \subset S$ is a finite ring extension then $S$ is a finite product of henselian local rings.

If $R \subset S$ is a finite local homomorphism of local rings, then $S$ is a henselian local ring.

If $R \to S$ is a finite type ring map, and $\mathfrak q$ is a prime of $S$ lying over $\mathfrak m$ at which $R \to S$ is quasi-finite, then $S_{\mathfrak q}$ is henselian.

If $R \to S$ is quasi-finite then $S_{\mathfrak q}$ is henselian for every prime $\mathfrak q$ lying over $\mathfrak m$.

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