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

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

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

3. 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.

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

Proof. Part (2) implies part (1) since $S$ as in part (1) is a finite product of its localizations at the primes lying over $\mathfrak m$. Part (2) follows from Lemma 10.153.3 part (10) since any finite $S$-algebra is also a finite $R$-algebra. If $R \to S$ and $\mathfrak q$ are as in (3), then $S_{\mathfrak q}$ is a local ring of a finite $R$-algebra by Lemma 10.153.3 part (11). Hence (3) follows from (1). Part (4) follows from part (3). $\square$

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