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

If $R \to S$ is a finite ring map then $S$ is a finite product of henselian local rings each finite over $R$.

If $R \to S$ is a finite ring map and $S$ is local, then $S$ is a henselian local ring and $R \to S$ is a (finite) local ring map.

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 and finite over $R$.

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

## Comments (0)

There are also: