Lemma 15.11.8. Let $(A, I)$ be a henselian pair and let $A \to B$ be an integral ring map. Then $(B, IB)$ is a henselian pair.

Proof. Immediate from the fourth characterization of henselian pairs in Lemma 15.11.6 and the fact that the composition of integral ring maps is integral. $\square$

Comment #7377 by comment_bot on

It may be useful to include the statement that the same holds for any $A \rightarrow B$ that is a universal homeomorphism on spectra (sorry if this is already in the Stacks Project but I missed it!). I think this follows from the characterization of Henselian pairs in terms of lifting idempotents.

Comment #7378 by Laurent Moret-Bailly on

@#7377: A universal homeomorphism is integral (EGA IV, 18.12.10). So perhaps this should be (resp. already is) in the Stacks Project.

Comment #7379 by Matthieu Romagny on

Yes, it is in the SP, see for instance Lemma 01WM.

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