The Stacks project

Lemma 59.32.5. If $R$ is henselian and $A$ is a finite $R$-algebra, then $A$ is a finite product of henselian local rings.

Proof. See Algebra, Lemma 10.153.4. $\square$

Comments (2)

Comment #7506 by Haohao Liu on

Is the ring local?

Comment #7512 by David Holmes on

Dear Haohao Liu, Henselian rings are local by definition in the stacks project, see 03QF. One could argue that the grammar is ambiguous (the def only tells us when a local ring is henselian, and could be interpreted as giving no information on when a non-local ring is henselian). But I think there it is quite standard to intrepret a definition written in this way as meaning that every local ring is henselian. Best wishes, David

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