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The Stacks project

Lemma 15.13.3. Let R be a ring and S a smooth R-algebra. Assume that A is an R-algebra and (A,I) is a henselian pair. Then any R-algebra map S \to A/I can be lifted to an R-algebra map S \to A.

Proof. Let \tau : S \to A/I be an R-algebra map. Observe that S \otimes _ R A is a smooth A-algebra by Algebra, Lemma 10.137.4. Thus by Lemma 15.9.14 we can lift the induced map S \otimes _ R A \to A/I to an A-algebra homorphism S \otimes _ R A \to A' where A \to A' is étale and induces an isomorphism A/I \to A'/IA'. Since (A, I) is henselian there is an A-algebra map A' \to A, see Lemma 15.11.6. The composition S \to S \otimes _ R A \to A' \to A is the desired lift. \square


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