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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