Lemma 10.154.5. Let $R \to S$ be a ring map with $S$ henselian local. Given

1. an $R$-algebra $A$ which is a filtered colimit of étale $R$-algebras,

2. a prime $\mathfrak q$ of $A$ lying over $\mathfrak p = R \cap \mathfrak m_ S$,

3. a $\kappa (\mathfrak p)$-algebra map $\tau : \kappa (\mathfrak q) \to S/\mathfrak m_ S$,

then there exists a unique homomorphism of $R$-algebras $f : A \to S$ such that $\mathfrak q = f^{-1}(\mathfrak m_ S)$ and $f \bmod \mathfrak q = \tau$.

Proof. Write $A = \mathop{\mathrm{colim}}\nolimits A_ i$ as a filtered colimit of étale $R$-algebras. Set $\mathfrak q_ i = A_ i \cap \mathfrak q$. We obtain $f_ i : A_ i \to S$ by applying Lemma 10.153.11. Set $f = \mathop{\mathrm{colim}}\nolimits f_ i$. $\square$

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