Lemma 15.38.3. Let $k$ be a field. Let $(A, \mathfrak m, \kappa )$ be a complete local $k$-algebra. If $\kappa /k$ is separable, then there exists a $k$-algebra map $\kappa \to A$ such that $\kappa \to A \to \kappa$ is $\text{id}_\kappa$.

Proof. By Algebra, Proposition 10.158.9 the extension $\kappa /k$ is formally smooth. By Lemma 15.37.2 $k \to \kappa$ is formally smooth in the sense of Definition 15.37.1. Then we get $\kappa \to A$ from Lemma 15.37.5. $\square$

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