Lemma 15.115.10. Let $A$ be a local ring annihilated by a prime $p$ whose maximal ideal is nilpotent. There exists a ring map $\sigma : \kappa _ A \to A$ which is a section to the residue map $A \to \kappa _ A$. If $A \to A'$ is a local homomorphism of local rings, then we can choose a similar ring map $\sigma ' : \kappa _{A'} \to A'$ compatible with $\sigma$ provided that the extension $\kappa _{A'}/\kappa _ A$ is separable.

Proof. Separable extensions are formally smooth by Algebra, Proposition 10.158.9. Thus the existence of $\sigma$ follows from the fact that $\mathbf{F}_ p \to \kappa _ A$ is separable. Similarly for the existence of $\sigma '$ compatible with $\sigma$. $\square$

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