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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