lift — Mathlib

English

For a local ring hom f: R →+* S into a field, the composition of lift f with the residue map equals f.

Русский

При локальном кольцо-гомоморфизме f: R →+* S в поле композиция lift f и остаточного отображения равна f.

LaTeX

$$$(\\mathrm{IsLocalRing.ResidueField.lift} f) \\circ (\\mathrm{IsLocalRing.ResidueField.residue} R) = f$$$

Lean4

/-- A local ring homomorphism into a field can be descended onto the residue field. -/
def lift {R S : Type*} [CommRing R] [IsLocalRing R] [Field S] (f : R →+* S) [IsLocalHom f] :
    IsLocalRing.ResidueField R →+* S :=
  Ideal.Quotient.lift _ f fun a ha => by_contradiction fun h => ha (isUnit_of_map_unit f a (isUnit_iff_ne_zero.mpr h))

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