Theorems · Definition · commutative algebra

IsLocalRing.ResidueField.lift

{R : Type u_4} →
  {S : Type u_5} →
    [inst : CommRing R] →
      [inst_1 : IsLocalRing R] → [inst_2 : Field S] → (f : R →+* S) → [IsLocalHom f] → IsLocalRing.ResidueField R →+* S

A local ring homomorphism into a field can be descended onto the residue field.

Defined in: Mathlib.RingTheory.LocalRing.ResidueField.Basic
Cited by: 2 results in Mathlib
Foundations: Depth 94 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommRing IsLocalRing Field IsLocalHom

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Cites8

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

CommRingstatement and proof · cited by 17,173
RingHomstatement and proof · cited by 10,189
Fieldstatement and proof · cited by 7,404
IsLocalRingstatement and proof · cited by 339
IsLocalRing.maximalIdealproof · cited by 297
IsLocalRing.ResidueFieldstatement · cited by 156
IsLocalHomstatement and proof · cited by 100
Ideal.Quotient.liftproof · cited by 19

Cited by3

Results whose statement or proof uses this declaration.

AlgebraicGeometry.Scheme.descResidueFieldproof · cited by 10
IsLocalRing.ResidueField.lift_comp_residuestatement · cited by 0
IsLocalRing.ResidueField.lift_residue_applystatement · cited by 0