Theorems · Theorem · commutative algebra

Algebra.isSeparable_residueField_iff

∀ {A : Type u_2} {B : Type u_3} [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] {p : Ideal A}
  {q : Ideal B} [inst_3 : q.LiesOver p] [inst_4 : p.IsMaximal] [inst_5 : q.IsMaximal]
  [inst_6 : Algebra (Localization.AtPrime p) (Localization.AtPrime q)]
  [inst_7 : Localization.AtPrime.IsLiesOverAlgebra p q],
  Algebra.IsSeparable p.ResidueField q.ResidueField ↔ Algebra.IsSeparable (A ⧸ p) (B ⧸ q)

Defined in

Mathlib.RingTheory.LocalRing.ResidueField.Instances

Cited by

0 results in Mathlib

Foundations

Depth 126 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraIdeal.LiesOverIdeal.IsMaximalIdeal.IsMaximalAlgebraLocalization.AtPrime.IsLiesOverAlgebra