Theorems · Theorem · commutative algebra

dvd_differentIdeal_of_not_isSeparable

∀ (A : Type u_1) {B : Type u_3} [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] [inst_3 : IsDomain A]
  [inst_4 : IsDedekindDomain A] [inst_5 : IsDedekindDomain B] [inst_6 : Module.IsTorsionFree A B]
  [inst_7 : Module.Finite A B] [Algebra.IsSeparable (FractionRing A) (FractionRing B)] {p : Ideal A} [p.IsMaximal],
  p ≠ ⊥ →
    ∀ (P : Ideal B) [P.IsMaximal] [inst_11 : P.LiesOver p],
      ¬Algebra.IsSeparable (A ⧸ p) (B ⧸ P) → P ∣ differentIdeal A B

Defined in

Mathlib.RingTheory.DedekindDomain.Different

Cited by

1 results in Mathlib

Foundations

Depth 201 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraIsDomainIsDedekindDomainIsDedekindDomainModule.IsTorsionFreeModule.FiniteAlgebra.IsSeparableIdeal.IsMaximalIdeal.IsMaximalIdeal.LiesOver