Theorems · Theorem · commutative algebra

not_dvd_differentIdeal_of_isCoprime_of_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 Q : Ideal B) [P.IsMaximal] [inst_11 : P.LiesOver p],
  IsCoprime P Q →
    P * Q = Ideal.map (algebraMap A B) p → ∀ [Algebra.IsSeparable (A ⧸ p) (B ⧸ P)], ¬P ∣ differentIdeal A B

Defined in

Mathlib.RingTheory.DedekindDomain.Different

Cited by

2 results in Mathlib

Foundations

Depth 199 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraIsDomainIsDedekindDomainIsDedekindDomainModule.IsTorsionFreeModule.FiniteAlgebra.IsSeparableIdeal.IsMaximalIdeal.IsMaximalIdeal.LiesOverAlgebra.IsSeparable