Theorems · Theorem · commutative algebra

differentialIdeal_le_iff

∀ {A : Type u_1} {K : Type u_2} {L : Type u} {B : Type u_3} [inst : CommRing A] [inst_1 : Field K] [inst_2 : CommRing B]
  [inst_3 : Field L] [inst_4 : Algebra A K] [inst_5 : Algebra B L] [inst_6 : Algebra A B] [inst_7 : Algebra K L]
  [inst_8 : Algebra A L] [inst_9 : IsScalarTower A K L] [inst_10 : IsScalarTower A B L] [inst_11 : IsDomain A]
  [IsFractionRing A K] [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsIntegralClosure B A L]
  [IsIntegrallyClosed A] [inst_17 : IsDedekindDomain B] [inst_18 : Module.IsTorsionFree A B]
  [inst_19 : IsFractionRing B L] {I : Ideal B},
  I ≠ ⊥ → (differentIdeal A B ≤ I ↔ Submodule.map (↑A (Algebra.trace K L)) (Submodule.restrictScalars A ↑(↑I)⁻¹) ≤ 1)

Defined in

Mathlib.RingTheory.DedekindDomain.Different

Cited by

2 results in Mathlib

Foundations

Depth 198 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingFieldCommRingFieldAlgebraAlgebraAlgebraAlgebraAlgebraIsScalarTowerIsScalarTowerIsDomainIsFractionRingFiniteDimensionalAlgebra.IsSeparableIsIntegralClosureIsIntegrallyClosedIsDedekindDomainModule.IsTorsionFreeIsFractionRing