Theorems · Theorem · commutative algebra

FractionalIdeal.mem_dual

∀ {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]
  [inst_12 : IsFractionRing A K] [inst_13 : FiniteDimensional K L] [inst_14 : Algebra.IsSeparable K L]
  [inst_15 : IsIntegralClosure B A L] [inst_16 : IsFractionRing B L] [inst_17 : IsIntegrallyClosed A]
  [inst_18 : IsDedekindDomain B] {I : FractionalIdeal (nonZeroDivisors B) L},
  I ≠ 0 → ∀ {x : L}, x ∈ FractionalIdeal.dual A K I ↔ ∀ a ∈ I, ((Algebra.traceForm K L) x) a ∈ (algebraMap A K).range

Defined in

Mathlib.RingTheory.DedekindDomain.Different

Cited by

3 results in Mathlib

Foundations

Depth 189 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingFieldCommRingFieldAlgebraAlgebraAlgebraAlgebraAlgebraIsScalarTowerIsScalarTowerIsDomainIsFractionRingFiniteDimensionalAlgebra.IsSeparableIsIntegralClosureIsFractionRingIsIntegrallyClosedIsDedekindDomain