Theorems · Theorem · commutative algebra

FractionalIdeal.trace_mem_dual_one

∀ (A : Type u_1) (K : Type u_2) (L : Type u) (B : Type u_5) [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] (C : Type u_3) (M : Type u_4) [inst_19 : CommRing C] [inst_20 : IsDedekindDomain C]
  [inst_21 : Field M] [inst_22 : Algebra C M] [inst_23 : IsFractionRing C M] [inst_24 : Algebra A C]
  [inst_25 : Algebra B C] [inst_26 : Algebra A M] [inst_27 : Algebra B M] [inst_28 : Algebra K M]
  [inst_29 : Algebra L M] [inst_30 : IsScalarTower A C M] [inst_31 : IsScalarTower A K M] [IsScalarTower B C M]
  [IsScalarTower B L M] [IsScalarTower K L M] [inst_35 : IsIntegralClosure C A M] [inst_36 : FiniteDimensional K M]
  [FiniteDimensional L M] [inst_38 : Algebra.IsSeparable K M],
  ∀ x ∈ FractionalIdeal.dual A K 1, (Algebra.trace L M) x ∈ FractionalIdeal.dual A K 1

Defined in

Mathlib.RingTheory.DedekindDomain.Different

Cited by

1 results in Mathlib

Foundations

Depth 191 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingFieldCommRingFieldAlgebraAlgebraAlgebraAlgebraAlgebraIsScalarTowerIsScalarTowerIsDomainIsFractionRingFiniteDimensionalAlgebra.IsSeparableIsIntegralClosureIsFractionRingIsIntegrallyClosedIsDedekindDomainCommRingIsDedekindDomainFieldAlgebraIsFractionRingAlgebraAlgebraAlgebraAlgebraAlgebraAlgebraIsScalarTowerIsScalarTowerIsScalarTowerIsScalarTowerIsScalarTowerIsIntegralClosureFiniteDimensionalFiniteDimensionalAlgebra.IsSeparable