Theorems · Theorem · commutative algebra

Submodule.traceDual_le_span_map_traceDual

∀ (A : Type u_1) (B : Type u_2) {K : Type u_3} {L : Type u_4} [inst : CommRing A] [inst_1 : Field K]
  [inst_2 : Algebra A K] [IsFractionRing A K] [inst_4 : CommRing B] [inst_5 : Field L] [inst_6 : Algebra B L]
  [inst_7 : Algebra A L] [inst_8 : Algebra K L] [FiniteDimensional K L] [inst_10 : IsScalarTower A K L] (R₁ : Type u_5)
  (R₂ : Type u_6) [inst_11 : CommRing R₁] [inst_12 : CommRing R₂] [IsDomain R₁] [inst_14 : Algebra A R₁]
  [inst_15 : Algebra A R₂] [inst_16 : Algebra R₁ B] [inst_17 : Algebra R₂ B] [inst_18 : Algebra R₁ L]
  [inst_19 : Algebra R₂ L] [IsScalarTower A R₁ L] [inst_21 : IsScalarTower R₁ B L] [IsScalarTower R₂ B L]
  [Module.Finite A R₂] {F₁ F₂ : IntermediateField K L} [inst_24 : Algebra R₁ ↥F₁] [inst_25 : Algebra R₂ ↥F₂]
  [Module.IsTorsionFree R₁ ↥F₁] [IsScalarTower A (↥F₂) L] [inst_28 : IsScalarTower A R₂ ↥F₂]
  [inst_29 : IsScalarTower R₁ (↥F₁) L] [IsScalarTower R₂ (↥F₂) L] [Algebra.IsSeparable K ↥F₂]
  [Algebra.IsSeparable (↥F₁) L] [Module.Free A R₂]
  [IsLocalization (Algebra.algebraMapSubmonoid R₂ (nonZeroDivisors A)) ↥F₂],
  F₁.LinearDisjoint ↥F₂ →
    F₁ ⊔ F₂ = ⊤ →
      Submodule.restrictScalars R₁ (Submodule.traceDual R₁ (↥F₁) 1) ≤
        Submodule.span R₁ (⇑(algebraMap (↥F₂) L) '' ↑(Submodule.traceDual A K 1))

Defined in

Mathlib.RingTheory.DedekindDomain.LinearDisjoint

Cited by

2 results in Mathlib

Foundations

Depth 185 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingFieldAlgebraIsFractionRingCommRingFieldAlgebraAlgebraAlgebraFiniteDimensionalIsScalarTowerCommRingCommRingIsDomainAlgebraAlgebraAlgebraAlgebraAlgebraAlgebraIsScalarTowerIsScalarTowerIsScalarTowerModule.FiniteAlgebraAlgebraModule.IsTorsionFreeIsScalarTowerIsScalarTowerIsScalarTowerIsScalarTowerAlgebra.IsSeparableAlgebra.IsSeparableModule.FreeIsLocalization