Theorems · Theorem · commutative algebra

Module.Basis.ofIsCoprimeDifferentIdeal_apply

∀ (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] [inst_3 : 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] [inst_9 : 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] [inst_13 : 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] [inst_20 : IsScalarTower A R₁ L] [inst_21 : IsScalarTower R₁ B L]
  [inst_22 : IsScalarTower R₂ B L] [inst_23 : Module.Finite A R₂] {F₁ F₂ : IntermediateField K L}
  [inst_24 : Algebra R₁ ↥F₁] [inst_25 : Algebra R₂ ↥F₂] [inst_26 : Module.IsTorsionFree R₁ ↥F₁]
  [inst_27 : IsScalarTower A (↥F₂) L] [inst_28 : IsScalarTower A R₂ ↥F₂] [inst_29 : IsScalarTower R₁ (↥F₁) L]
  [inst_30 : IsScalarTower R₂ (↥F₂) L] [inst_31 : Algebra.IsSeparable K ↥F₂] [inst_32 : Algebra.IsSeparable (↥F₁) L]
  [inst_33 : IsDomain A] [inst_34 : IsDedekindDomain B] [inst_35 : IsDedekindDomain R₁] [inst_36 : IsDedekindDomain R₂]
  [inst_37 : IsFractionRing B L] [inst_38 : IsFractionRing R₁ ↥F₁] [inst_39 : IsFractionRing R₂ ↥F₂]
  [inst_40 : IsIntegrallyClosed A] [inst_41 : IsIntegralClosure B R₁ L] [inst_42 : Module.IsTorsionFree R₁ B]
  [inst_43 : Module.IsTorsionFree R₂ B], ⋯

Defined in

Mathlib.RingTheory.DedekindDomain.LinearDisjoint

Cited by

1 results in Mathlib

Foundations

Depth 204 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingFieldAlgebraIsFractionRingCommRingFieldAlgebraAlgebraAlgebraFiniteDimensionalIsScalarTowerCommRingCommRingIsDomainAlgebraAlgebraAlgebraAlgebraAlgebraAlgebraIsScalarTowerIsScalarTowerIsScalarTowerModule.FiniteAlgebraAlgebraModule.IsTorsionFreeIsScalarTowerIsScalarTowerIsScalarTowerIsScalarTowerAlgebra.IsSeparableAlgebra.IsSeparableIsDomainIsDedekindDomainIsDedekindDomainIsDedekindDomainIsFractionRingIsFractionRingIsFractionRingIsIntegrallyClosedIsIntegralClosureModule.IsTorsionFreeModule.IsTorsionFreeAlgebraModule.FiniteModule.IsTorsionFreeModule.IsTorsionFreeModule.IsTorsionFreeModule.FiniteModule.FiniteIsScalarTowerModule.FiniteAlgebra.IsSeparableIsScalarTower