Theorems · Theorem · commutative algebra

isIntegral_discr_mul_of_mem_traceDual

∀ {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] [IsFractionRing A K]
  [IsIntegrallyClosed A] [FiniteDimensional K L] [IsIntegralClosure B A L] [Algebra.IsSeparable K L] (I : Submodule B L)
  {ι : Type u_4} [inst_16 : DecidableEq ι] [inst_17 : Fintype ι] {b : Module.Basis ι K L},
  (∀ (i : ι), IsIntegral A (b i)) →
    ∀ {a x : L}, a ∈ I → x ∈ Submodule.traceDual A K I → IsIntegral A (Algebra.discr K ⇑b • a * x)

If b is an A-integral basis of L with discriminant b, then d • a * x is integral over A for all a ∈ I and x ∈ Iᵛ.

Defined in

Mathlib.RingTheory.DedekindDomain.Different

Cited by

0 results in Mathlib

Foundations

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

Assumes

CommRingFieldCommRingFieldAlgebraAlgebraAlgebraAlgebraAlgebraIsScalarTowerIsScalarTowerIsFractionRingIsIntegrallyClosedFiniteDimensionalIsIntegralClosureAlgebra.IsSeparableDecidableEqFintype