Theorems · Theorem · commutative algebra

IsIntegralClosure.range_le_span_dualBasis

∀ {A : Type u_1} {K : Type u_2} [inst : CommRing A] [inst_1 : Field K] [inst_2 : Algebra A K] [IsFractionRing A K]
  {L : Type u_3} [inst_4 : Field L] (C : Type u_4) [inst_5 : CommRing C] [inst_6 : Algebra K L] [inst_7 : Algebra A L]
  [IsScalarTower A K L] [inst_9 : Algebra C L] [IsIntegralClosure C A L] [inst_11 : Algebra A C]
  [inst_12 : IsScalarTower A C L] [inst_13 : FiniteDimensional K L] [inst_14 : Algebra.IsSeparable K L] {ι : Type u_5}
  [inst_15 : Finite ι] [inst_16 : DecidableEq ι] (b : Module.Basis ι K L),
  (∀ (i : ι), IsIntegral A (b i)) →
    ∀ [IsIntegrallyClosed A],
      (↑A (Algebra.linearMap C L)).range ≤ Submodule.span A (Set.range ⇑((Algebra.traceForm K L).dualBasis ⋯ b))

Defined in

Mathlib.RingTheory.DedekindDomain.IntegralClosure

Cited by

2 results in Mathlib

Foundations

Depth 183 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingFieldAlgebraIsFractionRingFieldCommRingAlgebraAlgebraIsScalarTowerAlgebraIsIntegralClosureAlgebraIsScalarTowerFiniteDimensionalAlgebra.IsSeparableFiniteDecidableEqIsIntegrallyClosed