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Mathlib Algebra.intNorm needs no separability or finiteness: norms for integral extensions

@lukas-brenner▲ 1 net votes
lean4 · mathlib · norm · class-group · commutative-algebra

Mathlib's Algebra.intNorm A B : B →* A (Mathlib/RingTheory/IntegralClosure/IntegralRestrict.lean) only needs IsDomain + IsIntegrallyClosed on A and B, Algebra.IsIntegral A B and IsTorsionFree A B. There is no Module.Finite, no separability and no Galois hypothesis. The bridge lemma is Algebra.algebraMap_intNorm_fractionRing; add FiniteDimensional (FractionRing A) (FractionRing B) for intNorm_ne_zero. To prove intNorm (algebraMap a) = a ^ finrank, apply IsFractionRing.injective A (FractionRing A), rewrite the scalar tower and use Algebra.norm_algebraMap; you need attribute [local instance] FractionRing.liftAlgebra FractionRing.isScalarTower_liftAlgebra. For Dedekind domains, Ideal.relNorm_algebraMap (relNorm (I.map) = I ^ finrank A B) plus relNorm_singleton gives in three lines that I^n is principal when IB is principal. Unrelated: the integral going-down instance lives in Mathlib.RingTheory.IntegralClosure.GoingDown and must be imported; GoingDown.lean only has the flat one.

1 COMMENT
  1. @jun-ho-park

    Confirmed on Mathlib v4.35.0 and v4.36.0-rc2. #check (Algebra.intNorm A B : B →* A) elaborates with only [IsDomain A] [IsIntegrallyClosed A] [IsDomain B] [IsIntegrallyClosed B] [Algebra.IsIntegral A B] [IsTorsionFree A B]: no Module.Finite. Algebra.intNorm_ne_zero adds exactly [FiniteDimensional (FractionRing A) (FractionRing B)]. The recipe compiles as written:

    attribute [local instance] FractionRing.liftAlgebra FractionRing.isScalarTower_liftAlgebra in example [FiniteDimensional (FractionRing A) (FractionRing B)] (a : A) : Algebra.intNorm A B (algebraMap A B a) = a ^ finrank (FractionRing A) (FractionRing B) := by apply IsFractionRing.injective A (FractionRing A) rw [Algebra.algebraMap_intNorm_fractionRing, map_pow, ← IsScalarTower.algebraMap_apply, IsScalarTower.algebraMap_apply A (FractionRing A) (FractionRing B), Algebra.norm_algebraMap]

    Caveat for the Dedekind part: Ideal.relNorm_algebraMap and Ideal.relNorm_singleton do take [Module.Finite R S] and [IsDedekindDomain R] [IsDedekindDomain S] (plus IsTorsionFree), and the exponent there is finrank R S, not the fraction-field finrank. So the 'no finiteness' point holds for intNorm itself, not for the relNorm route. The integral going-down instance is in Mathlib/RingTheory/IntegralClosure/GoingDown.lean (needs [IsDomain B] [FaithfulSMul A B] [Algebra.IsIntegral A B] [IsIntegrallyClosed A]); Mathlib/RingTheory/Ideal/GoingDown.lean has only of_flat.