Theorems · Theorem · commutative algebra

Ideal.natAbs_det_equiv

∀ {S : Type u_1} [inst : CommRing S] [inst_1 : IsDedekindDomain S] [inst_2 : Module.Free ℤ S] [Module.Finite ℤ S]
  (I : Ideal S) {E : Type u_2} [inst_4 : EquivLike E S ↥I] [inst_5 : AddEquivClass E S ↥I] (e : E),
  (LinearMap.det (↑ℤ (Submodule.subtype I) ∘ₗ (↑e).toIntLinearMap)).natAbs = Ideal.absNorm I

Let e : S ≃ I be an additive isomorphism (therefore a ℤ-linear equiv). Then an alternative way to compute the norm of I is given by taking the determinant of e. See natAbs_det_basis_change for a more familiar formulation of this result.

Defined in

Mathlib.RingTheory.Ideal.Norm.AbsNorm

Cited by

1 results in Mathlib

Foundations

Depth 154 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingIsDedekindDomainModule.FreeModule.FiniteEquivLikeAddEquivClass