Theorems · Definition · number theory

ClassGroup.mkMMem

{R : Type u_1} →
  {S : Type u_2} →
    [inst : EuclideanDomain R] →
      [inst_1 : CommRing S] →
        [inst_2 : IsDomain S] →
          [inst_3 : Algebra R S] →
            {abv : AbsoluteValue R ℤ} →
              {ι : Type u_5} →
                [inst_4 : DecidableEq ι] →
                  [inst_5 : Fintype ι] →
                    (bS : Module.Basis ι R S) →
                      (adm : abv.IsAdmissible) →
                        [inst_6 : Infinite R] →
                          [inst_7 : DecidableEq R] →
                            [IsDedekindDomain S] →
                              { J // (algebraMap R S) (∏ m ∈ ClassGroup.finsetApprox bS adm, m) ∈ J } → ClassGroup S

ClassGroup.mkMMem is a specialization of ClassGroup.mk0 to (the finite set of) ideals that contain M := ∏ m ∈ finsetApprox L f abs, m. By showing this function is surjective, we prove that the class group is finite.

Defined in

Mathlib.NumberTheory.ClassNumber.Finite

Cited by

1 results in Mathlib

Foundations

Depth 200 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

EuclideanDomainCommRingIsDomainAlgebraDecidableEqFintypeInfiniteDecidableEqIsDedekindDomain