Theorems · Theorem · commutative algebra

ClassGroup.equivPic_symm_apply

∀ (R : Type u_5) [inst : CommRing R] [inst_1 : IsDomain R] (a : CommRing.Pic R),
  (ClassGroup.equivPic R).symm a =
    (ClassGroup.mulEquivUnitsSubmoduleQuotRange R).symm
      ((QuotientGroup.congr (Submodule.unitsToPic R (FractionRing R)).ker (Units.map ↑(Submodule.spanSingleton R)).range
          (MulEquiv.refl (Submodule R (FractionRing R))ˣ) ⋯)
        ((QuotientGroup.quotientKerEquivRange (Submodule.unitsToPic R (FractionRing R))).symm
          ((MulEquiv.subgroupCongr ⋯).symm ((MulEquiv.subgroupCongr ⋯).symm (Subgroup.topEquiv.symm a)))))

Defined in: Mathlib.RingTheory.PicardGroup
Cited by: 0 results in Mathlib
Foundations: Depth 125 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommRing IsDomain

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DFunLike.coestatement and proof · cited by 62,936
CommRingstatement and proof · cited by 17,173
Top.topstatement · cited by 9,680
Submodulestatement · cited by 7,192
Subgroupstatement · cited by 3,593
Unitsstatement · cited by 2,804
HasQuotient.Quotientstatement · cited by 2,301
IsDomainstatement and proof · cited by 2,196
MulEquivstatement · cited by 1,142
nonZeroDivisorsstatement · cited by 895
MulEquiv.symmstatement and proof · cited by 482
MonoidHom.rangestatement · cited by 314

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