Theorems · Theorem · commutative algebra

ClassGroup.equivPic_apply

∀ (R : Type u_5) [inst : CommRing R] [inst_1 : IsDomain R] (a : ClassGroup R),
  (ClassGroup.equivPic R) a =
    ↑((QuotientGroup.quotientKerEquivRange (Submodule.unitsToPic R (FractionRing R)))
        ((QuotientGroup.congr (Units.map ↑(Submodule.spanSingleton R)).range
            (Submodule.unitsToPic R (FractionRing R)).ker (MulEquiv.refl (Submodule R (FractionRing R))ˣ) ⋯)
          ((ClassGroup.mulEquivUnitsSubmoduleQuotRange R) 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
Setstatement · cited by 53,352
CommRingstatement and proof · cited by 17,173
Submodulestatement · cited by 7,192
Set.rangestatement · cited by 4,705
MonoidHomstatement · cited by 3,629
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

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