Documentation

Mathlib.Algebra.Algebra.Subalgebra.MulOpposite

Subalgebras of opposite rings #

For every ring A over a commutative ring R, we construct an equivalence between subalgebras of A / R and that of Aᵐᵒᵖ / R.

@[simp]
theorem Subalgebra.op_toSubsemiring {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R A) :
S.op.toSubsemiring = S.op
def Subalgebra.op {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R A) :

Pull a subalgebra back to an opposite subalgebra along MulOpposite.unop

Equations
  • S.op = { toSubsemiring := S.op, algebraMap_mem' := }
Instances For
    @[simp]
    theorem Subalgebra.op_coe {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R A) :
    S.op = MulOpposite.unop ⁻¹' S
    @[simp]
    theorem Subalgebra.mem_op {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] {x : Aᵐᵒᵖ} {S : Subalgebra R A} :
    @[simp]
    theorem Subalgebra.unop_toSubsemiring {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R Aᵐᵒᵖ) :
    S.unop.toSubsemiring = S.unop
    def Subalgebra.unop {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R Aᵐᵒᵖ) :

    Pull an subalgebra subring back to a subalgebra along MulOpposite.op

    Equations
    • S.unop = { toSubsemiring := S.unop, algebraMap_mem' := }
    Instances For
      @[simp]
      theorem Subalgebra.unop_coe {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R Aᵐᵒᵖ) :
      S.unop = MulOpposite.op ⁻¹' S
      @[simp]
      theorem Subalgebra.mem_unop {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] {x : A} {S : Subalgebra R Aᵐᵒᵖ} :
      x S.unop MulOpposite.op x S
      @[simp]
      theorem Subalgebra.unop_op {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R A) :
      S.op.unop = S
      @[simp]
      theorem Subalgebra.op_unop {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R Aᵐᵒᵖ) :
      S.unop.op = S

      Lattice results #

      theorem Subalgebra.op_le_iff {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] {S₁ : Subalgebra R A} {S₂ : Subalgebra R Aᵐᵒᵖ} :
      S₁.op S₂ S₁ S₂.unop
      theorem Subalgebra.le_op_iff {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] {S₁ : Subalgebra R Aᵐᵒᵖ} {S₂ : Subalgebra R A} :
      S₁ S₂.op S₁.unop S₂
      @[simp]
      theorem Subalgebra.op_le_op_iff {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] {S₁ : Subalgebra R A} {S₂ : Subalgebra R A} :
      S₁.op S₂.op S₁ S₂
      @[simp]
      theorem Subalgebra.unop_le_unop_iff {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] {S₁ : Subalgebra R Aᵐᵒᵖ} {S₂ : Subalgebra R Aᵐᵒᵖ} :
      S₁.unop S₂.unop S₁ S₂
      @[simp]
      theorem Subalgebra.opEquiv_apply {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R A) :
      Subalgebra.opEquiv S = S.op
      @[simp]
      theorem Subalgebra.opEquiv_symm_apply {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R Aᵐᵒᵖ) :
      (RelIso.symm Subalgebra.opEquiv) S = S.unop

      A subalgebra S of A / R determines a subring S.op of the opposite ring Aᵐᵒᵖ / R.

      Equations
      • Subalgebra.opEquiv = { toFun := Subalgebra.op, invFun := Subalgebra.unop, left_inv := , right_inv := , map_rel_iff' := }
      Instances For
        @[simp]
        theorem Subalgebra.op_bot {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] :
        .op =
        @[simp]
        theorem Subalgebra.unop_bot {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] :
        .unop =
        @[simp]
        theorem Subalgebra.op_top {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] :
        .op =
        @[simp]
        theorem Subalgebra.unop_top {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] :
        .unop =
        theorem Subalgebra.op_sup {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S₁ : Subalgebra R A) (S₂ : Subalgebra R A) :
        (S₁ S₂).op = S₁.op S₂.op
        theorem Subalgebra.unop_sup {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S₁ : Subalgebra R Aᵐᵒᵖ) (S₂ : Subalgebra R Aᵐᵒᵖ) :
        (S₁ S₂).unop = S₁.unop S₂.unop
        theorem Subalgebra.op_inf {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S₁ : Subalgebra R A) (S₂ : Subalgebra R A) :
        (S₁ S₂).op = S₁.op S₂.op
        theorem Subalgebra.unop_inf {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S₁ : Subalgebra R Aᵐᵒᵖ) (S₂ : Subalgebra R Aᵐᵒᵖ) :
        (S₁ S₂).unop = S₁.unop S₂.unop
        theorem Subalgebra.op_sSup {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Set (Subalgebra R A)) :
        (sSup S).op = sSup (Subalgebra.unop ⁻¹' S)
        theorem Subalgebra.unop_sSup {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Set (Subalgebra R Aᵐᵒᵖ)) :
        (sSup S).unop = sSup (Subalgebra.op ⁻¹' S)
        theorem Subalgebra.op_sInf {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Set (Subalgebra R A)) :
        (sInf S).op = sInf (Subalgebra.unop ⁻¹' S)
        theorem Subalgebra.unop_sInf {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Set (Subalgebra R Aᵐᵒᵖ)) :
        (sInf S).unop = sInf (Subalgebra.op ⁻¹' S)
        theorem Subalgebra.op_iSup {ι : Sort u_1} {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : ιSubalgebra R A) :
        (iSup S).op = ⨆ (i : ι), (S i).op
        theorem Subalgebra.unop_iSup {ι : Sort u_1} {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : ιSubalgebra R Aᵐᵒᵖ) :
        (iSup S).unop = ⨆ (i : ι), (S i).unop
        theorem Subalgebra.op_iInf {ι : Sort u_1} {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : ιSubalgebra R A) :
        (iInf S).op = ⨅ (i : ι), (S i).op
        theorem Subalgebra.unop_iInf {ι : Sort u_1} {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : ιSubalgebra R Aᵐᵒᵖ) :
        (iInf S).unop = ⨅ (i : ι), (S i).unop
        theorem Subalgebra.op_adjoin {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (s : Set A) :
        (Algebra.adjoin R s).op = Algebra.adjoin R (MulOpposite.unop ⁻¹' s)
        theorem Subalgebra.unop_adjoin {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (s : Set Aᵐᵒᵖ) :
        (Algebra.adjoin R s).unop = Algebra.adjoin R (MulOpposite.op ⁻¹' s)
        @[simp]
        theorem Subalgebra.linearEquivOp_apply_coe {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R A) :
        ∀ (a : S.toSubsemiring), (S.linearEquivOp a) = MulOpposite.op a
        @[simp]
        theorem Subalgebra.linearEquivOp_symm_apply_coe {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R A) :
        ∀ (a : S.op), (S.linearEquivOp.symm a) = MulOpposite.unop a
        def Subalgebra.linearEquivOp {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R A) :
        S ≃ₗ[R] S.op

        Bijection between a subalgebra S and its opposite.

        Equations
        • S.linearEquivOp = { toFun := S.addEquivOp.toFun, map_add' := , map_smul' := , invFun := S.addEquivOp.invFun, left_inv := , right_inv := }
        Instances For
          @[simp]
          theorem Subalgebra.algEquivOpMop_apply {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R A) :
          ∀ (a : S.toSubsemiring), S.algEquivOpMop a = MulOpposite.op (S.addEquivOp a)
          @[simp]
          theorem Subalgebra.algEquivOpMop_symm_apply_coe {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R A) :
          ∀ (a : (↥S.op)ᵐᵒᵖ), (S.algEquivOpMop.symm a) = MulOpposite.unop (MulOpposite.unop a)
          def Subalgebra.algEquivOpMop {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R A) :
          S ≃ₐ[R] (↥S.op)ᵐᵒᵖ

          Bijection between a subalgebra S and MulOpposite of its opposite.

          Equations
          • S.algEquivOpMop = { toEquiv := S.ringEquivOpMop.toEquiv, map_mul' := , map_add' := , commutes' := }
          Instances For
            @[simp]
            theorem Subalgebra.mopAlgEquivOp_apply_coe {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R A) :
            ∀ (a : (↥S.toSubsemiring)ᵐᵒᵖ), (S.mopAlgEquivOp a) = MulOpposite.op (MulOpposite.unop a)
            @[simp]
            theorem Subalgebra.mopAlgEquivOp_symm_apply {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R A) :
            ∀ (a : S.op), S.mopAlgEquivOp.symm a = MulOpposite.op (S.addEquivOp.symm a)
            def Subalgebra.mopAlgEquivOp {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R A) :
            (↥S)ᵐᵒᵖ ≃ₐ[R] S.op

            Bijection between MulOpposite of a subalgebra S and its opposite.

            Equations
            • S.mopAlgEquivOp = { toEquiv := S.mopRingEquivOp.toEquiv, map_mul' := , map_add' := , commutes' := }
            Instances For
              @[simp]
              theorem Subalgebra.op_toSubring {R : Type u_1} {A : Type u_2} [CommRing R] [Ring A] [Algebra R A] (S : Subalgebra R A) :
              S.op.toSubring = S.toSubring.op
              @[simp]
              theorem Subalgebra.unop_toSubring {R : Type u_1} {A : Type u_2} [CommRing R] [Ring A] [Algebra R A] (S : Subalgebra R Aᵐᵒᵖ) :
              S.unop.toSubring = S.toSubring.unop