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.

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def Subalgebra.op {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R A) :
Subalgebra R Aᵐᵒᵖ

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

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      @[simp]
      theorem Subalgebra.coe_op {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
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      @[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
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      @[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} :
      x ∈ S.op ↔ MulOpposite.unop x ∈ S
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      def Subalgebra.unop {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R Aᵐᵒᵖ) :
      Subalgebra R A

      Pull a subalgebra back to a subalgebra along MulOpposite.op

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          @[simp]
          theorem Subalgebra.coe_unop {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
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          @[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
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          @[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
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          @[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
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          @[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 #

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          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
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          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
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          @[simp]
          theorem Subalgebra.op_le_op_iff {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] {S₁ S₂ : Subalgebra R A} :
          S₁.op ≤ S₂.op ↔ S₁ ≤ S₂
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          @[simp]
          theorem Subalgebra.unop_le_unop_iff {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] {S₁ S₂ : Subalgebra R Aᵐᵒᵖ} :
          S₁.unop ≤ S₂.unop ↔ S₁ ≤ S₂
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          def Subalgebra.opEquiv {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] :
          Subalgebra R A ≃o Subalgebra R Aᵐᵒᵖ

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

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              theorem Subalgebra.opEquiv_apply {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R A) :
              opEquiv S = S.op
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              @[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 opEquiv) S = S.unop
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              @[simp]
              theorem Subalgebra.op_bot {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] :
              ⊥.op = ⊥
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              @[simp]
              theorem Subalgebra.unop_bot {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] :
              ⊥.unop = ⊥
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              @[simp]
              theorem Subalgebra.op_top {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] :
              ⊤.op = ⊤
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              @[simp]
              theorem Subalgebra.unop_top {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] :
              ⊤.unop = ⊤
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              theorem Subalgebra.op_sup {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S₁ S₂ : Subalgebra R A) :
              (S₁ ⊔ S₂).op = S₁.op ⊔ S₂.op
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              theorem Subalgebra.unop_sup {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S₁ S₂ : Subalgebra R Aᵐᵒᵖ) :
              (S₁ ⊔ S₂).unop = S₁.unop ⊔ S₂.unop
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              theorem Subalgebra.op_inf {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S₁ S₂ : Subalgebra R A) :
              (S₁ ⊓ S₂).op = S₁.op ⊓ S₂.op
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              theorem Subalgebra.unop_inf {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S₁ S₂ : Subalgebra R Aᵐᵒᵖ) :
              (S₁ ⊓ S₂).unop = S₁.unop ⊓ S₂.unop
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              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)
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              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)
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              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)
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              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)
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              theorem Subalgebra.op_iSup {ι : Sort u_1} {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : ι → Subalgebra R A) :
              (iSup S).op = ⨆ (i : ι), (S i).op
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              theorem Subalgebra.unop_iSup {ι : Sort u_1} {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : ι → Subalgebra R Aᵐᵒᵖ) :
              (iSup S).unop = ⨆ (i : ι), (S i).unop
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              theorem Subalgebra.op_iInf {ι : Sort u_1} {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : ι → Subalgebra R A) :
              (iInf S).op = ⨅ (i : ι), (S i).op
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              theorem Subalgebra.unop_iInf {ι : Sort u_1} {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : ι → Subalgebra R Aᵐᵒᵖ) :
              (iInf S).unop = ⨅ (i : ι), (S i).unop
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              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)
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              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)
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              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.

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                  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✝
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                  @[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✝
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                  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.

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                      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✝)
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                      @[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✝)
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                      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.

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                          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✝)
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                          @[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✝)
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                          @[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
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                          @[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