Algebra structures on the multiplicative opposite

Main definitions

• MulOpposite.instAlgebra: the algebra on Aᵐᵒᵖ
• AlgHom.op/AlgHom.unop: simultaneously convert the domain and codomain of a morphism to the opposite algebra.
• AlgHom.opComm: swap which side of a morphism lies in the opposite algebra.
• AlgEquiv.op/AlgEquiv.unop: simultaneously convert the source and target of an isomorphism to the opposite algebra.
• AlgEquiv.opOp: any algebra is isomorphic to the opposite of its opposite.
• AlgEquiv.toOpposite: in a commutative algebra, the opposite algebra is isomorphic to the original algebra.
• AlgEquiv.opComm: swap which side of an isomorphism lies in the opposite algebra.

instance MulOpposite.instAlgebra {R : Type u_1} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] : Algebra R Aᵐᵒᵖ

Equations

• MulOpposite.instAlgebra = Algebra.mk ((algebraMap R A).toOpposite ⋯) ⋯ ⋯

@[simp]

theorem MulOpposite.algebraMap_apply {R : Type u_1} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (c : R) : (algebraMap R Aᵐᵒᵖ) c = MulOpposite.op ((algebraMap R A) c)

def AlgEquiv.opOp (R : Type u_1) (A : Type u_3) [CommSemiring R] [Semiring A] [Algebra R A] : A ≃ₐ[R] Aᵐᵒᵖᵐᵒᵖ

An algebra is isomorphic to the opposite of its opposite.

Equations

• AlgEquiv.opOp R A = { toEquiv := (RingEquiv.opOp A).toEquiv, map_mul' := ⋯, map_add' := ⋯, commutes' := ⋯ }

@[simp]

theorem AlgEquiv.opOp_symm_apply (R : Type u_1) (A : Type u_3) [CommSemiring R] [Semiring A] [Algebra R A] (a✝ : Aᵐᵒᵖᵐᵒᵖ) : (AlgEquiv.opOp R A).symm a✝ = MulOpposite.unop (MulOpposite.unop a✝)

@[simp]

theorem AlgEquiv.opOp_apply (R : Type u_1) (A : Type u_3) [CommSemiring R] [Semiring A] [Algebra R A] (a✝ : A) : (AlgEquiv.opOp R A) a✝ = MulOpposite.op (MulOpposite.op a✝)

@[simp]

theorem AlgEquiv.toRingEquiv_opOp (R : Type u_1) (A : Type u_3) [CommSemiring R] [Semiring A] [Algebra R A] : ↑(AlgEquiv.opOp R A) = RingEquiv.opOp A

def AlgHom.fromOpposite {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) (hf : ∀ (x y : A), Commute (f x) (f y)) : Aᵐᵒᵖ →ₐ[R] B

An algebra homomorphism f : A →ₐ[R] B such that f x commutes with f y for all x, y defines an algebra homomorphism from Aᵐᵒᵖ.

Equations

• f.fromOpposite hf = { toFun := ⇑f ∘ MulOpposite.unop, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯, commutes' := ⋯ }

@[simp]

theorem AlgHom.fromOpposite_apply {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) (hf : ∀ (x y : A), Commute (f x) (f y)) : ⇑(f.fromOpposite hf) = ⇑f ∘ MulOpposite.unop

@[simp]

theorem AlgHom.toLinearMap_fromOpposite {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) (hf : ∀ (x y : A), Commute (f x) (f y)) : (f.fromOpposite hf).toLinearMap = f.toLinearMap ∘ₗ ↑(MulOpposite.opLinearEquiv R).symm

@[simp]

theorem AlgHom.toRingHom_fromOpposite {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) (hf : ∀ (x y : A), Commute (f x) (f y)) : ↑(f.fromOpposite hf) = (↑f).fromOpposite hf

def AlgHom.toOpposite {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) (hf : ∀ (x y : A), Commute (f x) (f y)) : A →ₐ[R] Bᵐᵒᵖ

An algebra homomorphism f : A →ₐ[R] B such that f x commutes with f y for all x, y defines an algebra homomorphism to Bᵐᵒᵖ.

Equations

• f.toOpposite hf = { toFun := MulOpposite.op ∘ ⇑f, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯, commutes' := ⋯ }

@[simp]

theorem AlgHom.toOpposite_apply {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) (hf : ∀ (x y : A), Commute (f x) (f y)) : ⇑(f.toOpposite hf) = MulOpposite.op ∘ ⇑f

@[simp]

theorem AlgHom.toLinearMap_toOpposite {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) (hf : ∀ (x y : A), Commute (f x) (f y)) : (f.toOpposite hf).toLinearMap = ↑(MulOpposite.opLinearEquiv R) ∘ₗ f.toLinearMap

@[simp]

theorem AlgHom.toRingHom_toOpposite {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) (hf : ∀ (x y : A), Commute (f x) (f y)) : ↑(f.toOpposite hf) = (↑f).toOpposite hf

def AlgHom.op {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] : (A →ₐ[R] B) ≃ (Aᵐᵒᵖ →ₐ[R] Bᵐᵒᵖ)

An algebra hom A →ₐ[R] B can equivalently be viewed as an algebra hom Aᵐᵒᵖ →ₐ[R] Bᵐᵒᵖ. This is the action of the (fully faithful) ᵐᵒᵖ-functor on morphisms.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem AlgHom.op_symm_apply_apply {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : Aᵐᵒᵖ →ₐ[R] Bᵐᵒᵖ) (a✝ : A) : (AlgHom.op.symm f) a✝ = MulOpposite.unop (f (MulOpposite.op a✝))

@[simp]

theorem AlgHom.op_apply_apply {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) (a✝ : Aᵐᵒᵖ) : (AlgHom.op f) a✝ = MulOpposite.op (f (MulOpposite.unop a✝))

theorem AlgHom.toRingHom_op {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) : (AlgHom.op f).toRingHom = RingHom.op f.toRingHom

@[reducible, inline]

abbrev AlgHom.unop {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] : (Aᵐᵒᵖ →ₐ[R] Bᵐᵒᵖ) ≃ (A →ₐ[R] B)

The 'unopposite' of an algebra hom Aᵐᵒᵖ →ₐ[R] Bᵐᵒᵖ. Inverse to RingHom.op.

Equations

• AlgHom.unop = AlgHom.op.symm

theorem AlgHom.toRingHom_unop {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : Aᵐᵒᵖ →ₐ[R] Bᵐᵒᵖ) : (AlgHom.unop f).toRingHom = RingHom.unop f.toRingHom

def AlgHom.opComm {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] : (A →ₐ[R] Bᵐᵒᵖ) ≃ (Aᵐᵒᵖ →ₐ[R] B)

Swap the ᵐᵒᵖ on an algebra hom to the opposite side.

Equations

• AlgHom.opComm = AlgHom.op.trans (AlgEquiv.refl.arrowCongr (AlgEquiv.opOp R B).symm)

@[simp]

theorem AlgHom.opComm_symm_apply_apply {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (a✝ : Aᵐᵒᵖ →ₐ[R] B) (a✝¹ : A) : (AlgHom.opComm.symm a✝) a✝¹ = MulOpposite.op (a✝ (MulOpposite.op a✝¹))

@[simp]

theorem AlgHom.opComm_apply_apply {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (a✝ : A →ₐ[R] Bᵐᵒᵖ) (a✝¹ : Aᵐᵒᵖ) : (AlgHom.opComm a✝) a✝¹ = MulOpposite.unop (a✝ (MulOpposite.unop a✝¹))

def AlgEquiv.op {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] : (A ≃ₐ[R] B) ≃ Aᵐᵒᵖ ≃ₐ[R] Bᵐᵒᵖ

An algebra iso A ≃ₐ[R] B can equivalently be viewed as an algebra iso Aᵐᵒᵖ ≃ₐ[R] Bᵐᵒᵖ. This is the action of the (fully faithful) ᵐᵒᵖ-functor on morphisms.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem AlgEquiv.op_symm_apply_apply {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : Aᵐᵒᵖ ≃ₐ[R] Bᵐᵒᵖ) (a✝ : A) : (AlgEquiv.op.symm f) a✝ = MulOpposite.unop (f (MulOpposite.op a✝))

@[simp]

theorem AlgEquiv.op_apply_apply {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A ≃ₐ[R] B) (a✝ : Aᵐᵒᵖ) : (AlgEquiv.op f) a✝ = MulOpposite.op (f (MulOpposite.unop a✝))

@[simp]

theorem AlgEquiv.op_symm_apply_symm_apply {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : Aᵐᵒᵖ ≃ₐ[R] Bᵐᵒᵖ) (a✝ : B) : (AlgEquiv.op.symm f).symm a✝ = MulOpposite.unop ((↑↑f).symm (MulOpposite.op a✝))

@[simp]

theorem AlgEquiv.op_apply_symm_apply {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A ≃ₐ[R] B) (a✝ : Bᵐᵒᵖ) : (AlgEquiv.op f).symm a✝ = MulOpposite.op ((↑↑f).symm (MulOpposite.unop a✝))

theorem AlgEquiv.toAlgHom_op {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A ≃ₐ[R] B) : ↑(AlgEquiv.op f) = AlgHom.op ↑f

theorem AlgEquiv.toRingEquiv_op {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A ≃ₐ[R] B) : (AlgEquiv.op f).toRingEquiv = RingEquiv.op f.toRingEquiv

@[reducible, inline]

abbrev AlgEquiv.unop {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] : (Aᵐᵒᵖ ≃ₐ[R] Bᵐᵒᵖ) ≃ A ≃ₐ[R] B

The 'unopposite' of an algebra iso Aᵐᵒᵖ ≃ₐ[R] Bᵐᵒᵖ. Inverse to AlgEquiv.op.

Equations

• AlgEquiv.unop = AlgEquiv.op.symm

theorem AlgEquiv.toAlgHom_unop {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : Aᵐᵒᵖ ≃ₐ[R] Bᵐᵒᵖ) : ↑(AlgEquiv.unop f) = AlgHom.unop ↑f

theorem AlgEquiv.toRingEquiv_unop {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : Aᵐᵒᵖ ≃ₐ[R] Bᵐᵒᵖ) : (AlgEquiv.unop f).toRingEquiv = RingEquiv.unop f.toRingEquiv

def AlgEquiv.opComm {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] : (A ≃ₐ[R] Bᵐᵒᵖ) ≃ Aᵐᵒᵖ ≃ₐ[R] B

Swap the ᵐᵒᵖ on an algebra isomorphism to the opposite side.

Equations

• AlgEquiv.opComm = AlgEquiv.op.trans (AlgEquiv.refl.equivCongr (AlgEquiv.opOp R B).symm)

@[simp]

theorem AlgEquiv.opComm_apply_apply {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (a✝ : A ≃ₐ[R] Bᵐᵒᵖ) (a✝¹ : Aᵐᵒᵖ) : (AlgEquiv.opComm a✝) a✝¹ = MulOpposite.unop (a✝ (MulOpposite.unop a✝¹))

@[simp]

theorem AlgEquiv.opComm_symm_apply_apply {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (a✝ : Aᵐᵒᵖ ≃ₐ[R] B) (a✝¹ : A) : (AlgEquiv.opComm.symm a✝) a✝¹ = MulOpposite.op (a✝ (MulOpposite.op a✝¹))

@[simp]

theorem AlgEquiv.opComm_apply_symm_apply {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (a✝ : A ≃ₐ[R] Bᵐᵒᵖ) (a✝¹ : B) : (AlgEquiv.opComm a✝).symm a✝¹ = (↑↑AlgEquiv.refl).symm ((↑↑((AlgEquiv.op a✝).trans (AlgEquiv.opOp R B).symm)).symm a✝¹)

@[simp]

theorem AlgEquiv.opComm_symm_apply_symm_apply {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (a✝ : Aᵐᵒᵖ ≃ₐ[R] B) (a✝¹ : Bᵐᵒᵖ) : (AlgEquiv.opComm.symm a✝).symm a✝¹ = MulOpposite.unop ((↑↑(AlgEquiv.refl.trans (a✝.trans (AlgEquiv.opOp R B)))).symm (MulOpposite.op a✝¹))

def AlgEquiv.toOpposite (R : Type u_1) (A : Type u_3) [CommSemiring R] [CommSemiring A] [Algebra R A] : A ≃ₐ[R] Aᵐᵒᵖ

A commutative algebra is isomorphic to its opposite.

Equations

• AlgEquiv.toOpposite R A = { toEquiv := (RingEquiv.toOpposite A).toEquiv, map_mul' := ⋯, map_add' := ⋯, commutes' := ⋯ }

@[simp]

theorem AlgEquiv.toOpposite_symm_apply (R : Type u_1) (A : Type u_3) [CommSemiring R] [CommSemiring A] [Algebra R A] (a✝ : Aᵐᵒᵖ) : (AlgEquiv.toOpposite R A).symm a✝ = MulOpposite.unop a✝

@[simp]

theorem AlgEquiv.toOpposite_apply (R : Type u_1) (A : Type u_3) [CommSemiring R] [CommSemiring A] [Algebra R A] (a✝ : A) : (AlgEquiv.toOpposite R A) a✝ = MulOpposite.op a✝

@[simp]

theorem AlgEquiv.toRingEquiv_toOpposite (R : Type u_1) (A : Type u_3) [CommSemiring R] [CommSemiring A] [Algebra R A] : ↑(AlgEquiv.toOpposite R A) = RingEquiv.toOpposite A

@[simp]

theorem AlgEquiv.toLinearEquiv_toOpposite (R : Type u_1) (A : Type u_3) [CommSemiring R] [CommSemiring A] [Algebra R A] : (AlgEquiv.toOpposite R A).toLinearEquiv = MulOpposite.opLinearEquiv R