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)

@[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.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 = a.unop.unop

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 = let __spread.0 := RingEquiv.opOp A; { toEquiv := __spread.0.toEquiv, map_mul' := ⋯, map_add' := ⋯, commutes' := ⋯ }

@[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

@[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

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

@[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

@[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

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

@[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

@[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 = (f (MulOpposite.op a)).unop

@[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 a.unop)

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.

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

@[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 = (a a_1.unop).unop

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 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 = ((↑↑f).symm (MulOpposite.op a)).unop

@[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 a.unop)

@[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 a.unop)

@[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 = (f (MulOpposite.op a)).unop

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.

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

@[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 = (a a_1.unop).unop

@[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 = ((↑↑(AlgEquiv.refl.trans (a.trans (AlgEquiv.opOp R B)))).symm (MulOpposite.op a_1)).unop

@[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)

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.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.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 = a.unop

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 = let __spread.0 := RingEquiv.toOpposite A; { toEquiv := __spread.0.toEquiv, map_mul' := ⋯, map_add' := ⋯, commutes' := ⋯ }

@[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