Homomorphisms of R-algebras

This file defines bundled homomorphisms of R-algebras.

Main definitions

• AlgHom R A B: the type of R-algebra morphisms from A to B.
• Algebra.ofId R A : R →ₐ[R] A: the canonical map from R to A, as an AlgHom.

Notations

• A →ₐ[R] B : R-algebra homomorphism from A to B.

structure AlgHom (R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] extends RingHom : Type (max v w)

Defining the homomorphism in the category R-Alg.

• toFun : A → B
• map_one' : self.toFun 1 = 1
• map_mul' : ∀ (x y : A), self.toFun (x * y) = self.toFun x * self.toFun y
• map_zero' : self.toFun 0 = 0
• map_add' : ∀ (x y : A), self.toFun (x + y) = self.toFun x + self.toFun y
• commutes' : ∀ (r : R), self.toFun ((algebraMap R A) r) = (algebraMap R B) r

def «term_→ₐ_» : Lean.TrailingParserDescr

Defining the homomorphism in the category R-Alg.

Equations

• «term_→ₐ_» = Lean.ParserDescr.trailingNode `term_→ₐ_ 25 26 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " →ₐ ") (Lean.ParserDescr.cat `term 25))

def «term_→ₐ[_]_» : Lean.TrailingParserDescr

Defining the homomorphism in the category R-Alg.

Equations

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

class AlgHomClass (F : Type u_1) (R : outParam (Type u_2)) (A : outParam (Type u_3)) (B : outParam (Type u_4)) [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] [FunLike F A B] extends RingHomClass : Prop

AlgHomClass F R A B asserts F is a type of bundled algebra homomorphisms from A to B.

• map_mul : ∀ (f : F) (x y : A), f (x * y) = f x * f y
• map_one : ∀ (f : F), f 1 = 1
• map_add : ∀ (f : F) (x y : A), f (x + y) = f x + f y
• map_zero : ∀ (f : F), f 0 = 0
• commutes : ∀ (f : F) (r : R), f ((algebraMap R A) r) = (algebraMap R B) r

instance AlgHomClass.linearMapClass {R : Type u_1} {A : Type u_2} {B : Type u_3} {F : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] [FunLike F A B] [AlgHomClass F R A B] : LinearMapClass F R A B

Equations

• ⋯ = ⋯

def AlgHomClass.toAlgHom {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] {F : Type u_5} [FunLike F A B] [AlgHomClass F R A B] (f : F) : A →ₐ[R] B

Turn an element of a type F satisfying AlgHomClass F α β into an actual AlgHom. This is declared as the default coercion from F to α →+* β.

Equations

• ↑f = let __spread.0 := ↑f; { toRingHom := { toMonoidHom := { toOneHom := { toFun := ⇑f, map_one' := ⋯ }, map_mul' := ⋯ }, map_zero' := ⋯, map_add' := ⋯ }, commutes' := ⋯ }

instance AlgHomClass.coeTC {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] {F : Type u_5} [FunLike F A B] [AlgHomClass F R A B] : CoeTC F (A →ₐ[R] B)

Equations

• AlgHomClass.coeTC = { coe := AlgHomClass.toAlgHom }

instance AlgHom.funLike {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] : FunLike (A →ₐ[R] B) A B

Equations

• AlgHom.funLike = { coe := fun (f : A →ₐ[R] B) => f.toFun, coe_injective' := ⋯ }

instance AlgHom.algHomClass {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] : AlgHomClass (A →ₐ[R] B) R A B

Equations

• ⋯ = ⋯

def AlgHom.Simps.apply {R : Type u} {α : Type v} {β : Type w} [CommSemiring R] [Semiring α] [Semiring β] [Algebra R α] [Algebra R β] (f : α →ₐ[R] β) : α → β

See Note [custom simps projection]

Equations

• AlgHom.Simps.apply f = ⇑f

@[simp]

theorem AlgHom.coe_coe {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] {F : Type u_1} [FunLike F A B] [AlgHomClass F R A B] (f : F) : ⇑↑f = ⇑f

@[simp]

theorem AlgHom.toFun_eq_coe {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) : f.toFun = ⇑f

def AlgHom.toMonoidHom' {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) : A →* B

Equations

• ↑f = ↑↑f

instance AlgHom.coeOutMonoidHom {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] : CoeOut (A →ₐ[R] B) (A →* B)

Equations

• AlgHom.coeOutMonoidHom = { coe := AlgHom.toMonoidHom' }

def AlgHom.toAddMonoidHom' {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) : A →+ B

Equations

• ↑f = ↑↑f

instance AlgHom.coeOutAddMonoidHom {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] : CoeOut (A →ₐ[R] B) (A →+ B)

Equations

• AlgHom.coeOutAddMonoidHom = { coe := AlgHom.toAddMonoidHom' }

@[simp]

theorem AlgHom.coe_mk {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] {f : A →+* B} (h : ∀ (r : R), f.toFun ((algebraMap R A) r) = (algebraMap R B) r) : ⇑{ toRingHom := f, commutes' := h } = ⇑f

theorem AlgHom.coe_mks {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] {f : A → B} (h₁ : f 1 = 1) (h₂ : ∀ (x y : A), { toFun := f, map_one' := h₁ }.toFun (x * y) = { toFun := f, map_one' := h₁ }.toFun x * { toFun := f, map_one' := h₁ }.toFun y) (h₃ : { toOneHom := { toFun := f, map_one' := h₁ }, map_mul' := h₂ }.toFun 0 = 0) (h₄ : ∀ (x y : A), { toOneHom := { toFun := f, map_one' := h₁ }, map_mul' := h₂ }.toFun (x + y) = { toOneHom := { toFun := f, map_one' := h₁ }, map_mul' := h₂ }.toFun x + { toOneHom := { toFun := f, map_one' := h₁ }, map_mul' := h₂ }.toFun y) (h₅ : ∀ (r : R), { toMonoidHom := { toOneHom := { toFun := f, map_one' := h₁ }, map_mul' := h₂ }, map_zero' := h₃, map_add' := h₄ }.toFun ((algebraMap R A) r) = (algebraMap R B) r) : ⇑{ toRingHom := { toMonoidHom := { toOneHom := { toFun := f, map_one' := h₁ }, map_mul' := h₂ }, map_zero' := h₃, map_add' := h₄ }, commutes' := h₅ } = f

@[simp]

theorem AlgHom.coe_ringHom_mk {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] {f : A →+* B} (h : ∀ (r : R), f.toFun ((algebraMap R A) r) = (algebraMap R B) r) : ↑{ toRingHom := f, commutes' := h } = f

@[simp]

theorem AlgHom.toRingHom_eq_coe {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) : f.toRingHom = ↑f

@[simp]

theorem AlgHom.coe_toRingHom {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) : ⇑↑f = ⇑f

@[simp]

theorem AlgHom.coe_toMonoidHom {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) : ⇑↑f = ⇑f

@[simp]

theorem AlgHom.coe_toAddMonoidHom {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) : ⇑↑f = ⇑f

theorem AlgHom.coe_fn_injective {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] : Function.Injective DFunLike.coe

theorem AlgHom.coe_fn_inj {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] {φ₁ : A →ₐ[R] B} {φ₂ : A →ₐ[R] B} : ⇑φ₁ = ⇑φ₂ ↔ φ₁ = φ₂

theorem AlgHom.coe_ringHom_injective {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] : Function.Injective RingHomClass.toRingHom

theorem AlgHom.coe_monoidHom_injective {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] : Function.Injective MonoidHomClass.toMonoidHom

theorem AlgHom.coe_addMonoidHom_injective {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] : Function.Injective AddMonoidHomClass.toAddMonoidHom

theorem AlgHom.congr_fun {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] {φ₁ : A →ₐ[R] B} {φ₂ : A →ₐ[R] B} (H : φ₁ = φ₂) (x : A) : φ₁ x = φ₂ x

theorem AlgHom.congr_arg {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) {x : A} {y : A} (h : x = y) : φ x = φ y

theorem AlgHom.ext {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] {φ₁ : A →ₐ[R] B} {φ₂ : A →ₐ[R] B} (H : ∀ (x : A), φ₁ x = φ₂ x) : φ₁ = φ₂

theorem AlgHom.ext_iff {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] {φ₁ : A →ₐ[R] B} {φ₂ : A →ₐ[R] B} : φ₁ = φ₂ ↔ ∀ (x : A), φ₁ x = φ₂ x

@[simp]

theorem AlgHom.mk_coe {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] {f : A →ₐ[R] B} (h₁ : f 1 = 1) (h₂ : ∀ (x y : A), { toFun := ⇑f, map_one' := h₁ }.toFun (x * y) = { toFun := ⇑f, map_one' := h₁ }.toFun x * { toFun := ⇑f, map_one' := h₁ }.toFun y) (h₃ : { toOneHom := { toFun := ⇑f, map_one' := h₁ }, map_mul' := h₂ }.toFun 0 = 0) (h₄ : ∀ (x y : A), { toOneHom := { toFun := ⇑f, map_one' := h₁ }, map_mul' := h₂ }.toFun (x + y) = { toOneHom := { toFun := ⇑f, map_one' := h₁ }, map_mul' := h₂ }.toFun x + { toOneHom := { toFun := ⇑f, map_one' := h₁ }, map_mul' := h₂ }.toFun y) (h₅ : ∀ (r : R), { toMonoidHom := { toOneHom := { toFun := ⇑f, map_one' := h₁ }, map_mul' := h₂ }, map_zero' := h₃, map_add' := h₄ }.toFun ((algebraMap R A) r) = (algebraMap R B) r) : { toRingHom := { toMonoidHom := { toOneHom := { toFun := ⇑f, map_one' := h₁ }, map_mul' := h₂ }, map_zero' := h₃, map_add' := h₄ }, commutes' := h₅ } = f

@[simp]

theorem AlgHom.commutes {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) (r : R) : φ ((algebraMap R A) r) = (algebraMap R B) r

theorem AlgHom.comp_algebraMap {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) : RingHom.comp (↑φ) (algebraMap R A) = algebraMap R B

theorem AlgHom.map_add {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) (r : A) (s : A) : φ (r + s) = φ r + φ s

theorem AlgHom.map_zero {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) : φ 0 = 0

theorem AlgHom.map_mul {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) (x : A) (y : A) : φ (x * y) = φ x * φ y

theorem AlgHom.map_one {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) : φ 1 = 1

theorem AlgHom.map_pow {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) (x : A) (n : ℕ) : φ (x ^ n) = φ x ^ n

theorem AlgHom.map_smul {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) (r : R) (x : A) : φ (r • x) = r • φ x

theorem AlgHom.map_sum {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) {ι : Type u_1} (f : ι → A) (s : Finset ι) : φ (Finset.sum s fun (x : ι) => f x) = Finset.sum s fun (x : ι) => φ (f x)

theorem AlgHom.map_finsupp_sum {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) {α : Type u_1} [Zero α] {ι : Type u_2} (f : ι →₀ α) (g : ι → α → A) : φ (Finsupp.sum f g) = Finsupp.sum f fun (i : ι) (a : α) => φ (g i a)

theorem AlgHom.map_bit0 {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) (x : A) : φ (bit0 x) = bit0 (φ x)

theorem AlgHom.map_bit1 {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) (x : A) : φ (bit1 x) = bit1 (φ x)

def AlgHom.mk' {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →+* B) (h : ∀ (c : R) (x : A), f (c • x) = c • f x) : A →ₐ[R] B

If a RingHom is R-linear, then it is an AlgHom.

Equations

• AlgHom.mk' f h = { toRingHom := { toMonoidHom := { toOneHom := { toFun := ⇑f, map_one' := ⋯ }, map_mul' := ⋯ }, map_zero' := ⋯, map_add' := ⋯ }, commutes' := ⋯ }

@[simp]

theorem AlgHom.coe_mk' {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →+* B) (h : ∀ (c : R) (x : A), f (c • x) = c • f x) : ⇑(AlgHom.mk' f h) = ⇑f

def AlgHom.id (R : Type u) (A : Type v) [CommSemiring R] [Semiring A] [Algebra R A] : A →ₐ[R] A

Identity map as an AlgHom.

Equations

• AlgHom.id R A = let __src := RingHom.id A; { toRingHom := __src, commutes' := ⋯ }

@[simp]

theorem AlgHom.coe_id (R : Type u) (A : Type v) [CommSemiring R] [Semiring A] [Algebra R A] : ⇑(AlgHom.id R A) = id

@[simp]

theorem AlgHom.id_toRingHom (R : Type u) (A : Type v) [CommSemiring R] [Semiring A] [Algebra R A] : ↑(AlgHom.id R A) = RingHom.id A

theorem AlgHom.id_apply {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (p : A) : (AlgHom.id R A) p = p

def AlgHom.comp {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Algebra R A] [Algebra R B] [Algebra R C] (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : A →ₐ[R] C

Composition of algebra homeomorphisms.

Equations

• AlgHom.comp φ₁ φ₂ = let __src := RingHom.comp φ₁.toRingHom ↑φ₂; { toRingHom := __src, commutes' := ⋯ }

@[simp]

theorem AlgHom.coe_comp {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Algebra R A] [Algebra R B] [Algebra R C] (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : ⇑(AlgHom.comp φ₁ φ₂) = ⇑φ₁ ∘ ⇑φ₂

theorem AlgHom.comp_apply {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Algebra R A] [Algebra R B] [Algebra R C] (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) (p : A) : (AlgHom.comp φ₁ φ₂) p = φ₁ (φ₂ p)

theorem AlgHom.comp_toRingHom {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Algebra R A] [Algebra R B] [Algebra R C] (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : ↑(AlgHom.comp φ₁ φ₂) = RingHom.comp ↑φ₁ ↑φ₂

@[simp]

theorem AlgHom.comp_id {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) : AlgHom.comp φ (AlgHom.id R A) = φ

@[simp]

theorem AlgHom.id_comp {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) : AlgHom.comp (AlgHom.id R B) φ = φ

theorem AlgHom.comp_assoc {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} {D : Type v₁} [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Semiring D] [Algebra R A] [Algebra R B] [Algebra R C] [Algebra R D] (φ₁ : C →ₐ[R] D) (φ₂ : B →ₐ[R] C) (φ₃ : A →ₐ[R] B) : AlgHom.comp (AlgHom.comp φ₁ φ₂) φ₃ = AlgHom.comp φ₁ (AlgHom.comp φ₂ φ₃)

def AlgHom.toLinearMap {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) : A →ₗ[R] B

R-Alg ⥤ R-Mod

Equations

• AlgHom.toLinearMap φ = { toAddHom := { toFun := ⇑φ, map_add' := ⋯ }, map_smul' := ⋯ }

@[simp]

theorem AlgHom.toLinearMap_apply {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) (p : A) : (AlgHom.toLinearMap φ) p = φ p

theorem AlgHom.toLinearMap_injective {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] : Function.Injective AlgHom.toLinearMap

@[simp]

theorem AlgHom.comp_toLinearMap {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Algebra R A] [Algebra R B] [Algebra R C] (f : A →ₐ[R] B) (g : B →ₐ[R] C) : AlgHom.toLinearMap (AlgHom.comp g f) = AlgHom.toLinearMap g ∘ₗ AlgHom.toLinearMap f

@[simp]

theorem AlgHom.toLinearMap_id {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] : AlgHom.toLinearMap (AlgHom.id R A) = LinearMap.id

@[simp]

theorem AlgHom.ofLinearMap_apply {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₗ[R] B) (map_one : f 1 = 1) (map_mul : ∀ (x y : A), f (x * y) = f x * f y) (a : A) : (AlgHom.ofLinearMap f map_one map_mul) a = f a

def AlgHom.ofLinearMap {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₗ[R] B) (map_one : f 1 = 1) (map_mul : ∀ (x y : A), f (x * y) = f x * f y) : A →ₐ[R] B

Promote a LinearMap to an AlgHom by supplying proofs about the behavior on 1 and *.

Equations

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

@[simp]

theorem AlgHom.ofLinearMap_toLinearMap {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) (map_one : (AlgHom.toLinearMap φ) 1 = 1) (map_mul : ∀ (x y : A), (AlgHom.toLinearMap φ) (x * y) = (AlgHom.toLinearMap φ) x * (AlgHom.toLinearMap φ) y) : AlgHom.ofLinearMap (AlgHom.toLinearMap φ) map_one map_mul = φ

@[simp]

theorem AlgHom.toLinearMap_ofLinearMap {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₗ[R] B) (map_one : f 1 = 1) (map_mul : ∀ (x y : A), f (x * y) = f x * f y) : AlgHom.toLinearMap (AlgHom.ofLinearMap f map_one map_mul) = f

@[simp]

theorem AlgHom.ofLinearMap_id {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (map_one : LinearMap.id 1 = 1) (map_mul : ∀ (x y : A), LinearMap.id (x * y) = LinearMap.id x * LinearMap.id y) : AlgHom.ofLinearMap LinearMap.id map_one map_mul = AlgHom.id R A

theorem AlgHom.map_smul_of_tower {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) {R' : Type u_1} [SMul R' A] [SMul R' B] [LinearMap.CompatibleSMul A B R' R] (r : R') (x : A) : φ (r • x) = r • φ x

theorem AlgHom.map_list_prod {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) (s : List A) : φ (List.prod s) = List.prod (List.map (⇑φ) s)

theorem AlgHom.End_toSemigroup_toMul_mul {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (φ₁ : A →ₐ[R] A) (φ₂ : A →ₐ[R] A) : φ₁ * φ₂ = AlgHom.comp φ₁ φ₂

theorem AlgHom.End_toOne_one {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] : 1 = AlgHom.id R A

instance AlgHom.End {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] : Monoid (A →ₐ[R] A)

Equations

• AlgHom.End = Monoid.mk ⋯ ⋯ npowRec ⋯ ⋯

@[simp]

theorem AlgHom.one_apply {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (x : A) : 1 x = x

@[simp]

theorem AlgHom.mul_apply {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (φ : A →ₐ[R] A) (ψ : A →ₐ[R] A) (x : A) : (φ * ψ) x = φ (ψ x)

theorem AlgHom.algebraMap_eq_apply {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) {y : R} {x : A} (h : (algebraMap R A) y = x) : (algebraMap R B) y = f x

theorem AlgHom.map_multiset_prod {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [CommSemiring A] [CommSemiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) (s : Multiset A) : φ (Multiset.prod s) = Multiset.prod (Multiset.map (⇑φ) s)

theorem AlgHom.map_prod {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [CommSemiring A] [CommSemiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) {ι : Type u_1} (f : ι → A) (s : Finset ι) : φ (Finset.prod s fun (x : ι) => f x) = Finset.prod s fun (x : ι) => φ (f x)

theorem AlgHom.map_finsupp_prod {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [CommSemiring A] [CommSemiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) {α : Type u_1} [Zero α] {ι : Type u_2} (f : ι →₀ α) (g : ι → α → A) : φ (Finsupp.prod f g) = Finsupp.prod f fun (i : ι) (a : α) => φ (g i a)

theorem AlgHom.map_neg {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Ring A] [Ring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) (x : A) : φ (-x) = -φ x

theorem AlgHom.map_sub {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Ring A] [Ring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) (x : A) (y : A) : φ (x - y) = φ x - φ y

def RingHom.toNatAlgHom {R : Type u_1} {S : Type u_2} [Semiring R] [Semiring S] (f : R →+* S) : R →ₐ[ℕ] S

Reinterpret a RingHom as an ℕ-algebra homomorphism.

Equations

• RingHom.toNatAlgHom f = { toRingHom := { toMonoidHom := { toOneHom := { toFun := ⇑f, map_one' := ⋯ }, map_mul' := ⋯ }, map_zero' := ⋯, map_add' := ⋯ }, commutes' := ⋯ }

def RingHom.toIntAlgHom {R : Type u_1} {S : Type u_2} [Ring R] [Ring S] [Algebra ℤ R] [Algebra ℤ S] (f : R →+* S) : R →ₐ[ℤ] S

Reinterpret a RingHom as a ℤ-algebra homomorphism.

Equations

• RingHom.toIntAlgHom f = { toRingHom := f, commutes' := ⋯ }

theorem RingHom.toIntAlgHom_injective {R : Type u_1} {S : Type u_2} [Ring R] [Ring S] [Algebra ℤ R] [Algebra ℤ S] : Function.Injective RingHom.toIntAlgHom

def RingHom.toRatAlgHom {R : Type u_1} {S : Type u_2} [Ring R] [Ring S] [Algebra ℚ R] [Algebra ℚ S] (f : R →+* S) : R →ₐ[ℚ] S

Reinterpret a RingHom as a ℚ-algebra homomorphism. This actually yields an equivalence, see RingHom.equivRatAlgHom.

Equations

• RingHom.toRatAlgHom f = { toRingHom := f, commutes' := ⋯ }

@[simp]

theorem RingHom.toRatAlgHom_toRingHom {R : Type u_1} {S : Type u_2} [Ring R] [Ring S] [Algebra ℚ R] [Algebra ℚ S] (f : R →+* S) : ↑(RingHom.toRatAlgHom f) = f

@[simp]

theorem AlgHom.toRingHom_toRatAlgHom {R : Type u_1} {S : Type u_2} [Ring R] [Ring S] [Algebra ℚ R] [Algebra ℚ S] (f : R →ₐ[ℚ] S) : RingHom.toRatAlgHom ↑f = f

@[simp]

theorem RingHom.equivRatAlgHom_symm_apply {R : Type u_1} {S : Type u_2} [Ring R] [Ring S] [Algebra ℚ R] [Algebra ℚ S] (self : R →ₐ[ℚ] S) : RingHom.equivRatAlgHom.symm self = self.toRingHom

@[simp]

theorem RingHom.equivRatAlgHom_apply {R : Type u_1} {S : Type u_2} [Ring R] [Ring S] [Algebra ℚ R] [Algebra ℚ S] (f : R →+* S) : RingHom.equivRatAlgHom f = RingHom.toRatAlgHom f

def RingHom.equivRatAlgHom {R : Type u_1} {S : Type u_2} [Ring R] [Ring S] [Algebra ℚ R] [Algebra ℚ S] : (R →+* S) ≃ (R →ₐ[ℚ] S)

The equivalence between RingHom and ℚ-algebra homomorphisms.

Equations

• RingHom.equivRatAlgHom = { toFun := RingHom.toRatAlgHom, invFun := AlgHom.toRingHom, left_inv := ⋯, right_inv := ⋯ }

def Algebra.ofId (R : Type u) (A : Type v) [CommSemiring R] [Semiring A] [Algebra R A] : R →ₐ[R] A

AlgebraMap as an AlgHom.

Equations

• Algebra.ofId R A = let __src := algebraMap R A; { toRingHom := __src, commutes' := ⋯ }

theorem Algebra.ofId_apply {R : Type u} (A : Type v) [CommSemiring R] [Semiring A] [Algebra R A] (r : R) : (Algebra.ofId R A) r = (algebraMap R A) r

instance Algebra.subsingleton_id {R : Type u} (A : Type v) [CommSemiring R] [Semiring A] [Algebra R A] : Subsingleton (R →ₐ[R] A)

This is a special case of a more general instance that we define in a later file.

Equations

• ⋯ = ⋯

theorem Algebra.ext_id {R : Type u} (A : Type v) [CommSemiring R] [Semiring A] [Algebra R A] (f : R →ₐ[R] A) (g : R →ₐ[R] A) : f = g

This ext lemma closes trivial subgoals create when chaining heterobasic ext lemmas.

instance Algebra.instMulDistribMulActionAlgHomUnitsToMonoidToMonoidWithZeroEndInstMonoid {R : Type u} (A : Type v) [CommSemiring R] [Semiring A] [Algebra R A] : MulDistribMulAction (A →ₐ[R] A) Aˣ

Equations

• Algebra.instMulDistribMulActionAlgHomUnitsToMonoidToMonoidWithZeroEndInstMonoid A = MulDistribMulAction.mk ⋯ ⋯

@[simp]

theorem Algebra.smul_units_def {R : Type u} (A : Type v) [CommSemiring R] [Semiring A] [Algebra R A] (f : A →ₐ[R] A) (x : Aˣ) : f • x = (Units.map ↑f) x

@[simp]

theorem MulSemiringAction.toAlgHom_apply {M : Type u_1} (R : Type u_3) (A : Type u_4) [CommSemiring R] [Semiring A] [Algebra R A] [Monoid M] [MulSemiringAction M A] [SMulCommClass M R A] (m : M) (a : A) : (MulSemiringAction.toAlgHom R A m) a = m • a

def MulSemiringAction.toAlgHom {M : Type u_1} (R : Type u_3) (A : Type u_4) [CommSemiring R] [Semiring A] [Algebra R A] [Monoid M] [MulSemiringAction M A] [SMulCommClass M R A] (m : M) : A →ₐ[R] A

Each element of the monoid defines an algebra homomorphism.

This is a stronger version of MulSemiringAction.toRingHom and DistribMulAction.toLinearMap.

Equations

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

theorem MulSemiringAction.toAlgHom_injective {M : Type u_1} (R : Type u_3) (A : Type u_4) [CommSemiring R] [Semiring A] [Algebra R A] [Monoid M] [MulSemiringAction M A] [SMulCommClass M R A] [FaithfulSMul M A] : Function.Injective (MulSemiringAction.toAlgHom R A)