mathlib documentation

algebra.algebra.prod

The R-algebra structure on products of R-algebras #

The R-algebra structure on Π i : I, A i when each A i is an R-algebra.

Main defintions #

@[protected, instance]
def prod.algebra (R : Type u_1) (A : Type u_2) (B : Type u_3) [comm_semiring R] [semiring A] [algebra R A] [semiring B] [algebra R B] :
algebra R (A × B)
Equations
@[simp]
theorem prod.algebra_map_apply {R : Type u_1} {A : Type u_2} {B : Type u_3} [comm_semiring R] [semiring A] [algebra R A] [semiring B] [algebra R B] (r : R) :
(algebra_map R (A × B)) r = ((algebra_map R A) r, (algebra_map R B) r)
def alg_hom.fst (R : Type u_1) (A : Type u_2) (B : Type u_3) [comm_semiring R] [semiring A] [algebra R A] [semiring B] [algebra R B] :
A × B →ₐ[R] A

First projection as alg_hom.

Equations
def alg_hom.snd (R : Type u_1) (A : Type u_2) (B : Type u_3) [comm_semiring R] [semiring A] [algebra R A] [semiring B] [algebra R B] :
A × B →ₐ[R] B

Second projection as alg_hom.

Equations
def alg_hom.prod {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [comm_semiring R] [semiring A] [algebra R A] [semiring B] [algebra R B] [semiring C] [algebra R C] (f : A →ₐ[R] B) (g : A →ₐ[R] C) :
A →ₐ[R] B × C

The pi.prod of two morphisms is a morphism.

Equations
@[simp]
theorem alg_hom.prod_apply {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [comm_semiring R] [semiring A] [algebra R A] [semiring B] [algebra R B] [semiring C] [algebra R C] (f : A →ₐ[R] B) (g : A →ₐ[R] C) (ᾰ : A) :
theorem alg_hom.coe_prod {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [comm_semiring R] [semiring A] [algebra R A] [semiring B] [algebra R B] [semiring C] [algebra R C] (f : A →ₐ[R] B) (g : A →ₐ[R] C) :
@[simp]
theorem alg_hom.fst_prod {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [comm_semiring R] [semiring A] [algebra R A] [semiring B] [algebra R B] [semiring C] [algebra R C] (f : A →ₐ[R] B) (g : A →ₐ[R] C) :
(alg_hom.fst R B C).comp (f.prod g) = f
@[simp]
theorem alg_hom.snd_prod {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [comm_semiring R] [semiring A] [algebra R A] [semiring B] [algebra R B] [semiring C] [algebra R C] (f : A →ₐ[R] B) (g : A →ₐ[R] C) :
(alg_hom.snd R B C).comp (f.prod g) = g
@[simp]
theorem alg_hom.prod_fst_snd {R : Type u_1} {A : Type u_2} {B : Type u_3} [comm_semiring R] [semiring A] [algebra R A] [semiring B] [algebra R B] :
(alg_hom.fst R A B).prod (alg_hom.snd R A B) = 1
@[simp]
theorem alg_hom.prod_equiv_apply {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [comm_semiring R] [semiring A] [algebra R A] [semiring B] [algebra R B] [semiring C] [algebra R C] (f : (A →ₐ[R] B) × (A →ₐ[R] C)) :
@[simp]
theorem alg_hom.prod_equiv_symm_apply {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [comm_semiring R] [semiring A] [algebra R A] [semiring B] [algebra R B] [semiring C] [algebra R C] (f : A →ₐ[R] B × C) :
def alg_hom.prod_equiv {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [comm_semiring R] [semiring A] [algebra R A] [semiring B] [algebra R B] [semiring C] [algebra R C] :
(A →ₐ[R] B) × (A →ₐ[R] C) (A →ₐ[R] B × C)

Taking the product of two maps with the same domain is equivalent to taking the product of their codomains.

Equations