Documentation

Mathlib.Algebra.Algebra.Prod

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

The R-algebra structure on (i : I) → A i when each A i is an R-algebra.

Main definitions #

instance Prod.algebra (R : Type u_1) (A : Type u_2) (B : Type u_3) [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] :
Algebra R (A × B)
@[simp]
theorem Prod.algebraMap_apply {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (r : R) :
↑(algebraMap R (A × B)) r = (↑(algebraMap R A) r, ↑(algebraMap R B) r)
def AlgHom.fst (R : Type u_1) (A : Type u_2) (B : Type u_3) [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] :
A × B →ₐ[R] A

First projection as AlgHom.

Instances For
    def AlgHom.snd (R : Type u_1) (A : Type u_2) (B : Type u_3) [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] :
    A × B →ₐ[R] B

    Second projection as AlgHom.

    Instances For
      @[simp]
      theorem AlgHom.prod_apply {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommSemiring 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) (x : A) :
      ↑(AlgHom.prod f g) x = (f x, g x)
      def AlgHom.prod {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommSemiring 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.

      Instances For
        theorem AlgHom.coe_prod {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommSemiring 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) :
        ↑(AlgHom.prod f g) = Pi.prod f g
        @[simp]
        theorem AlgHom.fst_prod {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommSemiring 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 AlgHom.snd_prod {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommSemiring 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 AlgHom.prod_fst_snd {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] :
        @[simp]
        theorem AlgHom.prodEquiv_symm_apply {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C] (f : A →ₐ[R] B × C) :
        AlgHom.prodEquiv.symm f = (AlgHom.comp (AlgHom.fst R B C) f, AlgHom.comp (AlgHom.snd R B C) f)
        @[simp]
        theorem AlgHom.prodEquiv_apply {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommSemiring 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)) :
        AlgHom.prodEquiv f = AlgHom.prod f.fst f.snd
        def AlgHom.prodEquiv {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommSemiring 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.

        Instances For