Theorems · Theorem · ring theory

GradedTensorProduct.mulHom_apply

∀ {R : Type u_1} {ι : Type u_2} {A : Type u_3} {B : Type u_4} [inst : CommSemiring ι] [inst_1 : DecidableEq ι]
  [inst_2 : CommRing R] [inst_3 : Ring A] [inst_4 : Ring B] [inst_5 : Algebra R A] [inst_6 : Algebra R B]
  (𝒜 : ι → Submodule R A) (ℬ : ι → Submodule R B) [inst_7 : GradedAlgebra 𝒜] [inst_8 : GradedAlgebra ℬ]
  [inst_9 : Module ι (Additive ℤˣ)] (x y : GradedTensorProduct R 𝒜 ℬ),
  ((GradedTensorProduct.mulHom 𝒜 ℬ) x) y =
    (GradedTensorProduct.auxEquiv R 𝒜 ℬ).symm
      (((TensorProduct.gradedMul R (fun x => ↥(𝒜 x)) fun x => ↥(ℬ x)) ((GradedTensorProduct.auxEquiv R 𝒜 ℬ) x))
        ((GradedTensorProduct.auxEquiv R 𝒜 ℬ) y))

Defined in

Mathlib.LinearAlgebra.TensorProduct.Graded.Internal

Cited by

0 results in Mathlib

Foundations

Depth 103 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringDecidableEqCommRingRingRingAlgebraAlgebraGradedAlgebraGradedAlgebraModule