Theorems · Definition · ring theory

GradedTensorProduct.auxEquiv

(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] →
                              GradedTensorProduct R 𝒜 ℬ ≃ₗ[R]
                                TensorProduct R (DirectSum ι fun i => ↥(𝒜 i)) (DirectSum ι fun i => ↥(ℬ i))

An auxiliary construction to move between the graded tensor product of internally-graded objects and the tensor product of direct sums.

Defined in

Mathlib.LinearAlgebra.TensorProduct.Graded.Internal

Cited by

7 results in Mathlib

Foundations

Depth 101 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringDecidableEqCommRingRingRingAlgebraAlgebraGradedAlgebraGradedAlgebra