Theorems · Definition · ring theory

GradedTensorProduct.lift

{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)] →
                                {C : Type u_5} →
                                  [inst_10 : Ring C] →
                                    [inst_11 : Algebra R C] →
                                      (f : A →ₐ[R] C) →
                                        (g : B →ₐ[R] C) →
                                          (∀ ⦃i j : ι⦄ (a : ↥(𝒜 i)) (b : ↥(ℬ j)),
                                              f ↑a * g ↑b = (-1) ^ (j * i) • (g ↑b * f ↑a)) →
                                            GradedTensorProduct R 𝒜 ℬ →ₐ[R] C

The forwards direction of the universal property; an algebra morphism out of the graded tensor product can be assembled from maps on each component that (anti)commute on pure elements of the corresponding graded algebras.

Defined in

Mathlib.LinearAlgebra.TensorProduct.Graded.Internal

Cited by

1 results in Mathlib

Foundations

Depth 115 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringDecidableEqCommRingRingRingAlgebraAlgebraGradedAlgebraGradedAlgebraModuleRingAlgebra