Theorems · Definition · linear algebra

TensorProduct.quotientTensorEquiv

{R : Type u_1} →
  {M : Type u_2} →
    (N : Type u_3) →
      [inst : CommRing R] →
        [inst_1 : AddCommGroup M] →
          [inst_2 : Module R M] →
            [inst_3 : AddCommGroup N] →
              [inst_4 : Module R N] →
                (m : Submodule R M) →
                  TensorProduct R (M ⧸ m) N ≃ₗ[R] TensorProduct R M N ⧸ (TensorProduct.map m.subtype LinearMap.id).range

Let M, N be R-modules, m ≤ M be an R-submodule. Then we have a linear isomorphism between tensor products of the quotient and the quotient of the tensor product: (M ⧸ m) ⊗[R] N ≃ₗ[R] (M ⊗[R] N) ⧸ (m ⊗ N).

Defined in

Mathlib.LinearAlgebra.TensorProduct.Quotient

Cited by

3 results in Mathlib

Foundations

Depth 91 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleAddCommGroupModule