Theorems · Definition · linear algebra

PiTensorProduct.piTensorHomMap

{ι : Type u_1} →
  {R : Type u_4} →
    [inst : CommSemiring R] →
      {s : ι → Type u_7} →
        [inst_1 : (i : ι) → AddCommMonoid (s i)] →
          [inst_2 : (i : ι) → Module R (s i)] →
            {t : ι → Type u_11} →
              [inst_3 : (i : ι) → AddCommMonoid (t i)] →
                [inst_4 : (i : ι) → Module R (t i)] →
                  (PiTensorProduct R fun i => s i →ₗ[R] t i) →ₗ[R]
                    (PiTensorProduct R fun i => s i) →ₗ[R] PiTensorProduct R fun i => t i

Let sᵢ and tᵢ be families of R-modules. Then there is an R-linear map between ⨂ᵢ Hom(sᵢ, tᵢ) and Hom(⨂ᵢ sᵢ, ⨂ tᵢ) defined by ⨂ᵢ fᵢ ↦ ⨂ᵢ aᵢ ↦ ⨂ᵢ fᵢ aᵢ. This is TensorProduct.homTensorHomMap for an arbitrary family of modules. Note that PiTensorProduct.piTensorHomMap (tprod R f) is equal to PiTensorProduct.map f.

Defined in

Mathlib.LinearAlgebra.PiTensorProduct.Basic

Cited by

7 results in Mathlib

Foundations

Depth 61 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringAddCommMonoidModuleAddCommMonoidModule