Theorems · Definition · commutative algebra

TensorProduct.congr

{R : Type u_1} →
  {R₂ : Type u_2} →
    [inst : CommSemiring R] →
      [inst_1 : CommSemiring R₂] →
        {σ₁₂ : R →+* R₂} →
          {M : Type u_7} →
            {N : Type u_8} →
              {M₂ : Type u_12} →
                {N₂ : Type u_14} →
                  [inst_2 : AddCommMonoid M] →
                    [inst_3 : AddCommMonoid N] →
                      [inst_4 : AddCommMonoid M₂] →
                        [inst_5 : AddCommMonoid N₂] →
                          [inst_6 : Module R M] →
                            [inst_7 : Module R N] →
                              [inst_8 : Module R₂ M₂] →
                                [inst_9 : Module R₂ N₂] →
                                  {σ₂₁ : R₂ →+* R} →
                                    [inst_10 : RingHomInvPair σ₁₂ σ₂₁] →
                                      [inst_11 : RingHomInvPair σ₂₁ σ₁₂] →
                                        (M ≃ₛₗ[σ₁₂] M₂) →
                                          (N ≃ₛₗ[σ₁₂] N₂) → TensorProduct R M N ≃ₛₗ[σ₁₂] TensorProduct R₂ M₂ N₂

If M and P are semilinearly equivalent and N and Q are semilinearly equivalent then M ⊗ N and P ⊗ Q are semilinearly equivalent.

Defined in

Mathlib.LinearAlgebra.TensorProduct.Map

Cited by

50 results in Mathlib

Foundations

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

Assumes

CommSemiringCommSemiringAddCommMonoidAddCommMonoidAddCommMonoidAddCommMonoidModuleModuleModuleModuleRingHomInvPairRingHomInvPair