Theorems · Definition · commutative algebra

IsLocalization.moduleTensorEquiv

{R : Type u_1} →
  [inst : CommSemiring R] →
    (S : Submonoid R) →
      (A : Type u_2) →
        [inst_1 : CommSemiring A] →
          [inst_2 : Algebra R A] →
            [IsLocalization S A] →
              (M₁ : Type u_5) →
                (M₂ : Type u_6) →
                  [inst_4 : AddCommMonoid M₁] →
                    [inst_5 : AddCommMonoid M₂] →
                      [inst_6 : Module R M₁] →
                        [inst_7 : Module R M₂] →
                          [inst_8 : Module A M₁] →
                            [inst_9 : Module A M₂] →
                              [inst_10 : IsScalarTower R A M₁] →
                                [IsScalarTower R A M₂] → TensorProduct A M₁ M₂ ≃ₗ[A] TensorProduct R M₁ M₂

If A is a localization of R, tensoring two A-modules over A is the same as tensoring them over R.

Defined in

Mathlib.RingTheory.Localization.BaseChange

Cited by

0 results in Mathlib

Foundations

Depth 64 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringCommSemiringAlgebraIsLocalizationAddCommMonoidAddCommMonoidModuleModuleModuleModuleIsScalarTowerIsScalarTower