Theorems · Definition · commutative algebra

IsBaseChange.tensorEquiv

{R : Type u_1} →
  {M : Type v₁} →
    {N : Type v₂} →
      {S : Type v₃} →
        [inst : AddCommMonoid M] →
          [inst_1 : AddCommMonoid N] →
            [inst_2 : CommSemiring R] →
              [inst_3 : CommSemiring S] →
                [inst_4 : Algebra R S] →
                  [inst_5 : Module R M] →
                    [inst_6 : Module R N] →
                      [inst_7 : Module S N] →
                        [inst_8 : IsScalarTower R S N] →
                          {f : M →ₗ[R] N} →
                            IsBaseChange S f →
                              (P : Type u_6) →
                                [inst_9 : AddCommGroup P] →
                                  [inst_10 : Module R P] →
                                    [inst_11 : Module S P] →
                                      [inst_12 : IsScalarTower R S P] → TensorProduct S P N ≃ₗ[S] TensorProduct R P M

Let R be a commutative ring, S be an R-algebra, M be an R-module, P be an S module, N be the base change of M to S, then P ⊗[S] N is isomorphic to P ⊗[R] M as S-modules.

Defined in

Mathlib.RingTheory.IsTensorProduct

Cited by

1 results in Mathlib

Foundations

Depth 77 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AddCommMonoidAddCommMonoidCommSemiringCommSemiringAlgebraModuleModuleModuleIsScalarTowerAddCommGroupModuleModuleIsScalarTower