Theorems · Definition · commutative algebra

Algebra.IsPushout.cancelBaseChange

(R : Type u_1) →
  (S : Type v₃) →
    [inst : CommSemiring R] →
      [inst_1 : CommSemiring S] →
        [inst_2 : Algebra R S] →
          (A : Type u_8) →
            (B : Type u_9) →
              [inst_3 : CommRing A] →
                [inst_4 : CommRing B] →
                  [inst_5 : Algebra R A] →
                    [inst_6 : Algebra R B] →
                      [inst_7 : Algebra A B] →
                        [inst_8 : Algebra S B] →
                          [inst_9 : IsScalarTower R A B] →
                            [inst_10 : IsScalarTower R S B] →
                              [Algebra.IsPushout R S A B] →
                                (M : Type u_10) →
                                  [inst_12 : AddCommGroup M] →
                                    [inst_13 : Module R M] →
                                      [inst_14 : Module A M] →
                                        [IsScalarTower R A M] → TensorProduct A B M ≃ₗ[S] TensorProduct R S M

If B = S ⊗[R] A, this is the canonical S-isomorphism: B ⊗[A] M ≃ₗ[S] S ⊗[R] M. This is the cancelling on the left version of TensorProduct.AlgebraTensorModule.cancelBaseChange.

Defined in

Mathlib.RingTheory.IsTensorProduct

Cited by

6 results in Mathlib

Foundations

Depth 86 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringCommSemiringAlgebraCommRingCommRingAlgebraAlgebraAlgebraAlgebraIsScalarTowerIsScalarTowerAlgebra.IsPushoutAddCommGroupModuleModuleIsScalarTower