Theorems · Definition · linear algebra

TensorProduct.AlgebraTensorModule.tensorTensorTensorComm

(R : Type uR) →
  (S : Type uS) →
    (A : Type uA) →
      (B : Type uB) →
        (M : Type uM) →
          (N : Type uN) →
            (P : Type uP) →
              (Q : Type uQ) →
                [inst : CommSemiring R] →
                  [inst_1 : CommSemiring A] →
                    [inst_2 : Semiring B] →
                      [inst_3 : Algebra R A] →
                        [inst_4 : Algebra R B] →
                          [inst_5 : AddCommMonoid M] →
                            [inst_6 : Module R M] →
                              [inst_7 : Module A M] →
                                [inst_8 : Module B M] →
                                  [IsScalarTower R A M] →
                                    [IsScalarTower R B M] →
                                      [inst_11 : SMulCommClass A B M] →
                                        [inst_12 : AddCommMonoid N] →
                                          [inst_13 : Module R N] →
                                            [inst_14 : AddCommMonoid P] →
                                              [inst_15 : Module A P] →
                                                [inst_16 : AddCommMonoid Q] →
                                                  [inst_17 : Module R Q] →
                                                    [inst_18 : Module R P] →
                                                      [inst_19 : IsScalarTower R A P] →
                                                        [inst_20 : Algebra A B] →
                                                          [IsScalarTower A B M] →
                                                            [inst_22 : CommSemiring S] →
                                                              [inst_23 : Algebra R S] →
                                                                [inst_24 : Algebra S B] →
                                                                  [inst_25 : Module S M] →
                                                                    [inst_26 : Module S N] →
                                                                      [IsScalarTower R S M] →
                                                                        [inst_28 : SMulCommClass A S M] →
                                                                          [inst_29 : SMulCommClass S A M] →
                                                                            [inst_30 : IsScalarTower S B M] →
                                                                              [inst_31 : IsScalarTower R S N] →
                                                                                TensorProduct A (TensorProduct S M N)
                                                                                    (TensorProduct R P Q) ≃ₗ[B]
                                                                                  TensorProduct S (TensorProduct A M P)
                                                                                    (TensorProduct R N Q)

Heterobasic version of tensorTensorTensorComm. Suppose we have towers of algebras R → S → B and R → A → B, and a B-module M, S-module N, A-module P, R-module Q, then `` (M ⊗ˢ N) ⎛ M ⎞ ⊗ˢ ⎛ N ⎞ ⊗ᴬ ≅ᴮ ⎜ ⊗ᴬ⎟ ⎜ ⊗ᴿ⎟ (P ⊗ᴿ Q) ⎝ P ⎠ ⎝ Q ⎠ ``

Defined in

Mathlib.LinearAlgebra.TensorProduct.Tower

Cited by

10 results in Mathlib

Foundations

Depth 76 from the axioms · uses propext, Quot.sound

Assumes

CommSemiringCommSemiringSemiringAlgebraAlgebraAddCommMonoidModuleModuleModuleIsScalarTowerIsScalarTowerSMulCommClassAddCommMonoidModuleAddCommMonoidModuleAddCommMonoidModuleModuleIsScalarTowerAlgebraIsScalarTowerCommSemiringAlgebraAlgebraModuleModuleIsScalarTowerSMulCommClassSMulCommClassIsScalarTowerIsScalarTower