Theorems · Definition · linear algebra

Submodule.tensorEquivSpan

{R : Type u_1} →
  (A : Type u_2) →
    {M : Type u_3} →
      [inst : CommSemiring R] →
        [inst_1 : CommSemiring A] →
          [inst_2 : Algebra R A] →
            [inst_3 : AddCommMonoid M] →
              [inst_4 : Module R M] →
                [inst_5 : Module A M] →
                  [IsScalarTower R A M] →
                    (p : Submodule R M) →
                      [Algebra.IsEpi R A] → [Module.Flat R A] → TensorProduct R A ↥p ≃ₗ[A] ↥(Submodule.span A ↑p)

If A is a flat epi R-algebra and p is an R-submodule of an A-module M then the natural surjection from A ⊗[R] p to span A p is an equivalence.

Defined in

Mathlib.LinearAlgebra.Span.TensorProduct

Cited by

2 results in Mathlib

Foundations

Depth 105 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringCommSemiringAlgebraAddCommMonoidModuleModuleIsScalarTowerAlgebra.IsEpiModule.Flat