Theorems · Definition · linear algebra

Submodule.tensorSpanEquivSpan

(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] →
                    [Algebra.IsEpi R A] →
                      [Module.Flat R A] →
                        (s : Set M) → TensorProduct R A ↥(Submodule.span R s) ≃ₗ[A] ↥(Submodule.span A s)

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

Defined in

Mathlib.LinearAlgebra.Span.TensorProduct

Cited by

1 results in Mathlib

Foundations

Depth 106 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringCommSemiringAlgebraAddCommMonoidModuleModuleIsScalarTowerAlgebra.IsEpiModule.Flat