Theorems · Definition · commutative algebra

Module.Flat.tensorSubmoduleAlgebraEquiv

{R : Type u} →
  {M : Type v} →
    {A : Type u_4} →
      [inst : CommSemiring R] →
        [inst_1 : AddCommMonoid M] →
          [inst_2 : Module R M] →
            [inst_3 : Semiring A] →
              [inst_4 : Algebra R A] →
                (e : TensorProduct R A M ≃ₗ[A] A) →
                  [Module.Flat R M] → [FaithfulSMul R A] → TensorProduct R A ↥(Module.Flat.submoduleAlgebra e) ≃ₗ[A] A

When a flat R-module M is embedded as a submodule of a faithful R-algebra A, the multiplication map induces an isomorphism A ⊗[R] M ≃ₗ[A] A.

Defined in

Mathlib.RingTheory.PicardGroup

Cited by

2 results in Mathlib

Foundations

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

Assumes

CommSemiringAddCommMonoidModuleSemiringAlgebraModule.FlatFaithfulSMul