Theorems · Definition · commutative algebra

LinearMap.tensorKerEquiv

{R : Type u_1} →
  (S : Type u_2) →
    [inst : CommRing R] →
      [inst_1 : CommRing S] →
        [inst_2 : Algebra R S] →
          (M : Type u_3) →
            [inst_3 : AddCommGroup M] →
              [inst_4 : Module R M] →
                [inst_5 : Module S M] →
                  [inst_6 : IsScalarTower R S M] →
                    {N : Type u_4} →
                      {P : Type u_5} →
                        [inst_7 : AddCommGroup N] →
                          [inst_8 : AddCommGroup P] →
                            [inst_9 : Module R N] →
                              [inst_10 : Module R P] →
                                (f : N →ₗ[R] P) →
                                  [Module.Flat R M] →
                                    TensorProduct R M ↥f.ker ≃ₗ[S]
                                      ↥((TensorProduct.AlgebraTensorModule.lTensor S M) f).ker

If M is R-flat, the canonical map M ⊗[R] ker f →ₗ[R] ker (𝟙 ⊗ f) is an isomorphism.

Defined in

Mathlib.RingTheory.Flat.Equalizer

Cited by

2 results in Mathlib

Foundations

Depth 109 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraAddCommGroupModuleModuleIsScalarTowerAddCommGroupAddCommGroupModuleModuleModule.Flat