Theorems · Definition · commutative algebra

AlgHom.tensorEqualizerEquiv

{R : Type u_1} →
  (S : Type u_2) →
    [inst : CommRing R] →
      [inst_1 : CommRing S] →
        [inst_2 : Algebra R S] →
          (T : Type u_3) →
            [inst_3 : CommRing T] →
              [inst_4 : Algebra R T] →
                [inst_5 : Algebra S T] →
                  [inst_6 : IsScalarTower R S T] →
                    {A : Type u_4} →
                      {B : Type u_5} →
                        [inst_7 : CommRing A] →
                          [inst_8 : CommRing B] →
                            [inst_9 : Algebra R A] →
                              [inst_10 : Algebra R B] →
                                (f g : A →ₐ[R] B) →
                                  [Module.Flat R T] →
                                    TensorProduct R T ↥(f.equalizer g) ≃ₐ[S]
                                      ↥((Algebra.TensorProduct.map (AlgHom.id S T) f).equalizer
                                          (Algebra.TensorProduct.map (AlgHom.id S T) g))

If T is R-flat, the canonical map T ⊗[R] eq(f, g) →ₐ[S] eq (𝟙 ⊗ f, 𝟙 ⊗ g) is an isomorphism.

Defined in

Mathlib.RingTheory.Flat.Equalizer

Cited by

1 results in Mathlib

Foundations

Depth 112 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraCommRingAlgebraAlgebraIsScalarTowerCommRingCommRingAlgebraAlgebraModule.Flat