Theorems · Definition · commutative algebra

Algebra.Generators.cotangentCompLocalizationAwayEquiv

{R : Type u_1} →
  {S : Type u_2} →
    {T : Type u_3} →
      {ι : Type u_4} →
        [inst : CommRing R] →
          [inst_1 : CommRing S] →
            [inst_2 : Algebra R S] →
              [inst_3 : CommRing T] →
                [inst_4 : Algebra R T] →
                  [inst_5 : Algebra S T] →
                    [inst_6 : IsScalarTower R S T] →
                      (g : S) →
                        [inst_7 : IsLocalization.Away g T] →
                          (P : Algebra.Generators R S ι) →
                            {x : ((Algebra.Generators.localizationAway T g).comp P).toExtension.Cotangent} →
                              (Algebra.Extension.Cotangent.map
                                      ((Algebra.Generators.localizationAway T g).ofComp P).toExtensionHom)
                                    x =
                                  Algebra.Generators.cMulXSubOneCotangent T g →
                                ((Algebra.Generators.localizationAway T g).comp P).toExtension.Cotangent ≃ₗ[T]
                                  TensorProduct S T P.toExtension.Cotangent ×
                                    (Algebra.Generators.localizationAway T g).toExtension.Cotangent

Let S be generated over R by P : R[X] → S with kernel I and let T be the localization of S away from g generated over S by S[Y] → T with kernel K. Denote by J the kernel of the induced R[X, Y] → T. Then J/J² ≃ₗ[T] T ⊗[S] (I/I²) × (K/K²). This is the splitting characterised by x ↦ (0, g * X - 1).

Defined in

Mathlib.RingTheory.Extension.Cotangent.LocalizationAway

Cited by

6 results in Mathlib

Foundations

Depth 148 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraCommRingAlgebraAlgebraIsScalarTowerIsLocalization.Away