Theorems · Definition · commutative algebra

Algebra.Generators.cotangentCompAwaySec

{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) →
                            ((Algebra.Generators.localizationAway T g).comp P).toExtension.Cotangent →
                              (Algebra.Generators.localizationAway T g).toExtension.Cotangent →ₗ[T]
                                ((Algebra.Generators.localizationAway T g).comp P).toExtension.Cotangent

In the notation of the module docstring: Since T is standard smooth of relative dimension one over S, K/K² is free of rank one generated by the image of g * X - 1. This is the section K/K² → J/J² defined by sending the image of g * X - 1 to x : J/J².

Defined in

Mathlib.RingTheory.Extension.Cotangent.LocalizationAway

Cited by

5 results in Mathlib

Foundations

Depth 144 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraCommRingAlgebraAlgebraIsScalarTowerIsLocalization.Away