Theorems · Definition · commutative algebra

retractionKerCotangentToTensorEquivSection

{R : Type u_1} →
  {P : Type u_2} →
    {S : Type u_3} →
      [inst : CommRing R] →
        [inst_1 : CommRing P] →
          [inst_2 : CommRing S] →
            [inst_3 : Algebra R P] →
              [inst_4 : Algebra P S] →
                [inst_5 : Algebra R S] →
                  [inst_6 : IsScalarTower R P S] →
                    Function.Surjective ⇑(algebraMap P S) →
                      { l // l ∘ₗ KaehlerDifferential.kerCotangentToTensor R P S = LinearMap.id } ≃
                        { g // (IsScalarTower.toAlgHom R P S).kerSquareLift.comp g = AlgHom.id R S }

Given a surjective algebra homomorphism f : P →ₐ[R] S with kernel I, there is a one-to-one correspondence between P-linear retractions of I/I² →ₗ[P] S ⊗[P] Ω[P/R] and algebra homomorphism sections of f‾ : P/I² → S.

Defined in

Mathlib.RingTheory.Smooth.Kaehler

Cited by

2 results in Mathlib

Foundations

Depth 107 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingCommRingAlgebraAlgebraAlgebraIsScalarTower