Theorems · Theorem · commutative algebra

toAlgHom_comp_sectionOfRetractionKerToTensorAux

∀ {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] (l : TensorProduct P S Ω[P⁄R] →ₗ[P] ↥(RingHom.ker (algebraMap P S)))
  (hl : l ∘ₗ KaehlerDifferential.kerToTensor R P S = LinearMap.id) (σ : S → P)
  (hσ : ∀ (x : S), (algebraMap P S) (σ x) = x) [inst_5 : Algebra R S] [inst_6 : IsScalarTower R P S]
  (hf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥),
  Function.Surjective ⇑(algebraMap P S) →
    (IsScalarTower.toAlgHom R P S).comp (sectionOfRetractionKerToTensorAux l hl σ hσ hf') = AlgHom.id R S

Defined in

Mathlib.RingTheory.Smooth.Kaehler

Cited by

1 results in Mathlib

Foundations

Depth 103 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingCommRingAlgebraAlgebraAlgebraIsScalarTower