Theorems · Definition · commutative algebra

KaehlerDifferential.tensorKaehlerEquivOfFormallyEtale

(R : Type u_1) →
  (S : Type u_2) →
    (T : Type u_3) →
      [inst : CommRing R] →
        [inst_1 : CommRing S] →
          [inst_2 : CommRing T] →
            [inst_3 : Algebra R S] →
              [inst_4 : Algebra R T] →
                [inst_5 : Algebra S T] →
                  [IsScalarTower R S T] → [Algebra.FormallyEtale S T] → TensorProduct S T Ω[S⁄R] ≃ₗ[T] Ω[T⁄R]

The canonical isomorphism T ⊗[S] Ω[S⁄R] ≃ₗ[T] Ω[T⁄R] for T a formally étale S-algebra. Also see S ⊗[R] Ω[A⁄R] ≃ₗ[S] Ω[S ⊗[R] A⁄S] at KaehlerDifferential.tensorKaehlerEquiv.

Defined in

Mathlib.RingTheory.Etale.Kaehler

Cited by

5 results in Mathlib

Foundations

Depth 119 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingCommRingAlgebraAlgebraAlgebraIsScalarTowerAlgebra.FormallyEtale