Theorems · Definition · commutative algebra

tensorKaehlerQuotKerSqEquiv

(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] →
                  [IsScalarTower R P S] →
                    TensorProduct (P ⧸ RingHom.ker (algebraMap P S) ^ 2) S
                        Ω[P ⧸ RingHom.ker (algebraMap P S) ^ 2⁄R] ≃ₗ[S]
                      TensorProduct P S Ω[P⁄R]

Given a tower of algebras S/P/R, with I = ker(P → S) and Q := P/I², there is an isomorphism of S-modules S ⊗[Q] Ω[Q/R] ≃ S ⊗[P] Ω[P/R].

Defined in

Mathlib.RingTheory.Smooth.Kaehler

Cited by

4 results in Mathlib

Foundations

Depth 101 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingCommRingAlgebraAlgebraAlgebraIsScalarTower