Theorems · Definition · commutative algebra

derivationQuotKerSq

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

Given a tower of algebras S/P/R, with I = ker(P → S), this is the R-derivative P/I² → S ⊗[P] Ω[P⁄R] given by [x] ↦ 1 ⊗ D x.

Defined in

Mathlib.RingTheory.Smooth.Kaehler

Cited by

2 results in Mathlib

Foundations

Depth 99 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingCommRingAlgebraAlgebraAlgebraIsScalarTower