Theorems · Theorem · commutative algebra

Algebra.Generators.Cotangent.exact

∀ {R : Type u₁} {S : Type u₂} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] {T : Type u₃}
  [inst_3 : CommRing T] [inst_4 : Algebra R T] [inst_5 : Algebra S T] [inst_6 : IsScalarTower R S T] {ι : Type w₁}
  {σ : Type w₂} (Q : Algebra.Generators S T ι) (P : Algebra.Generators R S σ),
  Function.Exact ⇑(LinearMap.liftBaseChange T (Algebra.Extension.Cotangent.map (Q.toComp P).toExtensionHom))
    ⇑(Algebra.Extension.Cotangent.map (Q.ofComp P).toExtensionHom)

Given representations 0 → I → R[X] → S → 0 and 0 → K → S[Y] → T → 0, we may consider the induced representation 0 → J → R[X, Y] → T → 0, and the sequence T ⊗[S] (I/I²) → J/J² → K/K² is exact.

Defined in

Mathlib.RingTheory.Kaehler.JacobiZariski

Cited by

6 results in Mathlib

Foundations

Depth 106 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraCommRingAlgebraAlgebraIsScalarTower