Theorems · Theorem · commutative algebra

Algebra.Generators.H1Cotangent.exact_liftBaseChange_map_of_flat

∀ {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) [Module.Flat S T],
  Function.Exact ⇑(LinearMap.liftBaseChange T (Algebra.Extension.H1Cotangent.map (Q.toComp P).toExtensionHom))
    ⇑(Algebra.Extension.H1Cotangent.map (Q.ofComp P).toExtensionHom)

When $T$ is flat over $S$, the left bottom part of the snake lemma diagram used in the construction of the connecting homomorphism Algebra.Generators.H1Cotangent.δ naturally extends via a base change map.

Defined in

Mathlib.RingTheory.Kaehler.JacobiZariski

Cited by

1 results in Mathlib

Foundations

Depth 117 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraCommRingAlgebraAlgebraIsScalarTowerModule.Flat