Theorems · Theorem · commutative algebra

Algebra.FormallySmooth.iff_split_surjection

∀ {R : Type u} {A : Type v} [inst : CommRing R] [inst_1 : CommRing A] [inst_2 : Algebra R A] {P : Type u_2}
  [inst_3 : CommRing P] [inst_4 : Algebra R P] [Algebra.FormallySmooth R P] (f : P →ₐ[R] A),
  Function.Surjective ⇑f → (Algebra.FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A)

Let P →ₐ[R] A be a surjection with kernel J, and P a formally smooth R-algebra, then A is formally smooth over R iff the surjection P ⧸ J ^ 2 →ₐ[R] A has a section. Geometric intuition: we require that a first-order thickening of Spec A inside Spec P admits a retraction.

Defined in

Mathlib.RingTheory.Smooth.Basic

Cited by

5 results in Mathlib

Foundations

Depth 128 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraCommRingAlgebraAlgebra.FormallySmooth