Theorems · Theorem · commutative algebra

Algebra.FormallySmooth.exists_kerProj_comp_eq_id

∀ {R : Type u_1} {A : Type u_2} [inst : CommRing R] [inst_1 : CommRing A] [inst_2 : Algebra R A] {S : Type u_3}
  [inst_3 : CommRing S] [inst_4 : Algebra R S] [Algebra.FormallySmooth R A] (f : S →ₐ[R] A)
  (hf : Function.Surjective ⇑f), ∃ g, (AdicCompletion.kerProj hf).comp g = AlgHom.id R A

If A is formally smooth over R, the projection from the adic completion of S at the kernel of f : S →ₐ[R] A has a section.

Defined in

Mathlib.RingTheory.Smooth.AdicCompletion

Cited by

1 results in Mathlib

Foundations

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

Assumes

CommRingCommRingAlgebraCommRingAlgebraAlgebra.FormallySmooth