lift — Mathlib

English

For a formally smooth R-algebra A and a surjective map g: B ⟶ B/I with nilpotent I, any map g: A → B/I lifts to a map A → B.

Русский

Для формально гладкого R-алгебры A и сюръективного отображения g: B → B/I с nilpotent I любая карта A → B/I может быть поднята до A → B.

LaTeX

$$There exists f: A → B such that (quotient map) ∘ f = g under FormallySmooth assumptions.$$

Lean4

/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
  (FormallySmooth.exists_lift I hI g).choose

Похожие утверждения