Theorems · Definition · commutative algebra

Algebra.FormallySmooth.liftOfSurjective

{R : Type u} →
  {A : Type v} →
    [inst : CommRing R] →
      [inst_1 : CommRing A] →
        [inst_2 : Algebra R A] →
          {B : Type u_1} →
            [inst_3 : CommRing B] →
              [inst_4 : Algebra R B] →
                {C : Type u_4} →
                  [inst_5 : CommRing C] →
                    [inst_6 : Algebra R C] →
                      [Algebra.FormallySmooth R A] →
                        (A →ₐ[R] C) →
                          (g : B →ₐ[R] C) → Function.Surjective ⇑g → IsNilpotent (RingHom.ker ↑g) → A →ₐ[R] B

For a formally smooth R-algebra A and a map f : A →ₐ[R] B ⧸ I with I nilpotent, this is an arbitrary lift A →ₐ[R] B.

Defined in

Mathlib.RingTheory.Smooth.Basic

Cited by

2 results in Mathlib

Foundations

Depth 111 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraCommRingAlgebraCommRingAlgebraAlgebra.FormallySmooth