Theorems · Definition · commutative algebra

Derivation.liftOfRightInverse

{R : Type u_1} →
  {A : Type u_2} →
    {M : Type u_3} →
      [inst : CommSemiring R] →
        [inst_1 : CommRing A] →
          [inst_2 : CommRing M] →
            [inst_3 : Algebra R A] →
              [inst_4 : Algebra R M] →
                {F : Type u_4} →
                  [inst_5 : FunLike F A M] →
                    [AlgHomClass F R A M] →
                      {f : F} →
                        {f_inv : M → A} →
                          Function.RightInverse f_inv ⇑f →
                            ⦃d : Derivation R A A⦄ → (∀ (x : A), f x = 0 → f (d x) = 0) → Derivation R M M

Lift a derivation via an algebra homomorphism f with a right inverse such that f(x) = 0 → f(d(x)) = 0. This gives the derivation f ∘ d ∘ f⁻¹. This is needed for an argument in [Rosenlicht, M. Integration in finite terms][Rosenlicht_1972].

Defined in

Mathlib.RingTheory.Derivation.Basic

Cited by

3 results in Mathlib

Foundations

Depth 29 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringCommRingCommRingAlgebraAlgebraFunLikeAlgHomClass