Theorems · Definition · commutative algebra

DualNumber.lift

{R : Type u_1} →
  {B : Type u_3} →
    {A : Type u_4} →
      [inst : CommSemiring R] →
        [inst_1 : Semiring A] →
          [inst_2 : Semiring B] →
            [inst_3 : Algebra R A] →
              [inst_4 : Algebra R B] →
                { fe // fe.2 * fe.2 = 0 ∧ ∀ (a : A), Commute fe.2 (fe.1 a) } ≃ (DualNumber A →ₐ[R] B)

A universal property of the dual numbers, providing a unique A[ε] →ₐ[R] B for every map f : A →ₐ[R] B and a choice of element e : B which squares to 0 and commutes with the range of f. This isomorphism is named to match the similar Complex.lift. Note that when f : R →ₐ[R] B := Algebra.ofId R B, the commutativity assumption is automatic, and we are free to choose any element e : B.

Defined in

Mathlib.Algebra.DualNumber

Cited by

9 results in Mathlib

Foundations

Depth 41 from the axioms · uses propext, Quot.sound

Assumes

CommSemiringSemiringSemiringAlgebraAlgebra