Theorems · Definition · category theory

Function.Exact.splitSurjectiveEquiv

{R : Type u_1} →
  {M : Type u_2} →
    {N : Type u_4} →
      {P : Type u_6} →
        [inst : Semiring R] →
          [inst_1 : AddCommGroup M] →
            [inst_2 : AddCommGroup N] →
              [inst_3 : AddCommGroup P] →
                [inst_4 : Module R M] →
                  [inst_5 : Module R N] →
                    [inst_6 : Module R P] →
                      {f : M →ₗ[R] N} →
                        {g : N →ₗ[R] P} →
                          Function.Exact ⇑f ⇑g →
                            Function.Injective ⇑f →
                              { l // g ∘ₗ l = LinearMap.id } ≃
                                { e // f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e }

Given an exact sequence 0 → M → N → P, giving a section P → N is equivalent to giving a splitting N ≃ M × P.

Defined in

Mathlib.Algebra.Exact.Basic

Cited by

4 results in Mathlib

Foundations

Depth 39 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringAddCommGroupAddCommGroupAddCommGroupModuleModuleModule