exact_iff_image_eq_kernel — Mathlib

English

For cg a kernel fork of g and cf a cokernel cof Fork of f, S.Exact iff cg.ι ≫ cf.π = 0.

Русский

Для cg — ядрофорк g и cf — cokernel cofFork f, точность S эквивалентна cg.ι ≫ cf.π = 0.

LaTeX

$$$S.Exact \\iff cg.\\iota \\\\ cf.\\pi = 0$$$

Lean4

/-- In an abelian category, a short complex `S` is exact
iff `imageSubobject S.f = kernelSubobject S.g`.
-/
theorem exact_iff_image_eq_kernel : S.Exact ↔ imageSubobject S.f = kernelSubobject S.g :=
  by
  rw [exact_iff_isIso_imageToKernel]
  constructor
  · intro
    exact Subobject.eq_of_comm (asIso (imageToKernel _ _ S.zero)) (by simp)
  · intro h
    exact ⟨Subobject.ofLE _ _ h.ge, by ext; simp, by ext; simp⟩

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