D₂ — Mathlib

English

If the π-entry misses the i12 index, the vertical differential D2 vanishes.

Русский

Если π‑запись не попадает в индекс i12, D2 = 0.

LaTeX

$$$K.d_{2} c_{12} i_{1} i_{2} i_{12} = 0$ under the condition $c_{2}.Rel i_{2} (c_{2}.next i_{2})$ not matching $i_{12}$$$

Lean4

/-- The vertical differential in the total complex. -/
noncomputable def D₂ (i₁₂ i₁₂' : I₁₂) :
    K.toGradedObject.mapObj (ComplexShape.π c₁ c₂ c₁₂) i₁₂ ⟶ K.toGradedObject.mapObj (ComplexShape.π c₁ c₂ c₁₂) i₁₂' :=
  GradedObject.descMapObj _ (ComplexShape.π c₁ c₂ c₁₂) (fun ⟨i₁, i₂⟩ _ => K.d₂ c₁₂ i₁ i₂ i₁₂')

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