isoCocycles₁_hom_comp_i — Mathlib

English

The inverse of isoCocycles₁ composed with iCocycles equals shortComplexH1 leftHomology data composed with cochainsIso₁ inverse.

Русский

Обратное isoCocycles₁ композиция c iCocycles равна длинному левому гомоморфизму короткого комплекса H1 с коцепи1 обратной стороны.

LaTeX

$$$(isoCocycles_1(A)).inv \\circ iCocycles(A,1) = (shortComplexH1(A)).moduleCatLeftHomologyData.i \\circ (cochainsIso_1(A)).inv$$

Lean4

@[reassoc (attr := simp), elementwise (attr := simp)]
theorem isoCocycles₁_hom_comp_i :
    (isoCocycles₁ A).hom ≫ (shortComplexH1 A).moduleCatLeftHomologyData.i = iCocycles A 1 ≫ (cochainsIso₁ A).hom := by
  simp [isoCocycles₁, iCocycles, HomologicalComplex.iCycles, iCycles]

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