isoShortComplexH1 — Mathlib

English

The short complex A --d01--> Fun(G,A) --d12--> Fun(G×G,A) is isomorphic to the 1st short complex of inhomogeneous cochains.

Русский

Короткий комплекс A --d01--> Fun(G,A) --d12--> Fun(G×G,A) изоморфен 1-му короткому комплексу инхоомогенных когностей.

LaTeX

$$shortComplexH1(A) ≅ (inhomogeneousCochains A).sc 1$$

Lean4

/-- The short complex `A --d₀₁--> Fun(G, A) --d₁₂--> Fun(G × G, A)` is isomorphic to the 1st
short complex associated to the complex of inhomogeneous cochains of `A`. -/
@[simps! hom inv]
def isoShortComplexH1 : (inhomogeneousCochains A).sc 1 ≅ shortComplexH1 A :=
  (inhomogeneousCochains A).isoSc' 0 1 2 (by simp) (by simp) ≪≫
    isoMk (cochainsIso₀ A) (cochainsIso₁ A) (cochainsIso₂ A) (comp_d₀₁_eq A) (comp_d₁₂_eq A)

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