homologySequence_exact₁ — Mathlib

English

A δ-mediated exact sequence gives a perfect resolution in a shifted context.

Русский

Крaткая последовательность, управляемая δ, обеспечивает разрешение после сдвига.

LaTeX

$$$(F.shift n_0).shifted\ δ_1$-exactness$$

Lean4

theorem homologySequence_exact₁ : (ShortComplex.mk _ _ (F.homologySequenceδ_comp T hT _ _ h)).Exact :=
  by
  refine ShortComplex.exact_of_iso ?_ (F.homologySequence_exact₂ _ (inv_rot_of_distTriang _ hT) n₁)
  refine ShortComplex.isoMk (-((F.shiftIso (-1) n₁ n₀ (by cutsat)).app _)) (Iso.refl _) (Iso.refl _) ?_ (by simp)
  dsimp
  simp only [homologySequenceδ, neg_comp, map_neg, comp_id,
    F.shiftIso_hom_app_comp_shiftMap_of_add_eq_zero T.mor₃ (-1) (neg_add_cancel 1) n₀ n₁ (by cutsat)]

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