Theorems · Theorem · category theory

CategoryTheory.Abelian.SpectralObject.isZero_spectralSequence_page_X_iff

∀ {C : Type u_1} {ι : Type u_2} {κ : Type u_3} [inst : CategoryTheory.Category.{v_1, u_1} C]
  [inst_1 : CategoryTheory.Abelian C] [inst_2 : Preorder ι] (X : CategoryTheory.Abelian.SpectralObject C ι)
  {c : ℤ → ComplexShape κ} {r₀ : ℤ} (data : CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore ι c r₀)
  [inst_3 : X.HasSpectralSequence data] (r : ℤ) (hr : r₀ ≤ r) (pq : κ) (i₀ i₁ i₂ i₃ : ι) (h₀ : i₀ = data.i₀ r pq ⋯)
  (h₁ : i₁ = data.i₁ pq) (h₂ : i₂ = data.i₂ pq) (h₃ : i₃ = data.i₃ r pq ⋯) (n₀ n₁ n₂ : ℤ),
  n₁ = data.deg pq →
    ∀ (hn₁ : autoParam (n₀ + 1 = n₁) CategoryTheory.Abelian.SpectralObject.isZero_spectralSequence_page_X_iff._auto_1)
      (hn₂ : autoParam (n₁ + 1 = n₂) CategoryTheory.Abelian.SpectralObject.isZero_spectralSequence_page_X_iff._auto_3),
      CategoryTheory.Limits.IsZero (((X.spectralSequence data).page r ⋯).X pq) ↔
        CategoryTheory.Limits.IsZero
          (X.E (CategoryTheory.homOfLE ⋯) (CategoryTheory.homOfLE ⋯) (CategoryTheory.homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯)

Defined in

Mathlib.Algebra.Homology.SpectralObject.SpectralSequence

Cited by

1 results in Mathlib

Foundations

Depth 131 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.AbelianPreorderCategoryTheory.Abelian.SpectralObject.HasSpectralSequence