Theorems · Definition · category theory

CategoryTheory.Abelian.Ext.contravariantSequence

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    [inst_1 : CategoryTheory.Abelian C] →
      [CategoryTheory.HasExt C] →
        {S : CategoryTheory.ShortComplex C} →
          S.ShortExact → C → (n₀ n₁ : ℕ) → 1 + n₀ = n₁ → CategoryTheory.ComposableArrows AddCommGrpCat 5

Given a short exact short complex S in an abelian category C and an object Y : C, this is the long exact sequence Ext S.X₃ Y n₀ → Ext S.X₂ Y n₀ → Ext S.X₁ Y n₀ → Ext S.X₃ Y n₁ → Ext S.X₂ Y n₁ → Ext S.X₁ Y n₁ when 1 + n₀ = n₁.

Defined in

Mathlib.Algebra.Homology.DerivedCategory.Ext.ExactSequences

Cited by

2 results in Mathlib

Foundations

Depth 115 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.AbelianCategoryTheory.HasExt