A characterization of long exact sequences coming from the snake lemma

Given an abelian category, we characterize the long exact sequences of length six which can be obtained from the snake lemma. Equivalently, these are the long exact sequences which arise as the homology of a triangle in the corresponding derived bounded category.

💡 Research Summary

The paper addresses a classical problem in homological algebra: determining precisely which six‑term long exact sequences in an abelian category arise from the Snake Lemma. The authors begin by recalling the standard formulation of the Snake Lemma, which starts with two short exact sequences
0 → A′ → A → A″ → 0, 0 → B′ → B → B″ → 0
and a morphism f : A → B that makes the obvious squares commute. By taking kernels and cokernels of the induced maps, one obtains a six‑term exact sequence
0 → Ker f → Ker g → Ker h → Coker f → Coker g → Coker h → 0.
The central question is: given an arbitrary six‑term exact sequence, when can we guarantee that it is of this form?

To answer this, the authors translate the problem into the language of derived categories. In the bounded derived category D⁽ᵇ⁾(𝒜) of an abelian category 𝒜, any distinguished triangle X → Y → Z → X