A Characterization of Long Exact Sequences Coming from the Snake Lemma

0 → Ker f1 → Ker f2 → Ker f3 → Coker f1 → Coker f2 → Coker f3 → 0, where f = (f1, f2, f3) is a suitable homomorphism between short exact sequences in A? This question is an instance of a more general problem. Let D(A) be the derived bounded category of A. We ignore the possible set-theoretic difficulties mentioned in [5, §2.2], because in what we intend to do with D(A) here they do not play any role. Now, one can ask when a long exact sequence ǫ : 0 → A1 −→ A2 −→ . . . −→ An → 0 in A arises as the homology sequence of a triangle X → Y → Z → X[1] in D(A). For n = 5 this has been completely resolved by Neeman [6]: the long exact sequence ǫ is the homology of a triangle if and only if the corresponding class in ExtA(A5, A1) vanishes. It is worth to mention that Neeman proved more. Namely, if ǫ comes from a triangle for n > 5, then the corresponding class in ExtA vanishes for every bit of the form 0 → K −→ Ai−1 −→ Ai −→ Ai+1 −→ C → 0. One may wonder what exactly happens for n = 6. We have a necessary condition for two classes in ExtA, but this is not sufficient as we illustrate