TITLE: distinguished triangles and cohomology
QUESTION [19 upvotes]: Start with A an abelian category and form the derived category D(A).
Take a triangle (not necessarily distinguished) and take it's cohomology. We obtain a long sequence (not necessarily exact). If the triangle is distinguished it is exact. How about the converse: if the long sequence in cohomology is exact does it follow that the triangle is distinguished? (My guess is no, but I can't find a counter-example).

REPLY [3 votes]: You might be interested in the paper: Vaknin A., Virtual Triangles// K-Theory, 22 (2001), no. 2, 161--197.