Homotopy Category of Chain Complexes - The Triangulated Structure

The Triangulated Structure

The shift A of a complex A is the following complex

(note that ),

where the differential is .

For the cone of a morphism f we take the mapping cone. There are natural maps

This diagram is called a triangle. The homotopy category K(A) is a triangulated category, if one defines distinguished triangles to be isomorphic (in K(A), i.e. homotopy equivalent) to the triangles above, for arbitrary A, B and f. The same is true for the bounded variants K+(A), K-(A) and Kb(A). Although triangles make sense in Kom(A) as well, that category is not triangulated with respect to these distinguished triangles; for example,

is not distinguished since the cone of the identity map is not isomorphic to the complex 0 (however, the zero map is a homotopy equivalence, so that this triangle is distinguished in K(A)). Less trivially, the rotation of a distinguished triangle is obviously not distinguished in Kom(A), but (less obviously so) is distinguished in K(A). See the references for details.