property of derived categories is that they are *triangulated categories*. A triangulated category is a $k$-linear category ${\mathcal{T}}$ together with a $k$-linear automorphism $\Sigma:{\mathcal{T}}\to {\mathcal{T}}$ called the *suspension functor* and with a collection of sequences of morphisms of the form $$X \stackrel{f}{\longrightarrow} Y \stackrel{g}{\longrightarrow} Z \stackrel{h}{\longrightarrow} \Sigma X,$$ where $gf$ and $hg$ vanish. The sequences belonging to the collection are called *distinguished triangles*, or simply *triangles*. They are required to satisfy several axioms, which are listed below and which can be found in any of the references given at the beginning of the section.

(T1)

:   The class of triangles is closed under isomorphism of complexes of length $4$. For any object $X$, the sequence $X\stackrel{id_X}{\longrightarrow} X \longrightarrow 0 \longrightarrow \Sigma X$ is a triangle. Any morphism $X \stackrel{f}{\longrightarrow} Y$ can be embedded into a triangle $X \stackrel{f}{\longrightarrow} Y \stackrel{g}{\longrightarrow} Z \stackrel{h}{\longrightarrow} \Sigma X$.

(T2)

:   The sequence $X \stackrel{f}{\longrightarrow} Y \stackrel{g}{\longrightarrow} Z \stackrel{h}{\longrightarrow} \Sigma X$ is a triangle if and only if $ Y \stackrel{g}{\longrightarrow} Z \stackrel{h}{\longrightarrow} \Sigma X \stackrel{-\Sigma f}{\longrightarrow} \Sigma Y$ is.

(T3)

:   For any commutative diagram $$\xymatrix{ X \ar[r]^f\ar[d]^u & Y\ar[r]^g\ar[d]^v & Z\ar[r]^h & \Sigma X\ar[d]^{\Sigma u} \\
                   X' \ar[r]^{f'} & Y'\ar[r]^{g'} & Z'\ar[r]^{h'} & 