“big squares” containing the top and bottom vertices are commutative diagrams.

A consequence of the axioms is that for any object $X$ of ${\mathcal{T}}$, the functor ${\operatorname{{\rm Hom}}_{{\mathcal{T}}}}(X,?):{\mathcal{T}}\to {\mbox{{\rm mod \!}}}k$ sends triangles to exact sequences.

The derived category ${\mathcal{D}}({\mathcal{A}})$ is a triangulated category whose suspension functor is $[1]$.

### Hereditary case

We now restrict to the case where ${\mathcal{A}}= {\mbox{{\rm mod \!}}}kQ$, for some finite quiver $Q$ without oriented cycles. The path algebra $kQ$ is then *hereditary*; in other words, the extension bifunctors ${\operatorname{\rm Ext}^{i}_{kQ}}(?,?)$ vanish for $i\geq 2$.

In this situation, we have a good description of the objects of the bounded derived category ${\mathcal{D}}^b({\mbox{{\rm mod \!}}}kQ)$.

All indecomposable objects of ${\mathcal{D}}^b({\mbox{{\rm mod \!}}}kQ)$ are isomorphic to indecomposable *stalk complexes*, that is, complexes $C$ for which there is an integer $i$ such that $C_j=0$ for $j\neq i$, and $C_i$ is an indecomposable $kQ$-module.

Thus all indecomposable objects of ${\mathcal{D}}^b({\mbox{{\rm mod \!}}}kQ)$ have the form $M[i]$, for $M$ an indecomposable $kQ$-module and $i$ an integer.

Another important feature in this case is the existence of an automorphism of the derived category called the *Auslander–Reiten translation* and denoted by $\tau$. We refer the reader to, for instance, [@K08 Section 3] 