the categories ${\mathcal{C}}^+({\mathcal{A}})$, ${\mathcal{C}}^-({\mathcal{A}})$ and ${\mathcal{C}}^b({\mathcal{A}})$ of complexes bounded on the left, on the right and on both sides, respectively, then one defines derived categories ${\mathcal{D}}^+({\mathcal{A}})$, ${\mathcal{D}}^-({\mathcal{A}})$ and ${\mathcal{D}}^b({\mathcal{A}})$. Of importance to us in the next section will be ${\mathcal{D}}^b({\mathcal{A}})$, called the *bounded derived category*.

The advantage of defining the derived category by working in $K({\mathcal{A}})$ instead of ${\mathcal{C}}({\mathcal{A}})$ is that it allows one to use a notion of “calculus of fractions” of morphisms, see for instance [@Keller07 Section 2.2]. Another advantage, relevant to our situation, is that if ${\mathcal{A}}$ is the module category of a finite dimensional algebra $A$, and if we denote by ${\operatorname{{\rm proj }}}A$ the full subcategory of ${\mathcal{A}}$ whose objects are projective modules, then ${\mathcal{D}}^-({\mathcal{A}})$ is equivalent to $K^-({\operatorname{{\rm proj }}}A)$. This latter category is often easier to work with.

The functor $J:{\mathcal{A}}\to {\mathcal{D}}^*({\mathcal{A}})$ sending an object $M$ to the complex $C$ with $C_0 = M$ and $C_j=0$ if $j\neq 0$ is fully faithful. Here, $*$ can be $+$, $-$, $b$ or an absence of symbol.

By an abuse of notation, if $M$ is an object of ${\mathcal{A}}$, then we denote still by $M$ its image by the functor $J$.

### Triangulated categories

An important 