that is, $f_{i+1}\circ d_i = d'_i\circ f_i$.

We denote by ${\mathcal{C}}({\mathcal{A}})$ the category of complexes of ${\mathcal{A}}$. It is an abelian category. It admits an automorphism called the *shift functor* and denoted by $[1]$, which is defined by $(C[1])_i = C_{i+1}$, and where the differential $\delta$ of $C[1]$ is defined by $\delta_i = -d_{i+1}$.

The *homology* of a complex $C$ at degree $i$ is the object $H_i(C) := \ker(d_i)/{\operatorname{\rm im\,}}(d_{i-1})$. It is easy to see that a morphism of complexes $f:C\to C'$ induces in each degree a morphism $H_i(f):H_i(C)\to H_i(C')$. A *quasi-isomorphism* is a morphism of complexes $f$ such that all induced morphisms in homology are isomorphisms.

The derived category of ${\mathcal{A}}$ is the category obtained when formally inverting all quasi-isomorphisms in ${\mathcal{C}}({\mathcal{A}})$. A convenient construction of the derived category is given by first defining the *homotopy category* $K({\mathcal{A}})$. This category is the quotient of ${\mathcal{C}}({\mathcal{A}})$ by the ideal of all *null-homotopic morphisms*, that is, morphisms of complexes $f:C\to C'$ such that there exist morphisms $s_i:C_i\to C'_{i-1}$ is ${\mathcal{A}}$ such that $f_i=d'_{i-1}s_i + s_{i+1}d_i$ for all $i\in {\mathbb{Z}}$.

The *derived category* ${\mathcal{D}}({\mathcal{A}})$ is then the category obtained from $K({\mathcal{A}})$ by formally inverting all quasi-isomorphisms.

If, instead of ${\mathcal{C}}({\mathcal{A}})$, one considers 