Derived category

Remarks

The localization can be computed in two steps: first pass to the homotopy category $K(\mathcal{A})$, then invert quasi-isomorphisms there by a calculus of fractions. A morphism $X \to Y$ in $D(\mathcal{A})$ is then represented by a roof $X \xleftarrow{s} X' \to Y$ with $s$ a quasi-isomorphism. For general $\mathcal{A}$ the morphisms between two objects might not form a set. They do form a set for module categories, and for bounded below complexes when $\mathcal{A}$ has enough injectives. Example: over $\Z$, let $C$ be $\Z \xrightarrow{\cdot 2} \Z$ in degrees $1$ and $0$, and let $\Z/2\Z$ sit in degree $0$. The quotient map $C_0 = \Z \to \Z/2\Z$ is a chain map, and it is a quasi-isomorphism since $H_0(C) \cong \Z/2\Z$ and $H_1(C) = 0$; so $C \cong \Z/2\Z$ in $D(\mathcal{A})$. Non-example: $C$ and $\Z/2\Z$ are not isomorphic in $K(\mathcal{A})$, since every chain map $\Z/2\Z \to C$ is zero, because $\Hom(\Z/2\Z, \Z) = 0$; so $K(\mathcal{A})$ is not the derived category. The bounded variants $D^b$, $D^+$ and $D^-$ are defined in the same way from bounded complexes.