The homotopy category $K(\mathcal{A})$ of an abelian category $\mathcal{A}$ has:

• Objects: Chain complexes in $\mathcal{A}$
• Morphisms: Chain maps modulo chain homotopy

Two chain maps $f, g: C_\bullet \to D_\bullet$ are identified if they are chain homotopic.

A chain map $f: C_\bullet \to D_\bullet$ is a quasi-isomorphism (qis) if it induces isomorphisms on all homology groups: $H_n(f): H_n(C_\bullet) \xrightarrow{\cong} H_n(D_\bullet) \quad \text{for all } n$

Quasi-isomorphisms are the "weak equivalences" we want to invert.

The derived category $D(\mathcal{A})$ is obtained from the homotopy category $K(\mathcal{A})$ by formally inverting all quasi-isomorphisms: $D(\mathcal{A}) = K(\mathcal{A})[\text{qis}^{-1}]$

This is a localization of $K(\mathcal{A})$ at the class of quasi-isomorphisms.

For practical computations, we often use subcategories:

• $D^+(\mathcal{A})$: complexes bounded below (cohomologically)
• $D^-(\mathcal{A})$: complexes bounded above
• $D^b(\mathcal{A})$: bounded complexes (finite homology)

For coherent sheaves on a scheme, $D^b(\text{Coh}(X))$ is the most important.