Derived category

Summary

The derived category $D(\mathcal{A})$ of an abelian category $\mathcal{A}$ is obtained from the category of chain complexes by declaring every quasi-isomorphism to be an isomorphism. In it, an object and any of its resolutions become isomorphic, so derived functors become ordinary functors between derived categories. It is a triangulated category, and $\operatorname{Ext}^n(X, Y)$ is the group of morphisms from $X$ to $Y[n]$. Derived categories are the standard setting for duality theorems, for sheaf theory and for modern algebraic geometry.