The derived (infinity,1)-category or derived category of an abelian category is the setting for homological algebra in : the (infinity,1)-categorical localization of the category of chain complexes in at the class of quasi-isomorphisms. The derived category is a fundamental example of a stable (infinity,1)-category. By the stable Dold-Kan correspondence, it may be viewed as a linearization of the stable (infinity,1)-category of spectra.
The derived (infinity,1)-category is presented by various dg-model structures on the category of chain complexes, as described at model structures on chain complexes. As such it has also a natural incarnation as a pretriangulated dg-category, which might be called the derived dg-category.
Like any stable (infinity,1)-category, the homotopy category of the derived (infinity,1)-category admits a canonical triangulated category structure. Often in the literature, the term derived category refers to the homotopy category, viewed only as a triangulated category. The loss of information can often be problematic, but for many purposes is not important.
Associated to is
The derived category of is equivalently
the (∞,1)-categorical homotopy category of .
In either case, this means that under the canonical localization functor
Hence the derived category is an approximation to the full simplicial localization of . It is or can be equipped with several further properties and structure that give a more accurate approximation. Notably every derived category is a triangulated category, which is a way of remembering the suspension and de-suspension operations on its objects – the suspension of chain complexes – hence its “stability”.
Derived categories were introduced by Jean-Louis Verdier in his thesis under the supervision of Alexandre Grothendieck. It was originally used to extend Serre duality to a relative context. See Hartshorne’s lecture notes “Residues and duality”.
The derived category is the homotopy category of with respect to these weak equivalences.
for the correspponding full subcategory of the derived category.
There are various ways to construct or express the derived category more explicitly in terms of various special objects or morphisms in the category of chain complexes.
The “homotopy category of chain complexes” is already a triangulated category. The derived category can be obtained as the construction of a homotopy category of a triangulated category with respect to a null system.
with the mapping cone .
The derived category is equivalently the localization of at the null system .
In the case that the underlying abelian category has enough injectives or enough projectives, the hom sets in the derived category may equivalently be obtained as homotopy-classes of chain maps from projective resolutions to injective resolutions of chain complexes.
In view of the existence of the injective and projective model structure on chain complexes this is a special case of the general fact that homotopy categories of model categories may be obtained by forming homotopy classes of maps in the model category from cofibrant resolutions to fibrant resolutions. But here we spell out an direct discussion of this fact for chain complexes.
If has enough injectives then the canonical functor
is an equivalence of categories.
Dually, if has enough projectives then the canonical functor
is an equivalence of categories.
For instance (Schapira, cor. 7.3.2).
The original reference is the thesis of Verdier:
and, similarly, in section 7 of
A pedagogical introduction is
A good survey of the more general topic of derived categories is
See in particular also the list of references given there.
Other lecture notes include
For the applications of derived categories in algebraic geometry, see
Aleksei Bondal, Dmitri Orlov, Reconstruction of a variety from the derived category and groups of autoequivalences, Compositio Mathematica 125 (03), 327-344. See also Bondal-Orlov reconstruction theorem.
Daniel Huybrechts, Fourier-Mukai Transforms in Algebraic Geometry, Oxford University Press, USA, 2006.