Derived category


In mathematics, the derived category D of an abelian category A is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on A. The construction proceeds on the basis that the objects of D should be chain complexes in A, with two such chain complexes considered isomorphic when there is a chain map that induces an isomorphism on the level of homology of the chain complexes. Derived functors can then be defined for chain complexes, refining the concept of hypercohomology. The definitions lead to a significant simplification of formulas otherwise described by complicated spectral sequences.
The development of the derived category, by Alexander Grothendieck and his student Jean-Louis Verdier shortly after 1960, now appears as one terminal point in the explosive development of homological algebra in the 1950s, a decade in which it had made remarkable strides. The basic theory of Verdier was written down in his dissertation, published finally in 1996 in [|Astérisque]. The axiomatics required an innovation, the concept of triangulated category, and the construction is based on localization of a category, a generalization of localization of a ring. The original impulse to develop the "derived" formalism came from the need to find a suitable formulation of Grothendieck's coherent duality theory. Derived categories have since become indispensable also outside of algebraic geometry, for example in the formulation of the theory of D-modules and microlocal analysis. Recently derived categories have also become important in areas nearer to physics, such as D-branes and mirror symmetry.
Unbounded derived categories were introduced by Spaltenstein in 1988.

Motivations

In coherent sheaf theory, pushing to the limit of what could be done with Serre duality without the assumption of a non-singular scheme, the need to take a whole complex of sheaves in place of a single dualizing sheaf became apparent. In fact the Cohen–Macaulay ring condition, a weakening of non-singularity, corresponds to the existence of a single dualizing sheaf; and this is far from the general case. From the top-down intellectual position, always assumed by Grothendieck, this signified a need to reformulate. With it came the idea that the 'real' tensor product and Hom functors would be those existing on the derived level; with respect to those, Tor and Ext become more like computational devices.
Despite the level of abstraction, derived categories became accepted over the following decades, especially as a convenient setting for sheaf cohomology. Perhaps the biggest advance was the formulation of the Riemann–Hilbert correspondence in dimensions greater than 1 in derived terms, around 1980. The Sato school adopted the language of derived categories, and the subsequent history of D-modules was of a theory expressed in those terms.
A parallel development was the category of spectra in homotopy theory. The homotopy category of spectra and the derived category of a ring are both examples of triangulated categories.

Definition

Let be an abelian category. The derived category is defined by a universal property with respect to the category of cochain complexes with terms in. The objects of are of the form
where each Xi is an object of and each of the composites is zero. The ith cohomology group of the complex is. If and are two objects in this category, then a morphism is defined to be a family of morphisms such that. Such a morphism induces morphisms on cohomology groups, and is called a quasi-isomorphism if each of these morphisms is an isomorphism in.
The universal property of the derived category is that it is a localization of the category of complexes with respect to quasi-isomorphisms. Specifically, the derived category is a category, together with a functor, having the following universal property: Suppose is another category and is a functor such that, whenever is a quasi-isomorphism in, its image is an isomorphism in ; then factors through. Any two categories having this universal property are equivalent.

Relation to the homotopy category

If and are two morphisms in, then a chain homotopy or simply homotopy is a collection of morphisms such that for every i. It is straightforward to show that two homotopic morphisms induce identical morphisms on cohomology groups. We say that is a chain homotopy equivalence if there exists such that and are chain homotopic to the identity morphisms on and, respectively. The homotopy category of cochain complexes is the category with the same objects as but whose morphisms are equivalence classes of morphisms of complexes with respect to the relation of chain homotopy. There is a natural functor which is the identity on objects and which sends each morphism to its chain homotopy equivalence class. Since every chain homotopy equivalence is a quasi-isomorphism, factors through this functor. Consequently can be equally well viewed as a localization of the homotopy category.
From the point of view of model categories, the derived category D is the true 'homotopy category' of the category of complexes, whereas K might be called the 'naive homotopy category'.

Constructing the derived category

There are several possible constructions of the derived category. When is a small category, then there is a direct construction of the derived category by formally adjoining inverses of quasi-isomorphisms. This is an instance of the general construction of a category by generators and relations.
When is a large category, this construction does not work for set theoretic reasons. This construction builds morphisms as equivalence classes of paths. If has a proper class of objects, all of which are isomorphic, then there is a proper class of paths between any two of these objects. The generators and relations construction therefore only guarantees that the morphisms between two objects form a proper class. However, the morphisms between two objects in a category are usually required to be sets, and so this construction fails to produce an actual category.
Even when is small, however, the construction by generators and relations generally results in a category whose structure is opaque, where morphisms are arbitrarily long paths subject to a mysterious equivalence relation. For this reason, it is conventional to construct the derived category more concretely even when set theory is not at issue.
These other constructions go through the homotopy category. The collection of quasi-isomorphisms in forms a multiplicative system. This is a collection of conditions that allow complicated paths to be rewritten as simpler ones. The Gabriel–Zisman theorem implies that localization at a multiplicative system has a simple description in terms of roofs. A morphism in may be described as a pair, where for some complex, is a quasi-isomorphism and is a chain homotopy equivalence class of morphisms. Conceptually, this represents. Two roofs are equivalent if they have a common overroof.
Replacing chains of morphisms with roofs also enables the resolution of the set-theoretic issues involved in derived categories of large categories. Fix a complex and consider the category whose objects are quasi-isomorphisms in with codomain and whose morphisms are commutative diagrams. Equivalently, this is the category of objects over whose structure maps are quasi-isomorphisms. Then the multiplicative system condition implies that the morphisms in from to are
assuming that this colimit is in fact a set. While is potentially a large category, in some cases it is controlled by a small category. This is the case, for example, if is a Grothendieck abelian category, with the essential point being that only objects of bounded cardinality are relevant. In these cases, the limit may be calculated over a small subcategory, and this ensures that the result is a set. Then may be defined to have these sets as its sets.
There is a different approach based on replacing morphisms in the derived category by morphisms in the homotopy category. A morphism in the derived category with codomain being a bounded below complex of injective objects is the same as a morphism to this complex in the homotopy category; this follows from termwise injectivity. By replacing termwise injectivity by a stronger condition, one gets a similar property that applies even to unbounded complexes. A complex is K-injective if, for every acyclic complex, we have. A straightforward consequence of this is that, for every complex, morphisms in are the same as such morphisms in. A theorem of Serpé, generalizing work of Grothendieck and of Spaltenstein, asserts that in a Grothendieck abelian category, every complex is quasi-isomorphic to a K-injective complex with injective terms, and moreover, this is functorial. In particular, we may define morphisms in the derived category by passing to K-injective resolutions and computing morphisms in the homotopy category. The functoriality of Serpé's construction ensures that composition of morphisms is well-defined. Like the construction using roofs, this construction also ensures suitable set theoretic properties for the derived category, this time because these properties are already satisfied by the homotopy category.

Derived Hom-sets

As noted before, in the derived category the hom sets are expressed through roofs, or valleys, where is a quasi-isomorphism. To get a better picture of what elements look like, consider an exact sequence
We can use this to construct a morphism by truncating the complex above, shifting it, and using the obvious morphisms above. In particular, we have the picture
where the bottom complex has concentrated in degree, the only non-trivial upward arrow is the equality morphism, and the only-nontrivial downward arrow is. This diagram of complexes defines a morphism
in the derived category. One application of this observation is the construction of the Atiyah-class.