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.

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 A be an abelian category. We obtain the derived category D in several steps:
The second step may be bypassed since a homotopy equivalence is in particular a quasi-isomorphism. But then the simple roof definition of morphisms must be replaced by a more complicated one using finite strings of morphisms. So the one-step construction is more efficient in a way, but more complicated.
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'.

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.

Remarks

For certain purposes one uses bounded-below, bounded-above or bounded complexes instead of unbounded ones. The corresponding derived categories are usually denoted D+, D− and Db, respectively.
If one adopts the classical point of view on categories, that there is a set of morphisms from one object to another, then one has to give an additional argument to prove this. If, for example, the abelian category A is small, i.e. has only a set of objects, then this issue will be no problem. Also, if A is a Grothendieck abelian category, then the derived category D is equivalent to a full subcategory of the homotopy category K, and hence has only a set of morphisms from one object to another. Grothendieck abelian categories include the category of modules over a ring, the category of sheaves of abelian groups on a topological space, and many other examples.
Composition of morphisms, i.e. roofs, in the derived category is accomplished by finding a third roof on top of the two roofs to be composed. It may be checked that this is possible and gives a well-defined, associative composition.
Since K is a triangulated category, its localization D is also triangulated. For an integer n and a complex X, define the complex X to be X shifted down by n, so that
with differential
By definition, a distinguished triangle in D is a triangle that is isomorphic in D to the triangle X → Y → Cone → X for some map of complexes f: X → Y. Here Cone denotes the mapping cone of f. In particular, for a short exact sequence
in A, the triangle X → Y → Z → X is distinguished in D. Verdier explained that the definition of the shift X is forced by requiring X to be the cone of the morphism X → 0.
By viewing an object of A as a complex concentrated in degree zero, the derived category D contains A as a full subcategory. Morphisms in the derived category include information about all Ext groups: for any objects X and Y in A and any integer j,

Projective and injective resolutions

One can easily show that a homotopy equivalence is a quasi-isomorphism, so the second step in the above construction may be omitted. The definition is usually given in this way because it reveals the existence of a canonical functor
In concrete situations, it is very difficult or impossible to handle morphisms in the derived category directly. Therefore, one looks for a more manageable category which is equivalent to the derived category. Classically, there are two approaches to this: projective and injective resolutions. In both cases, the restriction of the above canonical functor to an appropriate subcategory will be an equivalence of categories.
In the following we will describe the role of injective resolutions in the context of the derived category, which is the basis for defining right derived functors, which in turn have important applications in cohomology of sheaves on topological spaces or more advanced cohomology theories like étale cohomology or group cohomology.
In order to apply this technique, one has to assume that the abelian category in question has enough injectives, which means that every object X of the category admits a monomorphism to an injective object I. For example, every Grothendieck abelian category has enough injectives. Embedding X into some injective object I0, the cokernel of this map into some injective I1 etc., one constructs an injective resolution of X, i.e. an exact sequence
where the I* are injective objects. This idea generalizes to give resolutions of bounded-below complexes X, i.e. Xn = 0 for sufficiently small n. As remarked above, injective resolutions are not uniquely defined, but it is a fact that any two resolutions are homotopy equivalent to each other, i.e. isomorphic in the homotopy category. Moreover, morphisms of complexes extend uniquely to a morphism of two given injective resolutions.
This is the point where the homotopy category comes into play again: mapping an object X of A to injective resolution I* of A extends to a functor
from the bounded below derived category to the bounded below homotopy category of complexes whose terms are injective objects in A.
It is not difficult to see that this functor is actually inverse to the restriction of the canonical localization functor mentioned in the beginning. In other words, morphisms Hom in the derived category may be computed by resolving both X and Y and computing the morphisms in the homotopy category, which is at least theoretically easier. In fact, it is enough to resolve Y: for any complex X and any bounded below complex Y of injectives,
Dually, assuming that A has enough projectives, i.e. for every object X there is an epimorphism from a projective object P to X, one can use projective resolutions instead of injective ones.
In addition to these resolution techniques there are similar ones which apply to special cases, and which elegantly avoid the problem with bounded-above or -below restrictions: uses so-called K-injective and K-projective resolutions, and introduced so called cell-modules and semi-free modules, respectively.
More generally, carefully adapting the definitions, it is possible to define the derived category of an exact category.

The relation to derived functors

The derived category is a natural framework to define and study derived functors. In the following, let F: A → B be a functor of abelian categories. There are two dual concepts:
In the following we will describe right derived functors. So, assume that F is left exact. Typical examples are F: A → Ab given by X ↦ Hom or X ↦ Hom for some fixed object A, or the global sections functor on sheaves or the direct image functor. Their right derived functors are Extn, Extn, Hn or Rnf∗, respectively.
The derived category allows us to encapsulate all derived functors RnF in one functor, namely the so-called total derived functor RF: D+ → D+. It is the following composition: D+ ≅ K+ → K+ → D+, where the first equivalence of categories is described above. The classical derived functors are related to the total one via RnF = Hn. One might say that the RnF forget the chain complex and keep only the cohomologies, whereas RF does keep track of the complexes.
Derived categories are, in a sense, the "right" place to study these functors. For example, the Grothendieck spectral sequence of a composition of two functors
such that F maps injective objects in A to G-acyclics, is an expression of the following identity of total derived functors
J.-L. Verdier showed how derived functors associated with an abelian category A can be viewed as Kan extensions along embeddings of A into suitable derived categories .

Derived equivalence

It may happen that two abelian categories A and B are not equivalent, but their derived categories D and D are. Often this is an interesting relation between A and B. Such equivalences are related to the theory of t-structures in triangulated categories. Here are some examples.