Derived category

In mathematics, the derived category "D"("A") 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"("A") 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 (not completely faithfully) 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 (a summary had earlier appeared in SGA 4½). 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 formula_1 be an abelian category. (Examples include the category of modules over a ring and the category of sheaves of abelian groups on a topological space.) The derived category formula_2 is defined by a universal property with respect to the category formula_3 of cochain complexes with terms in formula_1. The objects of formula_3 are of the form
where each "X""i" is an object of formula_1 and each of the composites formula_8 is zero. The "i"th cohomology group of the complex is formula_9. If formula_10 and formula_11 are two objects in this category, then a morphism formula_12 is defined to be a family of morphisms formula_13 such that formula_14. Such a morphism induces morphisms on cohomology groups formula_15, and formula_16 is called a quasi-isomorphism if each of these morphisms is an isomorphism in formula_1.
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 formula_2 is a category, together with a functor formula_19, having the following universal property: Suppose formula_20 is another category (not necessarily abelian) and formula_21 is a functor such that, whenever formula_16 is a quasi-isomorphism in formula_3, its image formula_24 is an isomorphism in formula_20; then formula_26 factors through formula_27. Any two categories having this universal property are equivalent.
Relation to the homotopy category.
If formula_28 and formula_29 are two morphisms formula_30 in formula_3, then a chain homotopy or simply homotopy formula_32 is a collection of morphisms formula_33 such that formula_34 for every "i". It is straightforward to show that two homotopic morphisms induce identical morphisms on cohomology groups. We say that formula_35 is a chain homotopy equivalence if there exists formula_36 such that formula_37 and formula_38 are chain homotopic to the identity morphisms on formula_39 and formula_40, respectively. The homotopy category of cochain complexes formula_41 is the category with the same objects as formula_3 but whose morphisms are equivalence classes of morphisms of complexes with respect to the relation of chain homotopy equivalence. There is a natural functor formula_43 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, formula_27 factors through this functor. Consequently formula_2 can be equally well viewed as a localization of the homotopy category.
From the point of view of model categories, the derived category "D"("A") is the true 'homotopy category' of the category of complexes, whereas "K"("A") might be called the 'naive homotopy category'.
Constructing the derived category.
There are several possible constructions of the derived category. When formula_1 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 formula_1 is a large category, this construction does not work for set theoretic reasons. This construction builds morphisms as equivalence classes of paths. If formula_1 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 formula_1 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 formula_41 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 formula_30 in formula_2 may be described as a pair formula_53, where for some complex formula_54, formula_55 is a quasi-isomorphism and formula_56 is a chain homotopy equivalence class of morphisms. Conceptually, this represents formula_57. 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 formula_39 and consider the category formula_59 whose objects are quasi-isomorphisms in formula_41 with codomain formula_39 and whose morphisms are commutative diagrams. Equivalently, this is the category of objects over formula_39 whose structure maps are quasi-isomorphisms. Then the multiplicative system condition implies that the morphisms in formula_2 from formula_39 to formula_40 are
assuming that this colimit is in fact a set. While formula_59 is potentially a large category, in some cases it is controlled by a small category. This is the case, for example, if formula_1 is a Grothendieck abelian category (meaning that it satisfies AB5 and has a set of generators), 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 formula_2 may be defined to have these sets as its formula_70 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 formula_71 is "K"-injective if, for every acyclic complex formula_39, we have formula_73. A straightforward consequence of this is that, for every complex formula_39, morphisms formula_75 in formula_41 are the same as such morphisms in formula_2. 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 formula_78, where formula_79 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 formula_81 by truncating the complex above, shifting it, and using the obvious morphisms above. In particular, we have the picture
where the bottom complex has formula_83 concentrated in degree formula_84, the only non-trivial upward arrow is the equality morphism, and the only-nontrivial downward arrow is formula_85. 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 (see below) one uses "bounded-below" (formula_87 for formula_88), "bounded-above" (formula_87 for formula_90) or "bounded" (formula_87 for formula_92) complexes instead of unbounded ones. The corresponding derived categories are usually denoted "D+(A)", "D−(A)" and "Db(A)", respectively.
If one adopts the classical point of view on categories, that there is a set of morphisms from one object to another (not just a class), 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"("A") is equivalent to a full subcategory of the homotopy category "K"("A"), 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(A)" is a triangulated category, its localization "D(A)" is also triangulated. For an integer "n" and a complex "X", define the complex "X"["n"] to be "X" shifted down by "n", so that 
with differential
By definition, a distinguished triangle in "D(A)" is a triangle that is isomorphic in "D(A)" to the triangle "X" → "Y" → Cone("f") → "X"[1] for some map of complexes "f": "X" → "Y". Here Cone("f") denotes the mapping cone of "f". In particular, for a short exact sequence 
in "A", the triangle "X" → "Y" → "Z" → "X"[1] is distinguished in "D(A)". Verdier explained that the definition of the shift "X"[1] is forced by requiring "X"[1] 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(A)" 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 (dual) 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". (Neither the map nor the injective object has to be uniquely specified.) For example, every Grothendieck abelian category has enough injectives. Embedding "X" into some injective object "I"0, the cokernel of this map into some injective "I"1 etc., one constructs an "injective resolution" of "X", i.e. an exact (in general infinite) 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 (any) 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("X","Y") 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 (in a slightly different language) 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("X", "A") or "X" ↦ Hom("A", "X") for some fixed object "A", or the global sections functor on sheaves or the direct image functor. Their right derived functors are Ext"n"(–,"A"), Ext"n"("A",–), "H""n"("X", "F") or "R""n""f"∗ ("F"), respectively.
The derived category allows us to encapsulate all derived functors "RnF" in one functor, namely the so-called "total derived functor" "RF": "D"+("A") → "D"+("B"). It is the following composition: "D"+("A") ≅ "K"+(Inj("A")) → "K"+("B") → "D"+("B"), where the first equivalence of categories is described above. The classical derived functors are related to the total one via "RnF"("X") = "Hn"("RF"("X")). 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 (i.e. "R""i""G"("F"("I")) = 0 for all "i" > 0 and injective "I"), 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 [Mac Lane].
Derived equivalence.
It may happen that two abelian categories "A" and "B" are not equivalent, but their derived categories D("A") and D("B") 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.
References.
Four textbooks that discuss derived categories are: