Derived category

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In mathematics, the derived category D(C) of a category C is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on C (which therefore should already be an abelian category). The construction proceeds on the basis that the objects of D(C) should be chain complexes from C, identified in a certain way that in a sense absorbs the usual long exact sequences, provided by the snake lemma. There are in fact a few versions, depending on conditions bounding the chain complexes in various ways.

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 and became close to appearing a universal approach in mathematics. The basic theory of Verdier was written down in his dissertation, never to be published (a summary much later appeared in SGA4½). The axiomatics required an innovation, the concept of triangulated category, as well as localization of a category, and at least one notorious axiom (octahedral axiom). Such was the style of abstraction of the time. In fact there was a pressing concern, to get a neat formulation of coherent duality, that explains how the 'derived' way of thinking was ever launched. (It has later been hailed, for example by Manin, as a rectification of the deficiencies of the established Cartan-Eilenberg method of accepting derived functors such as the Tor functors and Ext functors as natural.)

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 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, the derived category methodology established itself over the following decades; and perhaps began to impose itself with the formulation of the Riemann-Hilbert correspondence in dimensions greater than 1 in derived terms, around 1980. The Sato school adopted it, and the subsequent history of D-modules was of a theory expressed in those terms.

A parallel development, speaking in fact the same language, was that of spectrum in homotopy theory. This was at the space level, rather than in the algebra.

1 Definition
2 Remarks
3 Projective and injective resolutions
4 The relation to derived functors
5 References
6 Notes

[edit] Definition

Let $\mathcal A$ be an abelian category. We obtain the derived category $D(\mathcal A)$ in several steps:

The basic object is the category $\operatorname{Kom}(\mathcal{A})$ of chain complexes in $\mathcal{A}$ . Its objects will be the objects of the derived category but its morphisms will be altered.
Pass to the homotopy category of chain complexes $K(\mathcal{A})$ by identifying morphisms which are chain homotopic.
Pass to the derived category $D(\mathcal{A})$ by localizing at the set of quasi-isomorphisms. Morphisms in the derived category may be explicitly described as roofs $A \stackrel{s}{\leftarrow} A' \stackrel{f}{\rightarrow} B$ , where s is a quasi-isomorphism and f is any morphism of chain complexes.

The second step may be bypassed since a homotopy equivalence is in particular a quasi-isomorphism, but the triangulated category structure of $D(\mathcal{A})$ arises in the homotopy category, so although the definition is more efficient the result is less powerful.

[edit] Remarks

For certain purposes (see below) one uses bounded-below (Aⁿ=0 for n<<0), bounded-above (Aⁿ=0 for n>>0) or bounded (Aⁿ=0 for |n|>>0) complexes instead of unbounded ones. The corresponding derived categories are usually denoted D⁺(A), D^-(A) and D^b(A), respectively.

If one adopts the classical point of view on categories, that morphisms have to be sets (not just classes), then one has to give an additional argument, why this is true. If, for example, the abelian category $\mathcal A$ is small, i.e. has only a set of objects, then this issue will be no problem.

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.

As the localization of K(A) (which is a triangulated category), the derived category is triangulated as well. Distinguished triangles are those quasi-isomorphic to triangles of the form $A \rightarrow Cone(f) \rightarrow B \rightarrow A[1]$ for two complexes A and B and a map f between them. This includes in particular triangles of the form $A \rightarrow B \rightarrow C \rightarrow A[1]$ for a short exact sequence

$0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$

in $\mathcal A$ .

[edit] Projective and injective resolutions

One can easily show, that a chain homotopy 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

$K(\mathcal A) \rightarrow D(\mathcal A)$ .

As can be read of the definition, 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 sheaf on topological spaces or more advanced cohomologies 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 A of the category admits an injective map to an injective object I. (Neither the map nor the injective object has to be uniquely specified). This assumption is often satisfied, for example it is true for the abelian category of R-modules over a fixed ring R or for sheaf of abelian groups on a topological space. Embedding A into some injective object I⁰, the cokernel of this map into some injective I¹ etc., one constructs an injective resolution of A, i.e. an exact (in general infinite) complex

$0 \rightarrow A \rightarrow I^0 \rightarrow I^1 \rightarrow ...$ ,

where the I^* are injective objects. This idea generalizes to give resolutions of bounded-below complexes A, i.e. Aⁿ = 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 morphism 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 A of $\mathcal A$ to (any) injective resolution $I *$ of A defines a functor

$D^+(\mathcal A) \rightarrow K^+(Inj(\mathcal A))$

from the bounded below derived category to the bounded below homotopy category of complexes whose terms are injective objects in $\mathcal A$ .

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(A,B) in the derived category may be computed by resolving both A and B and computing the morphisms in the homotopy category, which is at least theoretically easier.

Dually, assuming that $\mathcal A$ has enough projectives, i.e. for every object A there is a surjective map from a projective object P to A, 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: Spaltenstein^[1] uses so-called K-injective and K-projective resolutions, May^[2] and (in a slightly different language) Keller ^[3] introduced so called cell-modules and semi-free modules, respectively.

[edit] The relation to derived functors

The derived category is a natural framework to define and study derived functors. In the following, let $F : \mathcal A \rightarrow \mathcal B$ be a functor of abelian categories. There are two dual concepts:

right derived functors are "deriving" left exact functors and are calculated via injective resolutions
left derived functors come from right exact functors and are calculated via projective resolutions

In the following we will describe right derived functors. So, assume that F is left exact. Typical examples are $F : \mathcal A \rightarrow \mathbf{Ab}$ , $X \mapsto Hom(X, A)$ or $X \mapsto Hom(A, X)$ for some fixed object A. Their right derived functors are Extⁿ(-,A) and Extⁿ(A,-).

The derived category allows to encapsulate all derived functors RⁿF in one functor, namely the so-called total derived functor $RF: D^-(\mathcal A) \rightarrow D^-(\mathcal B)$ . It is the following composition: $D^-(\mathcal A) \stackrel{\cong}{\rightarrow} K^-(Inj(\mathcal A)) \stackrel{F}{\rightarrow} K^- (\mathcal B) \rightarrow D^-(\mathcal B)$ , where the first equivalence of categories is described above. The classical derived functors are related to the total one via $R n F (X) = H n (R F (X))$ . One might say that the RⁿF forget the chain complex and keep only the cohomologies, whereas R F does keep track of the complexes.

[edit] References

Three textbooks that discuss derived categories are:

Categories And Sheaves (Grundlehren Der Mathematischen Wissenschaften) by Masaki Kashiwara , Pierre Schapira ISBN 3-540-27949-0
Methods of Homological Algebra by Sergei I. Gelfand, Yuri I. Manin ISBN 3-540-43583-2
An introduction to homological algebra by Charles A. Weibel, ISBN 0-521-55987-1

[edit] Notes

^ Spaltenstein, N. Resolutions of unbounded complexes. Compositio Math. 65 (1988)
^ J. P. May, Derived categories from a topological point of view
^ Keller, Deriving DG-categories, Ann. Scient. Ecole Normale Sup. (4) 27 (1994)

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Category: Homological algebra