Derived category

In mathematics, the derived category D(C) of an abelian 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. The construction proceeds on the basis that the objects of D(C) should be chain complexes in C, 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 hyper-derived functors. 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 and became close to appearing as a universal approach to mathematics. The basic theory of Verdier was written down in his dissertation, published finally in 1996 in Astérisque (a summary much earlier appeared in SGA4½). The axiomatics required an innovation, the concept of triangulated category, and the construction is based on localization of a category, a generalisation 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.

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, 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.

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 then the simple roof definition of morphisms must be relaced by a more complicated one using finite strings of morphisms (technically, it is no longer a calculus of fractions), and the triangulated category structure of $D(\mathcal{A})$ arises in the homotopy category. So the one step construction is more efficient in a way but more complicated and the result is less powerful.

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 $A \rightarrow B \rightarrow \mathrm{Cone}(f) \rightarrow A[1]$ 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 $\mathcal A$.

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

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

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 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 a monomorphism 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 sheaves 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

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

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, 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 A of $\mathcal A$ to (any) injective resolution I ^* of A extends to a functor

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

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 epimorphism 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 (1988) uses so-called K-injective and K-projective resolutions, May (2006) and (in a slightly different language) Keller (1994) 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 (Keller 1996).

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 $F : \mathcal A \rightarrow \mathbf{Ab}$, $X \mapsto \mathrm{Hom}(X, A)$ or $X \mapsto \mathrm{Hom}(A, X)$ for some fixed object A, or the global sections functor on sheaves or the direct image functor. Their right derived functors are Extⁿ(–,A), Extⁿ(A,–), Hⁿ(X, F) or Rⁿf_∗ (F), respectively.

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: $D^+(\mathcal A) \stackrel{\cong}{\rightarrow} K^+(\mathrm{Inj}(\mathcal A)) \stackrel{F}{\rightarrow} K^+ (\mathcal B) \rightarrow D^+(\mathcal B)$, where the first equivalence of categories is described above. The classical derived functors are related to the total one via RⁿF(X) = Hⁿ(RF(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.

The derived categories is in a sense the "right" place to study these functors. For example, the Grothendieck spectral sequence of a composition of two functors

$\mathcal A \stackrel{F}{\rightarrow} \mathcal B \stackrel{G}{\rightarrow} \mathcal C, \,$

such that F maps injective objects in A to G-acyclics (i.e. RⁱG(F(I)) = 0 for all i > 0 and injective I), is an expression of the following identity of total derived functors

R(G○F) ≅ RG○RF.

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].

References

• Doorn, M.G.M. van (2001), "Derived category", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/D/d031280.htm 
• Keller, Bernhard (1996), "Derived categories and their uses", in Hazewinkel, M., Handbook of algebra, Amsterdam: North Holland, pp. 671–701, ISBN 0-444-82212-7, http://www.math.jussieu.fr/~keller/publ/dcu.ps 
• Keller, Bernhard (1994), "Deriving DG categories", Annales Scientifiques de l'École Normale Supérieure. Quatrième Série 27 (1): 63–102, ISSN 0012-9593, MR1258406, http://www.numdam.org/numdam-bin/fitem?id=ASENS_1994_4_27_1_63_0 
• May, J. P. (2006), Derived categories from a topological point of view, http://www.math.uchicago.edu/~may/MISC/DerivedCats.pdf 
• Spaltenstein, N. (1988), "Resolutions of unbounded complexes", Compositio Mathematica 65 (2): 121–154, ISSN 0010-437X, MR932640, http://www.numdam.org/numdam-bin/fitem?id=CM_1988__65_2_121_0 
• Verdier, Jean-Louis, "Des Catégories Dérivées des Catégories Abéliennes" (in French), Astérisque (Paris: Société Mathématique de France) 239, ISSN 0303-1179 

Three textbooks that discuss derived categories are:

• Manin, Yuri Ivanovich; Gelfand, Sergei I., Methods of Homological Algebra, Berlin, New York: Springer-Verlag, ISBN 978-3-540-43583-9 
• Schapira, Pierre; Kashiwara, Masaki, Categories and Sheaves, Grundlehren der mathematischen Wissenschaften, Berlin, New York: Springer-Verlag, ISBN 978-3-540-27949-5 
• Weibel, Charles A. (1994), An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, ISBN 978-0-521-55987-4, OCLC 36131259, MR1269324