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 much 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 A be an abelian category. (Some basic examples are the category of modules over a ring, or the category of sheaves of abelian groups on a topological space.) We obtain the derived category D(A) in several steps:

The basic object is the category Kom(A) of chain complexes

\cdots \to X^{-1} \xrightarrow{d^{-1}} X^0 \xrightarrow{d^0} X^1 \xrightarrow{d^1} X^2 \to \cdots

in 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(A) by identifying morphisms which are chain homotopic.
Pass to the derived category D(A) by localizing at the set of quasi-isomorphisms. Morphisms in the derived category may be explicitly described as roofs X ← X' → Y, where X' → X is a quasi-isomorphism and X' → Y 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 replaced by a more complicated one using finite strings of morphisms (technically, it is no longer a calculus of fractions). 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(A) is the true 'homotopy category' of the category of complexes, whereas K(A) might be called the 'naive homotopy category'.

Remarks

For certain purposes (see below) one uses bounded-below (Xⁿ = 0 for n << 0), bounded-above (Xⁿ = 0 for n >> 0) or bounded (Xⁿ = 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 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.^[1] 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^[2] the complex X[n] to be X shifted down by n, so that

X[n]^{i} = X^{n+i},

with differential

d_{X[n]} = (-1)^n d_X.

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

0 \rightarrow X \rightarrow Y \rightarrow Z \rightarrow 0

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.^[3]

By viewing an object of A as a complex concentrated in degree zero, the derived category D(A) contains A as a full subcategory. More interestingly, morphisms in the derived category include information about all Ext groups: for any objects X and Y in A and any integer j,

\text{Hom}_{D(\mathcal{A})}(X,Y[j]) = \text{Ext}^j_{\mathcal{A}}(X,Y).

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 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 I⁰, the cokernel of this map into some injective I¹ etc., one constructs an injective resolution of X, i.e. an exact (in general infinite) sequence

0 \rightarrow X \rightarrow I^0 \rightarrow I^1 \rightarrow \cdots, \,

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

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 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,

\mathrm{Hom}_{D(A)}(X, Y) = \mathrm{Hom}_{K(A)}(X, Y).

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: 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: A → B be a functor of abelian categories. There are two dual concepts:

right derived functors come from 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: 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 Extⁿ(–,A), Extⁿ(A,–), Hⁿ(X, F) or Rⁿf_∗ (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

\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].

Derived Equivalence

The bounded derived categories are equivalent as triangulated categories.

Examples include

Orlov's theorem that the bounded derived categories of coherent sheaves on two abelian varieties A and B such that there exists an isomorphism : $A \times \hat{A} \to B \times \hat{B}$
Homological mirror symmetry

Notes

↑ M. Kashiwara and P. Schapira. Categories and Sheaves. Springer-Verlag (2006). Theorem 14.3.1.
↑ S. Gelfand and Y. Manin. Methods of Homological Algebra. Springer-Verlag (2003). III.3.2.
↑ J.-L. Verdier. Astérisque 239. Soc. Math. de France (1996). Appendice to Ch. 1.

References

Doorn, M.G.M. van (2001), "Derived category", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
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, MR 1421815
Keller, Bernhard (1994), "Deriving DG categories", Annales Scientifiques de l'École Normale Supérieure. Quatrième Série 27 (1): 63–102, ISSN 0012-9593, MR 1258406
May, J. P. (2006), Derived categories from a topological point of view (PDF)
Spaltenstein, N. (1988), "Resolutions of unbounded complexes", Compositio Mathematica 65 (2): 121–154, ISSN 0010-437X, MR 932640
Verdier, Jean-Louis (1996), "Des Catégories Dérivées des Catégories Abéliennes", Astérisque (in French) (Paris: Société Mathématique de France) 239, ISSN 0303-1179, MR 1453167

Three textbooks that discuss derived categories are:

Gelfand, Sergei I.; Manin, Yuri Ivanovich (2003), Methods of Homological Algebra, Berlin, New York: Springer-Verlag, ISBN 978-3-540-43583-9, MR 1950475
Kashiwara, Masaki; Schapira, Pierre (2006), Categories and Sheaves, Grundlehren der mathematischen Wissenschaften, Berlin, New York: Springer-Verlag, ISBN 978-3-540-27949-5, MR 2182076
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, MR 1269324

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