Derived category

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.

Let ${\displaystyle {\mathcal {A}}}$ 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 ${\displaystyle D({\mathcal {A}})}$ is defined by a universal property with respect to the category ${\displaystyle \operatorname {Kom} ({\mathcal {A}})}$ of cochain complexes with terms in ${\displaystyle {\mathcal {A}}}$. The objects of ${\displaystyle \operatorname {Kom} ({\mathcal {A}})}$ are of the form

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

where each Xⁱ is an object of ${\displaystyle {\mathcal {A}}}$ and each of the composites ${\displaystyle d^{i+1}\circ d^{i}}$ is zero. The ith cohomology group of the complex is ${\displaystyle H^{i}(X^{\bullet })=\operatorname {ker} d^{i}/\operatorname {im} d^{i-1}}$. If ${\displaystyle (X^{\bullet },d_{X}^{\bullet })}$ and ${\displaystyle (Y^{\bullet },d_{Y}^{\bullet })}$ are two objects in this category, then a morphism ${\displaystyle f^{\bullet }\colon (X^{\bullet },d_{X}^{\bullet })\to (Y^{\bullet },d_{Y}^{\bullet })}$ is defined to be a family of morphisms ${\displaystyle f_{i}\colon X^{i}\to Y^{i}}$ such that ${\displaystyle f_{i+1}\circ d_{X}^{i}=d_{Y}^{i}\circ f_{i}}$. Such a morphism induces morphisms on cohomology groups ${\displaystyle H^{i}(f^{\bullet })\colon H^{i}(X^{\bullet })\to H^{i}(Y^{\bullet })}$, and ${\displaystyle f^{\bullet }}$ is called a quasi-isomorphism if each of these morphisms is an isomorphism in ${\displaystyle {\mathcal {A}}}$.

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 ${\displaystyle D({\mathcal {A}})}$ is a category, together with a functor ${\displaystyle Q\colon \operatorname {Kom} ({\mathcal {A}})\to D({\mathcal {A}})}$, having the following universal property: Suppose ${\displaystyle {\mathcal {C}}}$ is another category (not necessarily abelian) and ${\displaystyle F\colon \operatorname {Kom} ({\mathcal {A}})\to {\mathcal {C}}}$ is a functor such that, whenever ${\displaystyle f^{\bullet }}$ is a quasi-isomorphism in ${\displaystyle \operatorname {Kom} ({\mathcal {A}})}$, its image ${\displaystyle F(f^{\bullet })}$ is an isomorphism in ${\displaystyle {\mathcal {C}}}$; then ${\displaystyle F}$ factors through ${\displaystyle Q}$. Any two categories having this universal property are equivalent.

As noted before, in the derived category the hom sets are expressed through roofs, or valleys ${\displaystyle X\rightarrow Y'\leftarrow Y}$, where ${\displaystyle Y\to Y'}$ is a quasi-isomorphism. To get a better picture of what elements look like, consider an exact sequence

${\displaystyle 0\to {\mathcal {E}}_{n}{\overset {\phi _{n,n-1}}{\rightarrow }}{\mathcal {E}}_{n-1}{\overset {\phi _{n-1,n-2}}{\rightarrow }}\cdots {\overset {\phi _{1,0}}{\rightarrow }}{\mathcal {E}}_{0}\to 0}$

We can use this to construct a morphism ${\displaystyle \phi$ :{\mathcal {E}}_{0}\to {\mathcal {E}}_{n}[+(n-1)]} by truncating the complex above, shifting it, and using the obvious morphisms above. In particular, we have the picture

${\displaystyle {\begin{matrix}0&\to &{\mathcal {E}}_{n}&\to &0&\to &\cdots &\to &0&\to &0\\\uparrow &&\uparrow &&\uparrow &&\cdots &&\uparrow &&\uparrow \\0&\to &{\mathcal {E}}_{n}&\to &{\mathcal {E}}_{n-1}&\to &\cdots &\to &{\mathcal {E}}_{1}&\to &0\\\downarrow &&\downarrow &&\downarrow &&\cdots &&\downarrow &&\downarrow \\0&\to &0&\to &0&\to &\cdots &\to &{\mathcal {E}}_{0}&\to &0\end{matrix}}}$

where the bottom complex has ${\displaystyle {\mathcal {E}}_{0}}$ concentrated in degree ${\displaystyle 0}$, the only non-trivial upward arrow is the equality morphism, and the only-nontrivial downward arrow is ${\displaystyle \phi _{1,0}:{\mathcal {E}}_{1}\to {\mathcal {E}}_{0}}$. This diagram of complexes defines a morphism

${\displaystyle \phi \in \mathbf {RHom} ({\mathcal {E}}_{0},{\mathcal {E}}_{n}[+(n-1)])}$

in the derived category. One application of this observation is the construction of the Atiyah-class.[5]

For certain purposes (see below) one uses bounded-below (${\displaystyle X^{n}=0}$ for ${\displaystyle n\ll 0}$), bounded-above (${\displaystyle X^{n}=0}$ for ${\displaystyle n\gg 0}$) or bounded (${\displaystyle X^{n}=0}$ for ${\displaystyle |n|\gg 0}$) 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.[6] 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[7] the complex X[n] to be X shifted down by n, so that

${\displaystyle X[n]^{i}=X^{n+i},}$

with differential

${\displaystyle d_{X[n]}=(-1)^{n}d_{X}.}$

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

${\displaystyle 0\rightarrow X\rightarrow Y\rightarrow Z\rightarrow 0}$

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.[8]

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,

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

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

${\displaystyle K({\mathcal {A}})\rightarrow D({\mathcal {A}}).}$

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

${\displaystyle 0\rightarrow X\rightarrow I^{0}\rightarrow I^{1}\rightarrow \cdots ,\,}$

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

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

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,

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

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 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

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

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].

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.[9]

• Let ${\displaystyle \mathrm {Coh} (\mathbb {P} ^{1})}$ be an abelian category of coherent sheaves on the projective line over a field k. Let K₂-Rep be an abelian category of representations of the Kronecker quiver with two vertices. They are very different abelian categories, but their (bounded) derived categories are equivalent.
• Let Q be any quiver and P be a quiver obtained from Q by reversing some arrows. In general, the categories of representations of Q and P are different, but D^b(Q-Rep) is always equivalent to D^b(P-Rep).
• Let X be an abelian variety, Y its dual abelian variety. Then D^b(Coh(X)) is equivalent to D^b(Coh(Y)) by the theory of Fourier–Mukai transforms. Varieties with equivalent derived categories of coherent sheaves are sometimes called Fourier–Mukai partners.

• Homotopy category of chain complexes
• Derived noncommutative algebraic geometry
• Coherent sheaf cohomology
• Coherent duality
• Derived algebraic geometry

• Doorn, M.G.M. van (2001) [1994], "Derived category", Encyclopedia of Mathematics, EMS Press
• Keller, Bernhard (1996), "Derived categories and their uses", in Hazewinkel, M. (ed.), 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, Série 4, 27 (1): 63–102, doi:10.24033/asens.1689, 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 0932640
• Verdier, Jean-Louis (1996), "Des Catégories Dérivées des Catégories Abéliennes", Astérisque (in French), Paris: Société Mathématique de France, 239, ISSN 0303-1179, MR 1453167

Four 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. Vol. 38. Cambridge University Press. ISBN 978-0-521-55987-4. MR 1269324. OCLC 36131259.
• Yekutieli, Amnon (2019). Derived Categories. Cambridge Studies in Advanced Mathematics. Vol. 183. Cambridge University Press. ISBN 978-1108419338.