In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the derived category ''D''(''A'') of an
abelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ab ...
''A'' is a construction of
homological algebra
Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...
introduced to refine and in a certain sense to simplify the theory of
derived functor
In mathematics, certain functors may be ''derived'' to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics.
Motivation
It was noted in vari ...
s defined on ''A''. The construction proceeds on the basis that the
objects
Object may refer to:
General meanings
* Object (philosophy), a thing, being, or concept
** Object (abstract), an object which does not exist at any particular time or place
** Physical object, an identifiable collection of matter
* Goal, an ...
of ''D''(''A'') should be
chain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or module (mathematics), modules) and a sequence of group homomorphism, homomorphisms between consecutive groups such that the image (mathemati ...
es in ''A'', with two such chain complexes considered
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
when there is a
chain map
A chain is a serial assembly of connected pieces, called links, typically made of metal, with an overall character similar to that of a rope in that it is flexible and curved in compression but linear, rigid, and load-bearing in tension. A ...
that induces an isomorphism on the level of
homology
Homology may refer to:
Sciences
Biology
*Homology (biology), any characteristic of biological organisms that is derived from a common ancestor
* Sequence homology, biological homology between DNA, RNA, or protein sequences
*Homologous chrom ...
of the chain complexes. Derived functors can then be defined for chain complexes, refining the concept of
hypercohomology
In homological algebra, the hyperhomology or hypercohomology (\mathbb_*(-), \mathbb^*(-)) is a generalization of (co)homology functors which takes as input not objects in an abelian category \mathcal but instead chain complexes of objects, so objec ...
. The definitions lead to a significant simplification of formulas otherwise described (not completely faithfully) by complicated
spectral sequence
In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by , they hav ...
s.
The development of the derived category, by
Alexander Grothendieck and his student
Jean-Louis Verdier
Jean-Louis Verdier (; 2 February 1935 – 25 August 1989) was a French mathematician who worked, under the guidance of his doctoral advisor Alexander Grothendieck, on derived categories and Verdier duality. He was a close collaborator of Grothe ...
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
'' Astérisque'' is a mathematical journal published by Société Mathématique de France
Lactalis is a French multinational dairy products corporation, owned by the Besnier family and based in Laval, Mayenne, France. The company's former na ...
(a summary had earlier appeared in
SGA 4½). The axiomatics required an innovation, the concept of
triangulated category In mathematics, a triangulated category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category of an abelian category, as well as the stable homotopy cat ...
, and the construction is based on
localization of a category In mathematics, localization of a category consists of adding to a category inverse morphisms for some collection of morphisms, constraining them to become isomorphisms. This is formally similar to the process of localization of a ring; it in gene ...
, a generalization of
localization of a ring
In commutative algebra and algebraic geometry, localization is a formal way to introduce the "denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing ring/module ''R'', so that it consists of fractions \ ...
. The original impulse to develop the "derived" formalism came from the need to find a suitable formulation of Grothendieck's
coherent duality In mathematics, coherent duality is any of a number of generalisations of Serre duality, applying to coherent sheaves, in algebraic geometry and complex manifold theory, as well as some aspects of commutative algebra that are part of the 'local' the ...
theory. Derived categories have since become indispensable also outside of
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, for example in the formulation of the theory of
D-module
In mathematics, a ''D''-module is a module (mathematics), module over a ring (mathematics), ring ''D'' of differential operators. The major interest of such ''D''-modules is as an approach to the theory of linear partial differential equations. Sin ...
s and
microlocal analysis
In mathematical analysis, microlocal analysis comprises techniques developed from the 1950s onwards based on Fourier transforms related to the study of variable-coefficients-linear and nonlinear partial differential equations. This includes genera ...
. Recently derived categories have also become important in areas nearer to physics, such as
D-brane
In string theory, D-branes, short for ''Dirichlet membrane'', are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named. D-branes were discovered by Jin Dai, Leigh, and Polchi ...
s and
mirror symmetry
In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry.
In 2D ther ...
.
Motivations
In
coherent sheaf
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refe ...
theory, pushing to the limit of what could be done with
Serre duality In algebraic geometry, a branch of mathematics, Serre duality is a duality for the coherent sheaf cohomology of algebraic varieties, proved by Jean-Pierre Serre. The basic version applies to vector bundles on a smooth projective variety, but Al ...
without the assumption of a
non-singular
In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In cas ...
scheme A scheme is a systematic plan for the implementation of a certain idea.
Scheme or schemer may refer to:
Arts and entertainment
* ''The Scheme'' (TV series), a BBC Scotland documentary series
* The Scheme (band), an English pop band
* ''The Schem ...
, the need to take a whole complex of sheaves in place of a single ''dualizing sheaf'' became apparent. In fact the
Cohen–Macaulay ring
In mathematics, a Cohen–Macaulay ring is a commutative ring with some of the algebro-geometric properties of a smooth variety, such as local equidimensionality. Under mild assumptions, a local ring is Cohen–Macaulay exactly when it is a fini ...
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
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...
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 In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally when i ...
. Perhaps the biggest advance was the formulation of the
Riemann–Hilbert correspondence In mathematics, the term Riemann–Hilbert correspondence refers to the correspondence between regular singular flat connections on algebraic vector bundles and representations of the fundamental group, and more generally to one of several generaliz ...
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-module
In mathematics, a ''D''-module is a module (mathematics), module over a ring (mathematics), ring ''D'' of differential operators. The major interest of such ''D''-modules is as an approach to the theory of linear partial differential equations. Sin ...
s was of a theory expressed in those terms.
A parallel development was the category of
spectra in
homotopy theory
In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topolog ...
. The homotopy category of spectra and the derived category of a ring are both examples of
triangulated categories In mathematics, a triangulated category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category of an abelian category, as well as the stable homotopy cate ...
.
Definition
Let
be an
abelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ab ...
. (Examples include the category of
modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...
over a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
and the category of
sheaves of abelian groups on a topological space.) The derived category
is defined by a universal property with respect to the category
of
cochain complexes with terms in
. The objects of
are of the form
:
where each ''X''
''i'' is an object of
and each of the composites
is zero. The ''i''th cohomology group of the complex is
. If
and
are two objects in this category, then a morphism
is defined to be a family of morphisms
such that
. Such a morphism induces morphisms on cohomology groups
, and
is called a quasi-isomorphism if each of these morphisms is an isomorphism in
.
The universal property of the derived category is that it is a
localization
Localization or localisation may refer to:
Biology
* Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence
* Localization of sensation, ability to tell what part of the body is a ...
of the category of complexes with respect to quasi-isomorphisms. Specifically, the derived category
is a category, together with a functor
, having the following universal property: Suppose
is another category (not necessarily abelian) and
is a functor such that, whenever
is a quasi-isomorphism in
, its image
is an isomorphism in
; then
factors through
. Any two categories having this universal property are equivalent.
Relation to the homotopy category
If
and
are two morphisms
in
, then a chain homotopy or simply homotopy
is a collection of morphisms
such that
for every ''i''. It is straightforward to show that two homotopic morphisms induce identical morphisms on cohomology groups. We say that
is a chain homotopy equivalence if there exists
such that
and
are chain homotopic to the identity morphisms on
and
, respectively. The
homotopy category of cochain complexes is the category with the same objects as
but whose morphisms are equivalence classes of morphisms of complexes with respect to the relation of chain homotopy equivalence. There is a natural functor
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,
factors through this functor. Consequently
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
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
is a large category, this construction does not work for set theoretic reasons. This construction builds morphisms as equivalence classes of paths. If
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
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
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
in
may be described as a pair
, where for some complex
,
is a quasi-isomorphism and
is a chain homotopy equivalence class of morphisms. Conceptually, this represents
. 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
and consider the category
whose objects are quasi-isomorphisms in
with codomain
and whose morphisms are commutative diagrams. Equivalently, this is the category of objects over
whose structure maps are quasi-isomorphisms. Then the multiplicative system condition implies that the morphisms in
from
to
are
:
assuming that this colimit is in fact a set. While
is potentially a large category, in some cases it is controlled by a small category. This is the case, for example, if
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
may be defined to have these sets as its
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
is ''K''-injective if, for every acyclic complex
, we have
. A straightforward consequence of this is that, for every complex
, morphisms
in
are the same as such morphisms in
. 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
, where
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