Homological algebra is the branch of
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
which studies
homologyIn mathematics , homology is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group...
in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in
combinatorial topologyIn mathematics, combinatorial topology was an older name for algebraic topology, dating from the time when topological invariants of spaces were regarded as derived from combinatorial decompositions such as simplicial complexes...
(a precursor to
algebraic topologyAlgebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism...
) and
abstract algebraAbstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
(theory of
modulesIn abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the "scalars" may lie in an arbitrary ring...
and syzygies) at the end of the 19th century, chiefly by
Henri PoincaréJules Henri Poincaré was a French mathematician and theoretical physicist, and a philosopher of science...
and
David HilbertDavid Hilbert was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. He discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry...
.
The development of homological algebra was closely intertwined with the emergence of
category theoryIn mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions to objects linked in diagrams by morphisms or arrows....
. By and large, homological algebra is the study of homological
functorIn category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms in the category of small categories....
s and the intricate algebraic structures that they entail. The hidden fabric of mathematics is woven of
chain complexIn mathematics, chain complex and cochain complex are constructs originally used in the field of algebraic topology. They are algebraic means of representing the relationships between the cycles and boundaries in various dimensions of some "space". Here the "space" could be a topological space or...
es, which manifest themselves through their homology and
cohomologyIn mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological
invariantsIn mathematics, an invariant is something that does not change under a set of transformations. The property of being an invariant is invariance....
of
ringsIn mathematics, a ring is an algebraic structure consisting of a set together with two binary operations , where each operation combines two elements to form a third element...
, modules,
topological spaceTopological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
s, and other 'tangible' mathematical objects. A powerful tool for doing this is provided by
spectral sequenceIn the area of mathematics known as homological algebra, especially in algebraic topology and group cohomology, a spectral sequence is a means of computing homology groups by taking successive approximations...
s.
From its very origins, homological algebra has played an enormous role in algebraic topology. Its sphere of influence has gradually expanded and presently includes
commutative algebraCommutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
,
algebraic geometryAlgebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such...
,
algebraic number theoryIn mathematics, algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as...
,
representation theoryRepresentation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces. In essence, a representation makes an abstract algebraic object concrete by describing its elements by matrices and the algebraic...
,
mathematical physicsMathematical physics is the scientific discipline concerned with the interface of mathematics and physics. The Journal of Mathematical Physics defines it as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the...
,
operator algebraIn functional analysis, an operator algebra is an algebra of continuous linear operators on a topological vector space with the multiplication given by the composition of mappings...
s,
complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating functions of complex numbers...
, and the theory of
partial differential equationIn mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables...
s.
K-theoryIn mathematics, K-theory is a tool used in several disciplines. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It also has some applications in operator algebras...
is an independent discipline which draws upon methods of homological algebra, as does the
noncommutative geometryNoncommutative geometry, or NCG, is a branch of mathematics concerned with the possible spatial interpretations of algebraic structures for which the commutative law fails, that is, for which xy does not always equal yx. For example; 3 steps of 4 units and 4 steps of 3 units length might be...
of
Alain ConnesAlain Connes is a French mathematician, currently Professor at the College de France, IHÉS and Vanderbilt University.-Work:...
.
Chain complexes and homology
The
chain complexIn mathematics, chain complex and cochain complex are constructs originally used in the field of algebraic topology. They are algebraic means of representing the relationships between the cycles and boundaries in various dimensions of some "space". Here the "space" could be a topological space or...
is the central notion of homological algebra. It is a sequence of
abelian groupAn abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
s and
group homomorphismIn mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that...
s,
with the property that the composition of any two consecutive maps is zero:
The elements of
Cn are called
n-
chains and the homomorphisms
dn are called the
boundary maps or
differentials. The
chain groups Cn may be endowed with extra structure; for example, they may be
vector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s or
modulesIn abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the "scalars" may lie in an arbitrary ring...
over a fixed
ringIn mathematics, a ring is an algebraic structure consisting of a set together with two binary operations , where each operation combines two elements to form a third element...
R. The differentials must preserve the extra structure if it exists; for example, they must be linear maps or homomorphisms of
R-modules. For notational convenience, restrict attention to abelian groups (more correctly, to the
categoryIn mathematics, a category is an algebraic structure consisting of a collection of "objects", linked together by a collection of "arrows" that have two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. Objects and arrows may...
Ab of abelian groups); a celebrated
theorem by Barry MitchellMitchell's embedding theorem, also known as the Freyd-Mitchell theorem, is a mathematical result about abelian categories; it states that these categories, while rather abstractly defined, are all quite concrete categories of modules...
implies the results will generalize to any
abelian categoryIn 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 prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...
. Every chain complex defines two further sequences of abelian groups, the
cycles Zn = Ker
dn and the
boundaries Bn = Im
dn+1, where Ker
d and Im
d denote the
kernelIn mathematics, the word kernel has several meanings. Kernel may mean a subset associated with a mapping:* The kernel of a mapping is the set of elements that map to the zero element , as in kernel of a linear operator and kernel of a matrix...
and the
imageIn mathematics, the image of a subset of a function's domain under the function is the set of all outputs obtained when the function is evaluated at each element of the subset...
of
d. Since the composition of two consecutive boundary maps is zero, these groups are embedded into each other as
Subgroups of abelian groups are automatically
normalIn mathematics, more specifically in abstract algebra, a normal subgroup is a special kind of subgroup. Normal subgroups are important because they can be used to construct quotient groups from a given group....
; therefore we can define the
nth
homology group Hn(
C) as the factor group of the
n-cycles by the
n-boundaries,
A chain complex is called
acyclic or an
exact sequence if all its homology groups are zero.
Chain complexes arise in abundance in
algebraAbstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
and
algebraic topologyAlgebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism...
. For example, if
X is a
topological spaceTopological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
then the singular chains
Cn(
X) are formal
linear combinationIn mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics.Most of this article deals with linear combinations in the context of a vector space over a field, with some generalizations given at the end of the article.- Definition:Suppose that K is a...
s of continuous maps from the standard
n-
simplexIn geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope with n + 1 vertices, of which the simplex is the convex hull...
into
X; if
K is a
simplicial complexIn mathematics, a simplicial complex is a topological space of a particular kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts...
then the simplicial chains
Cn(
K) are formal linear combinations of the
n-simplices of
X; if
A =
F/
R is a presentation of an abelian group
A by generators and relations, where
F is a
free abelian groupIn abstract algebra, a free abelian group is an abelian group that has a "basis" in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. Hence, free abelian groups over a basis B are...
spanned by the generators and
R is the subgroup of relations, then letting
C1(
A) =
R,
C0(
A) =
F, and
Cn(
A) = 0 for all other
n defines a sequence of abelian groups. In all these cases, there are natural differentials
dn making
Cn into a chain complex, whose homology reflects the structure of the topological space
X, the simplicial complex
K, or the abelian group
A. In the case of topological spaces, we arrive at the notion of
singular homologyIn algebraic topology, a branch of mathematics, singular homology refers to the study of a certain set of topological invariants of a topological space X, the so-called homology groups . Singular homology is a particular example of a homology theory, which has now grown to be a rather broad...
, which plays a fundamental role in investigating the properties of such spaces, for example,
manifoldIn mathematics, more specifically in differential geometry and topology, a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a certain dimension, called the dimension of the manifold....
s.
On a philosophical level, homological algebra teaches us that certain chain complexes associated with algebraic or geometric objects (topological spaces, simplicial complexes,
R-modules) contain a lot of valuable algebraic information about them, with the homology being only the most readily available part. On a technical level, homological algebra provides the tools for manipulating complexes and extracting this information. Here are two general illustrations.
- Two objects X and Y are connected by a map f between them. Homological algebra studies the relation, induced by the map f, between chain complexes associated to X and Y and their homology. This is generalized to the case of several objects and maps connecting them. Phrased in the language of category theory
In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions to objects linked in diagrams by morphisms or arrows....
, homological algebra studies the functorial propertiesIn category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms in the category of small categories....
of various constructions of chain complexes and of the homology of these complexes.
- An object X admits multiple descriptions (for example, as a topological space and as a simplicial complex) or the complex is constructed using some 'presentation' of X, which involves non-canonical choices. It is important to know the effect of change in the description of X on chain complexes associated to X. Typically, the complex and its homology are functorial with respect to the presentation; and the homology (although not the complex itself) is actually independent of the presentation chosen, thus it is an invariant
In mathematics, an invariant is something that does not change under a set of transformations. The property of being an invariant is invariance....
of X.
Functoriality
A continuous map of topological spaces gives rise to a homomorphism between their
nth homology groups for all
n. This basic fact of
algebraic topologyAlgebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism...
finds a natural explanation through certain properties of chain complexes. Since it is very common to study
several topological spaces simultaneously, in homological algebra one is led to simultaneous consideration of multiple chain complexes.
A
morphism between two chain complexes, , is a family of homomorphisms of abelian groups
Fn:
Cn →
Dn that commute with the differentials, in the sense that
Fn -1 •
dnC =
dnD •
Fn for all
n. A morphism of chain complexes induces a morphism of their homology groups, consisting of the homomorphisms
Hn(
F):
Hn(
C) →
Hn(
D) for all
n. A morphism
F is called a
quasi-isomorphism if it induces an isomorphism on the
nth homology for all
n.
Many constructions of chain complexes arising in algebra and geometry, including
singular homologyIn algebraic topology, a branch of mathematics, singular homology refers to the study of a certain set of topological invariants of a topological space X, the so-called homology groups . Singular homology is a particular example of a homology theory, which has now grown to be a rather broad...
, have the following
functorialityIn category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms in the category of small categories....
property: if two objects
X and
Y are connected by a map
f, then the associated chain complexes are connected by a morphism
F =
C(
f) from to and moreover, the composition
g •
f of maps
f:
X →
Y and
g:
Y →
Z induces the morphism
C(
g •
f) from to that coincides with the composition
C(
g) •
C(
f). It follows that the homology groups are functorial as well, so that morphisms between algebraic or topological objects give rise to compatible maps between their homology.
The following definition arises from a typical situation in algebra and topology. A triple consisting of three chain complexes and two morphisms between them,
is called an
exact triple, or a
short exact sequence of complexes, and written as
if for any
n, the sequence
is a short exact sequence of abelian groups. By definition, this means that
fn is an injection,
gn is a surjection, and Im
fn = Ker
gn. One of the most basic theorems of homological algebra, sometimes known as the
zig-zag lemmaIn mathematics, particularly homological algebra, the zig-zag lemma asserts the existence of a particular long exact sequence in the homology groups of certain chain complexes...
, states that, in this case, there is a
long exact sequence in homology
where the homology groups of
L,
M, and
N cyclically follow each other, and
δn are certain homomorphisms determined by
f and
g, called the
connecting homomorphisms. Topological manifestations of this theorem include the Mayer–Vietoris sequence and the long exact sequence for
relative homologyIn algebraic topology, a branch of mathematics, the homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces. The relative homology is useful and important in several ways...
.
Foundational aspects
Cohomology theories have been defined for many different objects such as
topological spaceTopological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
s,
sheavesIn mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
,
groupIn mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
s,
ringIn mathematics, a ring is an algebraic structure consisting of a set together with two binary operations , where each operation combines two elements to form a third element...
s,
Lie algebraIn mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
s, and
C*-algebraC*-algebras are an important area of research in functional analysis, a branch of mathematics. The prototypical example of a C*-algebra is a complex algebra A of linear operators on a complex Hilbert space with two additional properties:* A is a topologically closed set in the norm topology of...
s. The study of modern
algebraic geometryAlgebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such...
would be almost unthinkable without
sheaf cohomologyIn mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F...
.
Central to homological algebra is the notion of
exact sequenceIn mathematics, especially in homological algebra and other applications of abelian category theory, as well as in differential geometry and group theory, an exact sequence is a sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the...
; these can be used to perform actual calculations. A classical tool of homological algebra is that of
derived functorIn 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 :...
; the most basic examples are functors Ext and Tor.
With a diverse set of applications in mind, it was natural to try to put the whole subject on a uniform basis. There were several attempts before the subject settled down. An approximate history can be stated as follows:
- Cartan
Henri Paul Cartan was a son of Élie Cartan, and was, as his father was, a distinguished and influential French mathematician.-Life:...
-EilenbergSamuel Eilenberg was a Polish and American mathematician of Jewish descent. He was born in Warsaw, Russian Empire and died in New York City, USA, where he had spent much of his career as a professor at Columbia University.He earned his Ph.D. from Warsaw University in 1936. His thesis advisor was...
: In their 1956 book "Homological Algebra", these authors used projective and injective module resolutions.
- 'Tohoku': The approach in a celebrated paper by Alexander Grothendieck
Alexander Grothendieck is considered one of the greatest mathematicians of the 20th century.He is most famous for his revolutionary advances in algebraic geometry, but he has also made major contributions to algebraic topology, number theory, category theory, Galois theory, descent theory,...
which appeared in the Second Series of the Tohoku Mathematical JournalThe Tohoku Mathematical Journal is a mathematical research journal published by Tohoku University in Japan. It was founded in August 1911 by Tsuruichi Hayashi....
in 1957, using the abelian categoryIn 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 prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...
concept (to include sheavesIn mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
of abelian groups).
- The derived category
In mathematics, the derived category D 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...
of Grothendieck and VerdierJean-Louis Verdier was a French mathematician who worked, under the guidance of Alexander Grothendieck, on derived categories and Verdier duality...
. Derived categories date back to Verdier's 1967 thesis. They are examples of triangulated categoriesA triangulated category is a mathematical category satisfying some axioms that are based on the properties of the homotopy category of spectra, and the derived category of an abelian category. A t-category, though similarly named, refers to a more specific concept.- History :The notion of a...
used in a number of modern theories.
These move from computability to generality.
The computational sledgehammer
par excellence is the
spectral sequenceIn the area of mathematics known as homological algebra, especially in algebraic topology and group cohomology, a spectral sequence is a means of computing homology groups by taking successive approximations...
; these are essential in the Cartan-Eilenberg and Tohoku approaches where they are needed, for instance, to compute the derived functors of a composition of two functors. Spectral sequences are less essential in the derived category approach, but still play a role whenever concrete computations are necessary.
There have been attempts at 'non-commutative' theories which extend first cohomology as
torsors (important in
Galois cohomologyIn mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups...
).