Homological algebra is the branch of
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 ...
that studies
homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in
combinatorial topology (a precursor to
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
) and
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
(theory 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 s ...
and
syzygies) at the end of the 19th century, chiefly by
Henri Poincaré
Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "Th ...
and
David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...
.
Homological algebra is the study of homological
functors and the intricate algebraic structures that they entail; its development was closely intertwined with the emergence of
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
. A central concept is that of
chain complexes, which can be studied through both their homology and
cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be view ...
.
Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological
invariants of
rings, modules,
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
s, and other 'tangible' mathematical objects. A powerful tool for doing this is provided by
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 h ...
s.
It has played an enormous role in algebraic topology. Its influence has gradually expanded and presently includes
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
,
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 ...
,
algebraic number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic o ...
,
representation theory,
mathematical physics
Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developm ...
,
operator algebras,
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, and the theory of
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to h ...
s.
''K''-theory is an independent discipline which draws upon methods of homological algebra, as does the
noncommutative geometry of
Alain Connes.
History
Homological algebra began to be studied in its most basic form in the 1800s as a branch of topology, but it wasn't until the 1940s that it became an independent subject with the study of objects such as the
ext functor and the
tor functor
In mathematics, the Tor functors are the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central concepts of homological algebra, in which ideas from algebraic topology are used to co ...
, among others.
Chain complexes and homology
The notion of
chain complex is central in homological algebra. An abstract chain complex is a sequence
of
abelian group
In mathematics, an 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 the order in which they are written. That is, the group operation is comm ...
s and
group homomorphisms,
with the property that the composition of any two consecutive
maps is zero:
:
The elements of ''C''
''n'' are called ''n''-chains and the homomorphisms ''d''
''n'' are called the boundary maps or differentials. The chain groups ''C''
''n'' may be endowed with extra structure; for example, they may be
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s or
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 s ...
over a fixed
ring ''R''. The differentials must preserve the extra structure if it exists; for example, they must be
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
s or homomorphisms of ''R''-modules. For notational convenience, restrict attention to abelian groups (more correctly, to the
category Ab of abelian groups); a celebrated
theorem by Barry Mitchell implies the results will generalize to any
abelian category. Every chain complex defines two further sequences of abelian groups, the cycles ''Z''
''n'' = Ker ''d''
''n'' and the boundaries ''B''
''n'' = Im ''d''
''n''+1, where Ker ''d'' and Im ''d'' denote the
kernel and the
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
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
normal; therefore we can define the ''n''th homology group ''H''
''n''(''C'') as the
factor group
Factor, a Latin word meaning "who/which acts", may refer to:
Commerce
* Factor (agent), a person who acts for, notably a mercantile and colonial agent
* Factor (Scotland), a person or firm managing a Scottish estate
* Factors of production, s ...
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
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
and
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
. For example, if ''X'' is a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
then the
singular chains ''C''
''n''(''X'') are formal
linear combinations of
continuous maps from the standard ''n''-
simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
into ''X''; if ''K'' is a
simplicial complex
In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial ...
then the
simplicial chains ''C''
''n''(''K'') are formal linear combinations of the ''n''-simplices of ''K''; if ''A'' = ''F''/''R'' is a presentation of an abelian group ''A'' by
generators and relations, where ''F'' is a
free abelian group spanned by the generators and ''R'' is the subgroup of relations, then letting ''C''
1(''A'') = ''R'', ''C''
0(''A'') = ''F'', and ''C''
''n''(''A'') = 0 for all other ''n'' defines a sequence of abelian groups. In all these cases, there are natural differentials ''d''
''n'' making ''C''
''n'' 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 homology, which plays a fundamental role in investigating the properties of such spaces, for example,
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
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 with ''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
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
, homological algebra studies the
functorial properties 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 with ''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 of ''X''.
Standard tools
Exact sequences
In the context of
group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...
, a sequence
:
of
groups
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
and
group homomorphisms is called exact if the
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
of each homomorphism is equal to the
kernel of the next:
:
Note that the sequence of groups and homomorphisms may be either finite or infinite.
A similar definition can be made for certain other
algebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set o ...
s. For example, one could have an exact sequence of
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s and
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
s, or 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 s ...
and
module homomorphisms. More generally, the notion of an exact sequence makes sense in any
category with
kernels and
cokernels.
Short exact sequence
The most common type of exact sequence is the short exact sequence. This is an exact sequence of the form
:
where ƒ is a
monomorphism and ''g'' is an
epimorphism. In this case, ''A'' is a
subobject of ''B'', and the corresponding
quotient is
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 i ...
to ''C'':
:
(where ''f(A)'' = im(''f'')).
A short exact sequence of abelian groups may also be written as an exact sequence with five terms:
:
where 0 represents the
zero object, such as the
trivial group or a zero-dimensional vector space. The placement of the 0's forces ƒ to be a monomorphism and ''g'' to be an epimorphism (see below).
Long exact sequence
A long exact sequence is an exact sequence indexed by the
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s.
The five lemma
Consider the following
commutative diagram
350px, The commutative diagram used in the proof of the five lemma.
In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the s ...
in any
abelian category (such as the category of
abelian group
In mathematics, an 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 the order in which they are written. That is, the group operation is comm ...
s or the category of
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s over a given
field) or in the category of
groups.
The five lemma states that, if the rows are
exact, ''m'' and ''p'' are
isomorphism
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 i ...
s, ''l'' is an
epimorphism, and ''q'' is a
monomorphism, then ''n'' is also an isomorphism.
The snake lemma
In an
abelian category (such as the category of
abelian group
In mathematics, an 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 the order in which they are written. That is, the group operation is comm ...
s or the category of
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s over a given
field), consider a
commutative diagram
350px, The commutative diagram used in the proof of the five lemma.
In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the s ...
:
where the rows are
exact sequences and 0 is the
zero object.
Then there is an exact sequence relating the
kernels and
cokernels of ''a'', ''b'', and ''c'':
:
Furthermore, if the morphism ''f'' is a
monomorphism, then so is the morphism ker ''a'' → ker ''b'', and if ''g is an
epimorphism, then so is coker ''b'' → coker ''c''.
Abelian categories
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 ...
, 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 In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab.
Properties
The zero object of ...
, Ab. The theory originated in a tentative attempt to unify several
cohomology theories
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
by
Alexander Grothendieck. Abelian categories are very ''stable'' categories, for example they are
regular and they satisfy the
snake lemma. The class of Abelian categories is closed under several categorical constructions, for example, the category of
chain complexes of an Abelian category, or the category of
functors from a
small category to an Abelian category are Abelian as well. These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in
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 ...
,
cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be view ...
and pure
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
. Abelian categories are named after
Niels Henrik Abel.
More concretely, a category is abelian if
*it has a
zero object,
*it has all binary
products and binary
coproducts, and
*it has all
kernels and
cokernels.
*all
monomorphisms and
epimorphisms are
normal.
The Ext functor
Let ''R'' be a
ring and let Mod
''R'' be 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 s ...
over ''R''. Let ''B'' be in Mod
''R'' and set ''T''(''B'') = Hom
''R''(''A,B''), for fixed ''A'' in Mod
''R''. This is a
left exact functor
In mathematics, particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations because they can be directly applied to presentations of objects. Mu ...
and thus has right
derived functors ''R
nT''. The Ext functor is defined by
:
This can be calculated by taking any
injective resolution
:
and computing
:
Then (''R
nT'')(''B'') is the
homology of this complex. Note that Hom
''R''(''A,B'') is excluded from the complex.
An alternative definition is given using the functor ''G''(''A'')=Hom
''R''(''A,B''). For a fixed module ''B'', this is a
contravariant left exact functor
In mathematics, particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations because they can be directly applied to presentations of objects. Mu ...
, and thus we also have right
derived functors ''R
nG'', and can define
:
This can be calculated by choosing any
projective resolution
:
and proceeding dually by computing
:
Then (''R
nG'')(''A'') is the homology of this complex. Again note that Hom
''R''(''A,B'') is excluded.
These two constructions turn out to yield
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 i ...
results, and so both may be used to calculate the Ext functor.
Tor functor
Suppose ''R'' is a
ring, and denoted by ''R''-Mod the
category of
left ''R''-modules and by Mod-''R'' the category of right ''R''-modules (if ''R'' is
commutative, the two categories coincide). Fix a module ''B'' in ''R''-Mod. For ''A'' in Mod-''R'', set ''T''(''A'') = ''A''⊗
''R''''B''. Then ''T'' is a
right exact functor
In mathematics, particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations because they can be directly applied to presentations of objects. Mu ...
from Mod-''R'' to the
category of abelian groups In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab.
Properties
The zero object of ...
Ab (in the case when ''R'' is commutative, it is a right exact functor from Mod-''R'' to Mod-''R'') and its
left derived functors ''L
nT'' are defined. We set
:
i.e., we take a
projective resolution
:
then remove the ''A'' term and tensor the projective resolution with ''B'' to get the complex
:
(note that ''A''⊗
''R''''B'' does not appear and the last arrow is just the zero map) and take the
homology of this complex.
Spectral sequence
Fix an
abelian category, such as a category of modules over a ring. A spectral sequence is a choice of a nonnegative integer ''r''
0 and a collection of three sequences:
# For all integers ''r'' ≥ ''r''
0, an object ''E
r'', called a ''sheet'' (as in a sheet of
paper
Paper is a thin sheet material produced by mechanically or chemically processing cellulose fibres derived from wood, rags, grasses or other vegetable sources in water, draining the water through fine mesh leaving the fibre evenly distribu ...
), or sometimes a ''page'' or a ''term'',
# Endomorphisms ''d
r'' : ''E
r'' → ''E
r'' satisfying ''d
r''
o ''d
r'' = 0, called ''boundary maps'' or ''differentials'',
# Isomorphisms of ''E
r+1'' with ''H''(''E
r''), the homology of ''E
r'' with respect to ''d
r''.
A doubly graded spectral sequence has a tremendous amount of data to keep track of, but there is a common visualization technique which makes the structure of the spectral sequence clearer. We have three indices, ''r'', ''p'', and ''q''. For each ''r'', imagine that we have a sheet of graph paper. On this sheet, we will take ''p'' to be the horizontal direction and ''q'' to be the vertical direction. At each lattice point we have the object
.
It is very common for ''n'' = ''p'' + ''q'' to be another natural index in the spectral sequence. ''n'' runs diagonally, northwest to southeast, across each sheet. In the homological case, the differentials have bidegree (−''r'', ''r'' − 1), so they decrease ''n'' by one. In the cohomological case, ''n'' is increased by one. When ''r'' is zero, the differential moves objects one space down or up. This is similar to the differential on a chain complex. When ''r'' is one, the differential moves objects one space to the left or right. When ''r'' is two, the differential moves objects just like a
knight
A knight is a person granted an honorary title of knighthood by a head of state (including the Pope) or representative for service to the monarch, the Christian denomination, church or the country, especially in a military capacity. Knighthood ...
's move in
chess
Chess is a board game for two players, called White and Black, each controlling an army of chess pieces in their color, with the objective to checkmate the opponent's king. It is sometimes called international chess or Western chess to dist ...
. For higher ''r'', the differential acts like a generalized knight's move.
Derived functor
Suppose we are given a covariant
left exact functor
In mathematics, particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations because they can be directly applied to presentations of objects. Mu ...
''F'' : A → B between two
abelian categories
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 abe ...
A and B. If 0 → ''A'' → ''B'' → ''C'' → 0 is a short exact sequence in A, then applying ''F'' yields the exact sequence 0 → ''F''(''A'') → ''F''(''B'') → ''F''(''C'') and one could ask how to continue this sequence to the right to form a long exact sequence. Strictly speaking, this question is ill-posed, since there are always numerous different ways to continue a given exact sequence to the right. But it turns out that (if A is "nice" enough) there is one
canonical way of doing so, given by the right derived functors of ''F''. For every ''i''≥1, there is a functor ''R
iF'': A → B, and the above sequence continues like so: 0 → ''F''(''A'') → ''F''(''B'') → ''F''(''C'') → ''R''
1''F''(''A'') → ''R''
1''F''(''B'') → ''R''
1''F''(''C'') → ''R''
2''F''(''A'') → ''R''
2''F''(''B'') → ... . From this we see that ''F'' is an exact functor if and only if ''R''
1''F'' = 0; so in a sense the right derived functors of ''F'' measure "how far" ''F'' is from being exact.
Functoriality
A
continuous map of topological spaces gives rise to a homomorphism between their ''n''th
homology groups for all ''n''. This basic fact of
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
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
that commute with the differentials, in the sense that
for all ''n''. A morphism of chain complexes induces a morphism
of their homology groups, consisting of the homomorphisms
for all ''n''. A morphism ''F'' is called a
quasi-isomorphism
In homological algebra, a branch of mathematics, a quasi-isomorphism or quism is a morphism ''A'' → ''B'' of chain complexes (respectively, cochain complexes) such that the induced morphisms
:H_n(A_\bullet) \to H_n(B_\bullet)\ (\text H^n(A^\bul ...
if it induces an isomorphism on the ''n''th homology for all ''n''.
Many constructions of chain complexes arising in algebra and geometry, including
singular homology, have the following
functoriality property: if two objects ''X'' and ''Y'' are connected by a map ''f'', then the associated chain complexes are connected by a morphism
and moreover, the composition
of maps ''f'': ''X'' → ''Y'' and ''g'': ''Y'' → ''Z'' induces the morphism
that coincides with the composition
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 ''f''
''n'' is an
injection, ''g''
''n'' is a
surjection, and Im ''f''
''n'' = Ker ''g''
''n''. One of the most basic theorems of homological algebra, sometimes known as the
zig-zag lemma, 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 ''δ''
''n'' are certain homomorphisms determined by ''f'' and ''g'', called the
connecting homomorphism
The snake lemma is a tool used in mathematics, particularly homological algebra, to construct long exact sequences. The snake lemma is valid in every abelian category and is a crucial tool in homological algebra and its applications, for instance ...
s. Topological manifestations of this theorem include the
Mayer–Vietoris sequence and the long exact sequence for
relative homology.
Foundational aspects
Cohomology theories have been defined for many different objects such as
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
s,
sheaves,
groups,
rings,
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
s, and
C*-algebras. The study of modern
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 ...
would be almost unthinkable without
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 whe ...
.
Central to homological algebra is the notion of
exact sequence; these can be used to perform actual calculations. A classical tool of homological algebra is that of
derived functor; the most basic examples are functors
Ext
Ext, ext or EXT may refer to:
* Ext functor, used in the mathematical field of homological algebra
* Ext (JavaScript library), a programming library used to build interactive web applications
* Exeter Airport (IATA airport code), in Devon, England
...
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-
Eilenberg: In their 1956 book "Homological Algebra", these authors used
projective and
injective module resolutions.
* 'Tohoku': The approach in a
celebrated paper by
Alexander Grothendieck which appeared in the Second Series of the ''
Tohoku Mathematical Journal
The ''Tohoku Mathematical Journal'' is a mathematical research journal published by Tohoku University in Japan. It was founded in August 1911 by Tsuruichi Hayashi.
History
Due to World War II the publication of the journal stopped in 1943 wi ...
'' in 1957, using the
abelian category concept (to include
sheaves of abelian groups).
* The
derived category of
Grothendieck and
Verdier. Derived categories date back to Verdier's 1967 thesis. They are 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 cat ...
used in a number of modern theories.
These move from computability to generality.
The computational sledgehammer ''par excellence'' is the
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 h ...
; 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 cohomology).
See also
*
Abstract nonsense In mathematics, abstract nonsense, general abstract nonsense, generalized abstract nonsense, and general nonsense are terms used by mathematicians to describe abstract methods related to category theory and homological algebra. More generally, "a ...
, a term for homological algebra and
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
*
Derivator In mathematics, derivators are a proposed frameworkpg 190-195 for homological algebra giving a foundation for both abelian and non-abelian homological algebra and various generalizations of it. They were introduced to address the deficiencies of ...
*
Homotopical algebra
In mathematics, homotopical algebra is a collection of concepts comprising the ''nonabelian'' aspects of homological algebra as well as possibly the abelian aspects as special cases. The ''homotopical'' nomenclature stems from the fact that a c ...
*
List of homological algebra topics
References
*
Henri Cartan,
Samuel Eilenberg, ''Homological algebra''. With an appendix by David A. Buchsbaum. Reprint of the 1956 original. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1999. xvi+390 pp.
*
*
Saunders Mac Lane, ''Homology''. Reprint of the 1975 edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995. x+422 pp.
*
Peter Hilton
Peter John Hilton (7 April 1923Peter Hilton, "On all Sorts of Automorphisms", ''The American Mathematical Monthly'', 92(9), November 1985, p. 6506 November 2010) was a British mathematician, noted for his contributions to homotopy theory and ...
; Stammbach, U. ''A course in homological algebra''. Second edition. Graduate Texts in Mathematics, 4. Springer-Verlag, New York, 1997. xii+364 pp.
* Gelfand, Sergei I.;
Yuri Manin, ''Methods of homological algebra''. Translated from Russian 1988 edition. Second edition. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003. xx+372 pp.
* Gelfand, Sergei I.; Yuri Manin, ''Homological algebra''. Translated from the 1989 Russian original by the authors. Reprint of the original English edition from the series Encyclopaedia of Mathematical Sciences (''Algebra'', V, Encyclopaedia Math. Sci., 38, Springer, Berlin, 1994). Springer-Verlag, Berlin, 1999. iv+222 pp.
*
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