cochain

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OR:

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 ...
, a chain complex is an
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 ...
that consists of 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 com ...
s (or modules) and a sequence of homomorphisms between consecutive groups such that 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-dimension ...
of each homomorphism is included in the
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learni ...
of the next. Associated to a chain complex is its homology, which describes how the images are included in the kernels. A cochain complex is similar to a chain complex, except that its homomorphisms are in the opposite direction. The homology of a cochain complex is called its cohomology. In
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 ...
, the singular chain complex of 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 poi ...
X is constructed using
continuous maps In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in va ...
from a
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 dimensio ...
to X, and the homomorphisms of the chain complex capture how these maps restrict to the boundary of the simplex. The homology of this chain complex is called the singular homology of X, and is a commonly used invariant of a topological space. Chain complexes are studied in
homological algebra Homological algebra is the branch of 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 topolo ...
, but are used in several areas of mathematics, including
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 ...
,
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory t ...
,
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
and
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 ...
. They can be defined more generally in abelian categories.

# Definitions

A chain complex $\left(A_\bullet, d_\bullet\right)$ is a sequence of abelian groups or modules ..., ''A''0, ''A''1, ''A''2, ''A''3, ''A''4, ... connected by homomorphisms (called boundary operators or differentials) , such that the composition of any two consecutive maps is the zero map. Explicitly, the differentials satisfy , or with indices suppressed, . The complex may be written out as follows. ::$\cdots \xleftarrow A_0 \xleftarrow A_1 \xleftarrow A_2 \xleftarrow A_3 \xleftarrow A_4 \xleftarrow \cdots$ The cochain complex $\left(A^\bullet, d^\bullet\right)$ is the
dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatica ...
notion to a chain complex. It consists of a sequence of abelian groups or modules ..., ''A''0, ''A''1, ''A''2, ''A''3, ''A''4, ... connected by homomorphisms satisfying . The cochain complex may be written out in a similar fashion to the chain complex. ::$\cdots \xrightarrow A^0 \xrightarrow A^1 \xrightarrow A^2 \xrightarrow A^3 \xrightarrow A^4 \xrightarrow \cdots$ The index ''n'' in either ''A''''n'' or ''A''''n'' is referred to as the degree (or dimension). The difference between chain and cochain complexes is that, in chain complexes, the differentials decrease dimension, whereas in cochain complexes they increase dimension. All the concepts and definitions for chain complexes apply to cochain complexes, except that they will follow this different convention for dimension, and often terms will be given the prefix ''co-''. In this article, definitions will be given for chain complexes when the distinction is not required. A bounded chain complex is one in which
almost all In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the math ...
the ''A''''n'' are 0; that is, a finite complex extended to the left and right by 0. An example is the chain complex defining the
simplicial homology In algebraic topology, simplicial homology is the sequence of homology groups of a simplicial complex. It formalizes the idea of the number of holes of a given dimension in the complex. This generalizes the number of connected components (the case ...
of a finite
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 ...
. A chain complex is bounded above if all modules above some fixed degree ''N'' are 0, and is bounded below if all modules below some fixed degree are 0. Clearly, a complex is bounded both above and below if and only if the complex is bounded. The elements of the individual groups of a (co)chain complex are called (co)chains. The elements in the kernel of ''d'' are called (co)cycles (or closed elements), and the elements in the image of ''d'' are called (co)boundaries (or exact elements). Right from the definition of the differential, all boundaries are cycles. The ''n''-th (co)homology group ''H''''n'' (''H''''n'') is the group of (co)cycles modulo (co)boundaries in degree ''n'', that is, ::$H_n = \ker d_/\mbox d_ \quad \left\left(H^n = \ker d^/\mbox d^ \right\right)$

## Exact sequences

An exact sequence (or exact complex) is a chain complex whose homology groups are all zero. This means all closed elements in the complex are exact. A short exact sequence is a bounded exact sequence in which only the groups ''A''''k'', ''A''''k''+1, ''A''''k''+2 may be nonzero. For example, the following chain complex is a short exact sequence. :$\cdots \xrightarrow \; 0 \; \xrightarrow \; \mathbf \; \xrightarrow \; \mathbf \twoheadrightarrow \mathbf/p\mathbf \; \xrightarrow \; 0 \; \xrightarrow \cdots$ In the middle group, the closed elements are the elements pZ; these are clearly the exact elements in this group.

## Chain maps

A chain map ''f'' between two chain complexes $\left(A_\bullet, d_\right)$ and $\left(B_\bullet, d_\right)$ is a sequence $f_\bullet$ of homomorphisms $f_n : A_n \rightarrow B_n$ for each ''n'' that commutes with the boundary operators on the two chain complexes, so $d_ \circ f_n = f_ \circ d_$. This is written out in 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 ...
. : A chain map sends cycles to cycles and boundaries to boundaries, and thus induces a map on homology $\left(f_\bullet\right)_*:H_\bullet\left(A_\bullet, d_\right) \rightarrow H_\bullet\left(B_\bullet, d_\right)$. A continuous map ''f'' between topological spaces ''X'' and ''Y'' induces a chain map between the singular chain complexes of ''X'' and ''Y'', and hence induces a map ''f''* between the singular homology of ''X'' and ''Y'' as well. When ''X'' and ''Y'' are both equal to the ''n''-sphere, the map induced on homology defines the degree of the map ''f''. The concept of chain map reduces to the one of boundary through the construction of the
cone A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines con ...
of a chain map.

## Chain homotopy

A chain homotopy offers a way to relate two chain maps that induce the same map on homology groups, even though the maps may be different. Given two chain complexes ''A'' and ''B'', and two chain maps , a chain homotopy is a sequence of homomorphisms such that . The maps may be written out in a diagram as follows, but this diagram is not commutative. : The map ''hd''''A'' + ''d''''B''''h'' is easily verified to induce the zero map on homology, for any ''h''. It immediately follows that ''f'' and ''g'' induce the same map on homology. One says ''f'' and ''g'' are chain homotopic (or simply homotopic), and this property defines an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
between chain maps. Let ''X'' and ''Y'' be topological spaces. In the case of singular homology, a
homotopy In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a defo ...
between continuous maps induces a chain homotopy between the chain maps corresponding to ''f'' and ''g''. This shows that two homotopic maps induce the same map on singular homology. The name "chain homotopy" is motivated by this example.

# Examples

## Singular homology

Let ''X'' be a topological space. Define ''C''''n''(''X'') for
natural Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans a ...
''n'' to be the free abelian group formally generated by singular n-simplices in ''X'', and define the boundary map $\partial_n: C_n\left(X\right) \to C_\left(X\right)$ to be :: where the hat denotes the omission of a vertex. That is, the boundary of a singular simplex is the alternating sum of restrictions to its faces. It can be shown that ∂2 = 0, so $\left(C_\bullet, \partial_\bullet\right)$ is a chain complex; the singular homology $H_\bullet\left(X\right)$ is the homology of this complex. Singular homology is a useful invariant of topological spaces up to
homotopy equivalence In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a defo ...
. The degree zero homology group is a free abelian group on the path-components of ''X''.

## de Rham cohomology

The differential ''k''-forms on any smooth manifold ''M'' form a real vector space called Ω''k''(''M'') under addition. The
exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The re ...
''d'' maps Ω''k''(''M'') to Ω''k''+1(''M''), and ''d'' = 0 follows essentially from
symmetry of second derivatives In mathematics, the symmetry of second derivatives (also called the equality of mixed partials) refers to the possibility of interchanging the order of taking partial derivatives of a function :f\left(x_1,\, x_2,\, \ldots,\, x_n\right) of ''n'' ...
, so the vector spaces of ''k''-forms along with the exterior derivative are a cochain complex. :$\Omega^0\left(M\right)\ \stackrel\ \Omega^1\left(M\right) \to \Omega^2\left(M\right) \to \Omega^3\left(M\right) \to \cdots$ The cohomology of this complex is called the de Rham cohomology of ''M''. The homology group in dimension zero is isomorphic to the vector space of locally constant functions from ''M'' to R. Thus for a compact manifold, this is the real vector space whose dimension is the number of connected components of ''M''. Smooth maps between manifolds induce chain maps, and smooth homotopies between maps induce chain homotopies.

# Category of chain complexes

Chain complexes of ''K''-modules with chain maps form a
category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce ...
Ch''K'', where ''K'' is a commutative ring. If ''V'' = ''V''$_*$ and ''W'' = ''W''$_*$ are chain complexes, their tensor product $V \otimes W$ is a chain complex with degree ''n'' elements given by :$\left(V \otimes W\right)_n = \bigoplus_ V_i \otimes W_j$ and differential given by : $\partial \left(a \otimes b\right) = \partial a \otimes b + \left(-1\right)^ a \otimes \partial b$ where ''a'' and ''b'' are any two homogeneous vectors in ''V'' and ''W'' respectively, and $\left, a\$ denotes the degree of ''a''. This tensor product makes the category Ch''K'' into a symmetric monoidal category. The identity object with respect to this monoidal product is the base ring ''K'' viewed as a chain complex in degree 0. The
braiding A braid (also referred to as a plait) is a complex structure or pattern formed by interlacing two or more strands of flexible material such as textile yarns, wire, or hair. The simplest and most common version is a flat, solid, three-strande ...
is given on simple tensors of homogeneous elements by :$a \otimes b \mapsto \left(-1\right)^ b \otimes a$ The sign is necessary for the braiding to be a chain map. Moreover, the category of chain complexes of ''K''-modules also has
internal Hom In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory an ...
: given chain complexes ''V'' and ''W'', the internal Hom of ''V'' and ''W'', denoted Hom(''V'',''W''), is the chain complex with degree ''n'' elements given by $\Pi_\text_K \left(V_i,W_\right)$ and differential given by : $\left(\partial f\right)\left(v\right) = \partial\left(f\left(v\right)\right) - \left(-1\right)^ f\left(\partial\left(v\right)\right)$. We have a
natural isomorphism In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natura ...
:$\text\left(A\otimes B, C\right) \cong \text\left(A,\text\left(B,C\right)\right)$

# Further examples

* Amitsur complex *A complex used to define Bloch's higher Chow groups * Buchsbaum–Rim complex * Čech complex *
Cousin complex Most generally, in the lineal kinship system used in the English-speaking world, a cousin is a type of familial relationship in which two relatives are two or more familial generations away from their most recent common ancestor. Commonly, ...
* Eagon–Northcott complex * Gersten complex * Graph complex *
Koszul complex In mathematics, the Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul (see Lie algebra cohomology). It turned out to be a useful general construction in homological algebra. As a tool, its ...
* Moore complex *
Schur complex Schur is a German or Jewish surname. Notable people with the surname include: * Alexander Schur (born 1971), German footballer * Dina Feitelson-Schur (1926–1992), Israeli educator * Friedrich Schur (1856-1932), German mathematician * Fritz Sch ...