Chain complex explained

In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is contained in the kernel of the next. Associated to a chain complex is its homology, which is (loosely speaking) a measure of the failure of a chain complex to be exact.

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, the singular chain complex of a topological space X is constructed using continuous maps from a simplex 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, but are used in several areas of mathematics, including abstract algebra, Galois theory, differential geometry and algebraic geometry. They can be defined more generally in abelian categories.

Definitions

A chain complex

(A_\bullet,d_\bullet)

is a sequence of abelian groups or modules

… ,A_0,A_1,A_2,...

connected by homomorphisms (called boundary operators or differentials)

d_n:A_n\toA_n-1

, such that the composition of any two consecutive maps is the zero map. Explicitly, the differentials satisfy

d_n\circd_n+1=0

for all

, or, concisely,

d²⁼⁰

. The complex may be written out as follows:

… \xleftarrow{d₀}A_{0
\xleftarrow{d}_1}A_{1
\xleftarrow{d}_2}A_{2
\xleftarrow{d}_3}A_{3
\xleftarrow{d}_4}A_{4
\xleftarrow{d}_{5}
…}

The cochain complex

(A^\bullet,d^\bullet)

is the dual notion to a chain complex. It consists of a sequence of abelian groups or modules

… ,A^0,A^1,A^2,...

connected by homomorphisms (coboundary operators)

d^n:A^n\toAⁿ⁺¹

satisfying

dⁿ⁺¹\circdⁿ=0

. The cochain complex may be written out in a similar fashion to the chain complex:

… \xrightarrow{d^-1} A⁰\xrightarrow{d^0}
A¹\xrightarrow{d^1}
A²\xrightarrow{d^2}
A³\xrightarrow{d^3}
A⁴\xrightarrow{d^{4}
…}

In both cases, the index

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 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 of a finite simplicial complex. A chain complex is bounded above if all modules above some fixed degree

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

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 (Hⁿ) is the group of (co)cycles modulo (co)boundaries in degree n, that is,

H_n=\kerd_n/imd_n+1 \left(Hⁿ=\kerdⁿ/imd^n-1\right)

Exact sequences

See main article: Exact sequence. 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.

… \xrightarrow{} 0 \xrightarrow{} Z \xrightarrow{ x p} Z \twoheadrightarrowZ/pZ \xrightarrow{} 0 \xrightarrow{} …

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

(A_\bullet,d_A,\bullet)

and

(B_\bullet,d_B,\bullet)

is a sequence

f_\bullet

of homomorphisms

f_n:A_n → B_n

for each n that commutes with the boundary operators on the two chain complexes, so

d_B,n\circf_n=f_n-1\circd_A,n

. This is written out in the following commutative diagram.

A chain map sends cycles to cycles and boundaries to boundaries, and thus induces a map on homology

(f_\bullet)_*:H_\bullet(A_\bullet,d_A,\bullet) → H_\bullet(B_\bullet,d_B,\bullet)

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 of a chain map.

Chain homotopy

See also: Homotopy category of chain complexes. 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_Bh 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 between chain maps.

Let X and Y be topological spaces. In the case of singular homology, a homotopy 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

See main article: Singular homology. Let X be a topological space. Define C_n(X) for natural n to be the free abelian group formally generated by singular n-simplices in X, and define the boundary map

\partial_n:C_n(X)\toC_n-1(X)

to be

\partial_n:(\sigma:[v_0,\ldots,v_n]\toX)\mapsto

	n
(\sum
	i=0

(-1)ⁱ\sigma:[v_0,\ldots,\hatv_i,\ldots,v_n]\toX)

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

(C_\bullet,\partial_\bullet)

is a chain complex; the singular homology

H_\bullet(X)

is the homology of this complex.

Singular homology is a useful invariant of topological spaces up to homotopy equivalence. The degree zero homology group is a free abelian group on the path-components of X.

de Rham cohomology

See main article: 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 d maps Ω^k(M) to Ω^k+1(M), and d = 0 follows essentially from symmetry of second derivatives, so the vector spaces of k-forms along with the exterior derivative are a cochain complex.

0\stackrel{\subset}{\to} {\Re^c

} \stackrel\ \stackrel\ \stackrel\ \stackrel\ \Omega^3(M) \to \cdots

The cohomology of this complex is called the de Rham cohomology of M. Locally constant functions are designated with its isomorphism

\Re^c

with c the count of mutually disconnected components of M. This way the complex was extended to leave the complex exact at zero-form level using the subset operator.

Smooth maps between manifolds induce chain maps, and smooth homotopies between maps induce chain homotopies.

Category of chain complexes

Chain complexes of

-modules with chain maps as morphisms form a category

Ch_K

, where

is a commutative ring.

V=V_*

and

W=W_*

are chain complexes, their tensor product

V ⊗ W

is a chain complex with degree

elements given by

(V ⊗ W)_n=oplus_\{i,j\mid

} V_i \otimes W_j

and differential given by

\partial(a ⊗ b)=\partiala ⊗ b+(-1)^{\left|a\right|}a ⊗ \partialb

where

and

are any two homogeneous vectors in

and

respectively, and

\left|a\right|

denotes the degree of

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

viewed as a chain complex in degree

. The braiding is given on simple tensors of homogeneous elements by

a ⊗ b\mapsto(-1)^{\left|a\right|\left|b\right|}b ⊗ a

The sign is necessary for the braiding to be a chain map.

Moreover, the category of chain complexes of

-modules also has internal Hom: given chain complexes

and

, the internal Hom of

and

, denoted

Hom(V,W)

, is the chain complex with degree

elements given by

\Pi_iHom_K(V_i,W_i+n)

and differential given by

(\partialf)(v)=\partial(f(v))-(-1)^{\left|f\right|}f(\partial(v))

.We have a natural isomorphism

Hom(A ⊗ B,C)\congHom(A,Hom(B,C))

Further examples

Amitsur complex
A complex used to define Bloch's higher Chow groups
Buchsbaum–Rim complex
Čech complex
Cousin complex
Eagon–Northcott complex
Gersten complex
Graph complex^[1]
Koszul complex
Moore complex
Schur complex

References

- Book: Hatcher, Allen . Allen Hatcher . 2002 . Algebraic Topology . Cambridge . . 0-521-79540-0.

Notes and References

Web site: Graph complex.