Editing Chain complex (section)

==Definitions==
A '''chain complex''' <math>(A_\bullet, d_\bullet)</math> is a sequence of abelian groups or modules <math>\cdots, A_0, A_1, A_2,\dots</math> connected by homomorphisms (called '''boundary operators''' or '''differentials''') <math>d_n:A_n\to A_{n-1}</math>, such that the composition of any two consecutive maps is the zero map. Explicitly, the differentials satisfy <math>d_n\circ d_{n+1}=0</math> for all <math>n</math>, or, concisely, <math>d^2=0</math>. The complex may be written out as follows:

::<math>
\cdots
\xleftarrow{ d_0 } 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}
\cdots
</math>

The '''cochain complex''' <math>(A^\bullet, d^\bullet)</math> is the [[dual (category theory)|dual]] notion to a chain complex. It consists of a sequence of abelian groups or modules <math>\cdots, A^0, A^1, A^2,\dots</math> connected by homomorphisms <math>d^n:A^n\to A^{n+1}</math> satisfying <math>d^{n+1}\circ d^{n}=0</math>. The cochain complex may be written out in a similar fashion to the chain complex:

::<math>
\cdots
\xrightarrow{ d^{-1} }
A^0 \xrightarrow{d^0}
A^1 \xrightarrow{d^1}
A^2 \xrightarrow{d^2}
A^3 \xrightarrow{d^3}
A^4 \xrightarrow{d^4}
\cdots
</math>

In both cases, the index <math>n</math> 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#cardinality|almost all]] the <math>A_n</math> 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 <math>N</math> 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 <math>d</math> 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''<sub>''n''</sub> (''H''<sup>''n''</sup>) is the group of (co)cycles [[modulo (jargon)#structures|modulo]] (co)boundaries in degree ''n'', that is,

::<math>H_n = \ker d_{n}/\mbox{im } d_{n+1} \quad \left(H^n = \ker d^{n}/\mbox{im } d^{n-1} \right)</math>

===Exact sequences===
{{main|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''<sub>''k''</sub>, ''A''<sub>''k''+1</sub>, ''A''<sub>''k''+2</sub> may be nonzero. For example, the following chain complex is a short exact sequence.
:<math>
\cdots
\xrightarrow{} \; 0 \;
\xrightarrow{} \; \mathbf{Z} \;
\xrightarrow{\times p} \; \mathbf{Z}
\twoheadrightarrow \mathbf{Z}/p\mathbf{Z} \;
\xrightarrow{} \; 0 \;
\xrightarrow{}
\cdots
</math>
In the middle group, the closed elements are the elements p'''Z'''; these are clearly the exact elements in this group.

===Chain maps===
A '''chain map''' ''f'' between two chain complexes <math>(A_\bullet, d_{A,\bullet})</math> and <math>(B_\bullet, d_{B,\bullet})</math> is a sequence <math>f_\bullet</math> of homomorphisms <math>f_n : A_n \rightarrow B_n</math> for each ''n'' that commutes with the boundary operators on the two chain complexes, so <math> d_{B,n} \circ f_n = f_{n-1} \circ d_{A,n}</math>. This is written out in the following [[commutative diagram]].
:[[Image:Chain map.svg|650 px|class=skin-invert]]

A chain map sends cycles to cycles and boundaries to boundaries, and thus induces a map on homology <math>(f_\bullet)_*:H_\bullet(A_\bullet, d_{A,\bullet}) \rightarrow H_\bullet(B_\bullet, d_{B,\bullet})</math>.

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''<sub>*</sub> between the singular homology of ''X'' and ''Y'' as well. When ''X'' and ''Y'' are both equal to the [[n-sphere|''n''-sphere]], the map induced on homology defines the [[Degree of a continuous mapping#From Sn to Sn|degree]] of the map ''f''.

The concept of chain map reduces to the one of boundary through the construction of the [[Mapping cone (homological algebra)|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 {{nowrap|''f'', ''g'' : ''A'' → ''B''}}, a '''chain homotopy''' is a sequence of homomorphisms {{nowrap|''h''<sub>''n''</sub> : ''A''<sub>''n''</sub> → ''B''<sub>''n''+1</sub>}} such that {{nowrap|1=''hd''<sub>''A''</sub> + ''d''<sub>''B''</sub>''h'' = ''f'' − ''g''}}. The maps may be written out in a diagram as follows, but this diagram is not commutative.

:[[Image:Chain homotopy between chain complexes.svg|650 px|class=skin-invert]]

The map ''hd''<sub>''A''</sub> + ''d''<sub>''B''</sub>''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]] between chain maps.

Let ''X'' and ''Y'' be topological spaces. In the case of singular homology, a [[homotopy]] between continuous maps {{nowrap|''f'', ''g'' : ''X'' → ''Y''}} 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.