Editing Homology (mathematics) (section)

== Homology of chain complexes ==
To take the homology of a [[chain complex]], one starts with a '''chain complex,''' which is a sequence <math>  (C_\bullet, d_\bullet)</math> of [[Abelian group|abelian groups]] <math>C_{n}</math> (whose elements are called [[Chain (algebraic topology)|chains]]) and [[Group homomorphism|group homomorphisms]] <math>d_n</math> (called [[Boundary map|boundary maps]]) such that the composition of any two consecutive [[Map (mathematics)|maps]] is zero:
: <math> C_\bullet: \cdots \longrightarrow 
C_{n+1} \stackrel{d_{n+1}}{\longrightarrow}
C_n \stackrel{d_n}{\longrightarrow}
C_{n-1} \stackrel{d_{n-1}}{\longrightarrow}
\cdots, \quad d_n \circ d_{n+1}=0.</math><!--''d''<sub>''n''+1</sub> o ''d''<sub>''n''</sub> = 0 for all ''n''.-->

The <math>n</math>th homology group <math>H_{n}</math> of this chain complex is then the [[quotient group]] <math>H_n = Z_n/B_n</math> of cycles [[Quotient group|modulo]] boundaries, where the <math>n
</math>th group of '''cycles''' <math>Z_n</math> is given by the [[Kernel (algebra)|kernel]] subgroup <math>Z_n := \ker d_n :=\{c \in C_n \,|\; d_n(c) = 0\}</math>, and the <math>n</math>th group of '''boundaries''' <math>B_n</math> is given by the [[Image (mathematics)|image]] subgroup <math>B_n := \mathrm{im}\, d_{n+1} :=\{d_{n+1}(c)\,|\; c\in C_{n+1}\}</math>. One can optionally endow chain complexes with additional structure, for example by additionally taking the groups <math>C_n</math> to be [[Module (mathematics)|modules]] over a [[Ring (mathematics)|coefficient ring]] <math>R</math>, and taking the boundary maps <math>d_n</math> to be <math>R</math>-[[Module homomorphism|module homomorphisms]], resulting in homology groups <math>H_{n}</math> that are also [[Quotient module|quotient modules]]. Tools from [[homological algebra]] can be used to relate homology groups of different chain complexes.