Editing Homological algebra (section)

== Chain complexes and homology ==
{{main|Chain complex}}
The notion of [[chain complex]] is central in homological algebra. An abstract '''chain complex''' is a sequence <math>  (C_\bullet, d_\bullet)</math> of [[abelian group]]s and [[group homomorphism]]s, 
with the property that the composition of any two consecutive [[map (mathematics)|map]]s 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> 
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''d''<sub>''n''+1</sub> o ''d''<sub>''n''</sub> = 0 for all ''n''. 
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The elements of ''C''<sub>''n''</sub> are called ''n''-'''chains''' and the homomorphisms ''d''<sub>''n''</sub> are called the '''boundary maps''' or '''differentials'''. The '''chain groups''' ''C''<sub>''n''</sub> may be endowed with extra structure; for example, they may be [[vector space]]s or [[module (mathematics)|modules]] over a fixed [[ring (mathematics)|ring]] ''R''. The differentials must preserve the extra structure if it exists; for example, they must be [[linear map]]s or homomorphisms of ''R''-modules. For notational convenience, restrict attention to abelian groups (more correctly, to the [[category (mathematics)|category]] '''Ab''' of abelian groups); a celebrated [[Mitchell's embedding theorem|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''<sub>''n''</sub>&nbsp;=&nbsp;Ker ''d''<sub>''n''</sub> and the '''boundaries''' ''B''<sub>''n''</sub>&nbsp;=&nbsp;Im ''d''<sub>''n''+1</sub>, where Ker&nbsp;''d'' and Im&nbsp;''d'' denote the [[kernel (algebra)|kernel]] and the [[image (mathematics)|image]] of ''d''. Since the composition of two consecutive boundary maps is zero, these groups are embedded into each other as

: <math> B_n \subseteq Z_n \subseteq C_n. </math>

[[Subgroup]]s of abelian groups are automatically [[normal subgroup|normal]]; therefore we can define the ''n''th '''homology group''' ''H''<sub>''n''</sub>(''C'') as the [[factor group]] of the ''n''-cycles by the ''n''-boundaries,

: <math> H_n(C) = Z_n/B_n = \operatorname{Ker}\, d_n/ \operatorname{Im}\, d_{n+1}. </math>

A chain complex is called '''acyclic''' or an '''[[exact sequence]]''' if all its homology groups are zero.

Chain complexes arise in abundance in [[abstract algebra|algebra]] and [[algebraic topology]]. For example, if ''X'' is a [[topological space]] then the [[singular chain]]s ''C''<sub>''n''</sub>(''X'') are formal [[linear combination]]s of [[continuous map]]s from the standard ''n''-[[simplex]] into ''X''; if ''K'' is a [[simplicial complex]] then the [[Chain (algebraic topology)|simplicial chain]]s ''C''<sub>''n''</sub>(''K'') are formal linear combinations of the ''n''-simplices of ''K''; if ''A''&nbsp;=&nbsp;''F''/''R'' is a presentation of an abelian group ''A'' by [[Presentation of a group|generators and relations]], where ''F'' is a [[free abelian group]] spanned by the generators and ''R'' is the subgroup of relations, then letting ''C''<sub>1</sub>(''A'')&nbsp;=&nbsp;''R'', ''C''<sub>0</sub>(''A'')&nbsp;=&nbsp;''F'', and ''C''<sub>''n''</sub>(''A'')&nbsp;=&nbsp;0 for all other ''n'' defines a sequence of abelian groups. In all these cases, there are natural differentials ''d''<sub>''n''</sub> making  ''C''<sub>''n''</sub> 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]]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]], homological algebra studies the [[functor|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 <math>C_\bullet(X)</math> 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  <math>H_\bullet(C)</math> 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 (mathematics)|invariant]] of ''X''.