Chain complexes and homology Homological algebra

the notion of chain complex central in homological algebra. abstract chain complex sequence



(

c

∙


,

d

∙


)


{\displaystyle (c_{\bullet },d_{\bullet })}

of abelian groups , group homomorphisms, property composition of 2 consecutive maps zero:

c

∙


:
⋯
⟶

c

n
+
1






⟶



d

n
+
1







c

n






⟶



d

n







c

n
−
1






⟶



d

n
−
1






⋯
,


d

n


∘

d

n
+
1


=
0.


{\displaystyle 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.}

the elements of cn called n-chains , homomorphisms dn called boundary maps or differentials. chain groups cn may endowed structure; example, may vector spaces or modules on fixed ring r. differentials must preserve structure if exists; example, must linear maps or homomorphisms of r-modules. notational convenience, restrict attention abelian groups (more correctly, category ab of abelian groups); celebrated theorem barry mitchell implies results generalize abelian category. every chain complex defines 2 further sequences of abelian groups, cycles zn = ker dn , boundaries bn = im dn+1, ker d , im d denote kernel , image of d. since composition of 2 consecutive boundary maps zero, these groups embedded each other as

b

n


⊆

z

n


⊆

c

n


.


{\displaystyle b_{n}\subseteq z_{n}\subseteq c_{n}.}

subgroups of abelian groups automatically normal; therefore can define nth homology group hn(c) factor group of n-cycles n-boundaries,

h

n


(
c
)
=

z

n



/


b

n


=
ker


d

n



/

im


d

n
+
1


.


{\displaystyle h_{n}(c)=z_{n}/b_{n}=\operatorname {ker} \,d_{n}/\operatorname {im} \,d_{n+1}.}

a chain complex called acyclic or exact sequence if homology groups zero.

chain complexes arise in abundance in algebra , algebraic topology. example, if x topological space singular chains cn(x) formal linear combinations of continuous maps standard n-simplex x; if k simplicial complex simplicial chains cn(k) formal linear combinations of n-simplices of k; if a = f/r presentation of abelian group generators , relations, f free abelian group spanned generators , r subgroup of relations, letting c1(a) = r, c0(a) = f, , cn(a) = 0 other n defines sequence of abelian groups. in these cases, there natural differentials dn making cn chain complex, homology reflects structure of topological space x, simplicial complex k, or abelian group a. in case of topological spaces, arrive @ notion of singular homology, plays fundamental role in investigating properties of such spaces, example, manifolds.

on philosophical level, homological algebra teaches chain complexes associated algebraic or geometric objects (topological spaces, simplicial complexes, r-modules) contain lot of valuable algebraic information them, homology being readily available part. on technical level, homological algebra provides tools manipulating complexes , extracting information. here 2 general illustrations.

two objects x , y connected map f between them. homological algebra studies relation, induced map f, between chain complexes associated x , y , homology. generalized case of several objects , maps connecting them. phrased in language of category theory, homological algebra studies functorial properties of various constructions of chain complexes , of homology of these complexes.
an object x admits multiple descriptions (for example, topological space , simplicial complex) or complex




c

∙


(
x
)


{\displaystyle c_{\bullet }(x)}

constructed using presentation of x, involves non-canonical choices. important know effect of change in description of x on chain complexes associated x. typically, complex , homology




h

∙


(
c
)


{\displaystyle h_{\bullet }(c)}

functorial respect presentation; , homology (although not complex itself) independent of presentation chosen, invariant of x.