View source for Complex (in homological algebra)

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One of the basic concepts of homological algebra. Let  $  A $
be an Abelian category. A graded object is a sequence  $  K = ( K _ {n} ) _ {n \in \mathbf Z }  $
of objects  $  K _ {n} $
in  $  A $. 
A sequence  $  \alpha = ( a _ {n} ) $
of morphisms  $  a _ {n} :  K _ {n} ^ { \prime } \rightarrow K _ {n} $
is called a morphism  $  a :  K ^ { \prime } \rightarrow K _ {n} $
of graded objects. One defines the object  $  K ( h) $
by setting  $  K ( h) _ {n} = K _ {n+} h $. 
A morphism of graded objects  $  K ^ { \prime } \rightarrow K ( h) $
is called a morphism of degree  $  h $
from  $  K ^ { \prime } $
into  $  K $. 
A graded object is said to be positive if  $  K _ {n} = 0 $
for all  $  n < 0 $, 
bounded from below if  $  K ( h) $
is positive for some  $  h $
and finite or bounded if  $  K _ {n} = 0 $
for all but a finite number of integers  $  n $. 
A chain complex in a category  $  A $
consists of a graded object  $  K $
and a morphism  $  d :  K \rightarrow K $
of degree  $  - 1 $
such that  $  d  ^ {2} = 0 $. 
More precisely:  $  d = ( d _ {n} ) $, 
where  $  d _ {n} :  K _ {n} \rightarrow K _ {n-} 1 $
and  $  d _ {n-} 1 d _ {n} = 0 $
for any  $  n $. 
A morphism of chain complexes

$$ 
( K ^ { \prime } , d ^ { \prime } )  \rightarrow  ( K , d )
$$

is a morphism  $  a :  K ^ { \prime } \rightarrow K $
of graded objects for which  $  a d ^ { \prime } = d a $. 
A cochain complex is defined in a dual manner (as a graded object with a morphism  $  d $
of degree  $  + 1 $).

Most frequently, complexes are considered in categories of Abelian groups, modules or sheaves of Abelian groups on a topological space. Thus, a complex of Abelian groups is a graded differential group the differential of which has degree  $  - 1 $
or  $  + 1 $.

Associated with each complex  $  K $
are the three graded objects:

the boundaries  $  B = B ( K) $, 
where  $  B _ {n} =  \mathop{\rm Im} ( K _ {n+} 1 \rightarrow ^ {d _ {n+} 1 } K _ {n} ) $;

the cycles  $  Z = Z ( K) $, 
where  $  Z _ {n} =  \mathop{\rm Ker} ( K _ {n} \rightarrow ^ {d _ {n} } K _ {n-} 1 ) $; 
and

the  $  n $-
dimensional homology objects (classes)  $  H = H ( K) $, 
where  $  H _ {n} = Z _ {n} / B _ {n} $(
see [[Homology of a complex|Homology of a complex]]).

For a cochain complex, the analogous objects are called coboundaries, cocycles and cohomology objects (notations  $  B  ^ {n} $, 
$  Z  ^ {n} $
and  $  H  ^ {n} $, 
respectively).

If  $  H ( K) = 0 $, 
then the complex  $  K $
is said to be acyclic.

A morphism  $  a :  K ^ { \prime } \rightarrow K $
of complexes induces morphisms

$$ 
Z ( K ^ { \prime } )  \rightarrow  Z ( K) ,\ \ 
B ( K ^ { \prime } )  \rightarrow  B ( K) ,
$$

and hence a homology or cohomology morphism

$$ 
H ( a) :  H ( K ^ { \prime } )  \rightarrow  H ( K) .
$$

Two morphisms  $  a , b :  K ^ { \prime } \rightarrow K $
are said to be homotopic (denoted by  $  a \simeq b $) 
if there is a morphism  $  s :  K ^ { \prime } \rightarrow K ( 1) $(
or  $  s :  K ^ { \prime } \rightarrow K ( - 1 ) $
for cochain complexes) of graded objects (called a homotopy), such that

$$ 
a - b  =  ds + sd  ^  \prime 
$$

(which implies that  $  H ( a) = H ( b) $). 
A complex  $  K $
is said to be contractible if  $  1 _ {K} \simeq 0 $, 
in which case the complex  $  K $
is acyclic.

If  $  0 \rightarrow K ^ { \prime } \rightarrow K \rightarrow K ^ { \prime\prime } \rightarrow 0 $
is an exact sequence of complexes, then there exists a connecting morphism  $  \partial  :  H ( K ^ { \prime } ) \rightarrow H ( K) $
of degree  $  - 1 $(
$  + 1 $) 
that is natural with respect to morphisms of exact sequences and is such that the long homology sequence (that is, the sequence

$$ 
\dots \rightarrow  H _ {n} ( K ^ { \prime } )  \rightarrow  H _ {n} ( K)  \rightarrow \ 
H _ {n} ( K ^ { \prime\prime } )   \mathop \rightarrow \limits ^  \partial  
$$

$$ 
 \mathop \rightarrow \limits ^  \partial    H _ {n-} 1 ( K ^ { \prime } )  \rightarrow  H _ {n-} 1 ( K)  \rightarrow  H _ {n-} 1 ( K ^ { \prime\prime } )  \rightarrow \dots
$$

for a chain complex, and the sequence

$$ 
\dots \rightarrow  H  ^ {n} ( K ^ { \prime } )  \rightarrow  H  ^ {n} ( K)  \rightarrow \ 
H  ^ {n} ( K ^ { \prime\prime } )   \mathop \rightarrow \limits ^  \partial  
$$

$$ 
 \mathop \rightarrow \limits ^  \partial    H  ^ {n+} 1 ( K ^ { \prime } )  \rightarrow  H  ^ {n+} 1 ( K)  \rightarrow  H  ^ {n+} 1 ( K ^ { \prime\prime } )  \rightarrow \dots
$$

for a cochain complex) is exact.

The cone of a morphism  $  a :  K ^ { \prime } \rightarrow K $
of chain complexes is the complex  $  MC ( a) $
defined as follows:

$$ 
MC ( A) _ {n}  =  K _ {n} \oplus K _ {n-} 1  ^  \prime 
$$

with

$$ 
d ( a) _ {n+} 1  = \ 
\left (

\begin{array}{cr}
d _ {n+} 1  &a _ {n}  \\
 0  &- d _ {n} ^ { \prime }  \\
\end{array}

\right )
:  MC ( a) _ {n+} 1  \rightarrow  MC ( a) _ {n} .
$$

The direct sum decomposition of the complex  $  MC ( a) $
leads to an exact sequence of complexes

$$ 
0  \rightarrow  K  \rightarrow  MC ( a)  \rightarrow  K ^ { \prime } ( - 1 )  \rightarrow  0 ,
$$

for which the associated long homology sequence is isomorphic to the sequence

$$ 
\dots \rightarrow  H _ {n} ( K)  \rightarrow  H _ {n} ( MC ( a) )  \rightarrow \ 
H _ {n-} 1 ( K ^ { \prime } )  \rightarrow ^ { {H _ n-} 1 ( a) }
$$

$$ 
\rightarrow ^ { {H _ n-} 1 ( a) }  H _ {n-} 1 ( K)  \rightarrow  H _ {n-} 1 ( MC ( a) )  \rightarrow \dots .
$$

Hence the chain complex  $  MC ( a) $
is acyclic if and only if  $  H ( a) $
is an isomorphism. Analogous notions and facts hold for cochain complexes.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Bass,   "Algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411080.png" />-theory" , Benjamin  (1968)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Cartan,   S. Eilenberg,   "Homological algebra" , Princeton Univ. Press  (1956)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P.J. Hilton,   U. Stammbach,   "A course in homological algebra" , Springer  (1971)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. MacLane,   "Homology" , Springer  (1963)</TD></TR></table>