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One of the basic concepts of homological algebra. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c0241101.png" /> be an Abelian category. A graded object is a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c0241102.png" /> of objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c0241103.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c0241104.png" />. A sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c0241105.png" /> of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c0241106.png" /> is called a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c0241107.png" /> of graded objects. One defines the object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c0241108.png" /> by setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c0241109.png" />. A morphism of graded objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411010.png" /> is called a morphism of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411012.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411013.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411014.png" />. A graded object is said to be positive if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411015.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411016.png" />, bounded from below if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411017.png" /> is positive for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411018.png" /> and finite or bounded if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411019.png" /> for all but a finite number of integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411020.png" />. A chain complex in a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411021.png" /> consists of a graded object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411022.png" /> and a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411023.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411024.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411025.png" />. More precisely: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411026.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411028.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411029.png" />. A morphism of chain complexes
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411030.png" /></td> </tr></table>
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is a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411031.png" /> of graded objects for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411032.png" />. A cochain complex is defined in a dual manner (as a graded object with a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411033.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411034.png" />).
+
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
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411035.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411036.png" />.
+
$$
 +
( K ^ { \prime } , d ^ { \prime } )  \rightarrow  ( K , d )
 +
$$
  
Associated with each complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411037.png" /> are the three graded objects:
+
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 $).
  
the boundaries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411038.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411039.png" />;
+
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 $.
  
the cycles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411040.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411041.png" />; and
+
Associated with each complex  $  K $
 +
are the three graded objects:
  
the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411042.png" />-dimensional homology objects (classes) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411043.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411044.png" /> (see [[Homology of a complex|Homology of a complex]]).
+
the boundaries  $  B = B ( K) $,  
 +
where $  B _ {n} = \mathop{\rm Im} ( K _ {n+} 1 \rightarrow ^ {d _ {n+} 1 } K _ {n} ) $;
  
For a cochain complex, the analogous objects are called coboundaries, cocycles and cohomology objects (notations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411047.png" />, respectively).
+
the cycles  $  Z = Z ( K) $,  
 +
where  $  Z _ {n} = \mathop{\rm Ker} ( K _ {n} \rightarrow ^ {d _ {n} } K _ {n-} 1 ) $;
 +
and
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411048.png" />, then the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411049.png" /> is said to be acyclic.
+
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]]).
  
A morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411050.png" /> of complexes induces morphisms
+
For a cochain complex, the analogous objects are called coboundaries, cocycles and cohomology objects (notations  $  B  ^ {n} $,
 +
$  Z  ^ {n} $
 +
and  $  H  ^ {n} $,
 +
respectively).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411051.png" /></td> </tr></table>
+
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
 
and hence a homology or cohomology morphism
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411052.png" /></td> </tr></table>
+
$$
 +
H ( a) : H ( K ^ { \prime } )  \rightarrow  H ( K) .
 +
$$
  
Two morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411053.png" /> are said to be homotopic (denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411054.png" />) if there is a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411055.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411056.png" /> for cochain complexes) of graded objects (called a homotopy), such that
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411057.png" /></td> </tr></table>
+
$$
 +
a - = ds + sd  ^  \prime
 +
$$
  
(which implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411058.png" />). A complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411059.png" /> is said to be contractible if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411060.png" />, in which case the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411061.png" /> is acyclic.
+
(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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411062.png" /> is an exact sequence of complexes, then there exists a connecting morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411063.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411064.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411065.png" />) that is natural with respect to morphisms of exact sequences and is such that the long homology sequence (that is, the sequence
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411066.png" /></td> </tr></table>
+
$$
 +
\dots \rightarrow  H _ {n} ( K ^ { \prime } )  \rightarrow  H _ {n} ( K)  \rightarrow \
 +
H _ {n} ( K ^ { \prime\prime } )  \mathop \rightarrow \limits ^  \partial 
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411067.png" /></td> </tr></table>
+
$$
 +
\mathop \rightarrow \limits ^  \partial    H _ {n-} 1 ( K ^ { \prime } )  \rightarrow  H _ {n-} 1 ( K)  \rightarrow  H _ {n-} 1 ( K ^ { \prime\prime } )  \rightarrow \dots
 +
$$
  
 
for a chain complex, and the sequence
 
for a chain complex, and the sequence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411068.png" /></td> </tr></table>
+
$$
 +
\dots \rightarrow  H  ^ {n} ( K ^ { \prime } )  \rightarrow  H  ^ {n} ( K)  \rightarrow \
 +
H  ^ {n} ( K ^ { \prime\prime } )  \mathop \rightarrow \limits ^  \partial 
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411069.png" /></td> </tr></table>
+
$$
 +
\mathop \rightarrow \limits ^  \partial    H  ^ {n+} 1 ( K ^ { \prime } )  \rightarrow  H  ^ {n+} 1 ( K)  \rightarrow  H  ^ {n+} 1 ( K ^ { \prime\prime } )  \rightarrow \dots
 +
$$
  
 
for a cochain complex) is exact.
 
for a cochain complex) is exact.
  
The cone of a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411070.png" /> of chain complexes is the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411071.png" /> defined as follows:
+
The cone of a morphism $  a : K ^ { \prime } \rightarrow K $
 +
of chain complexes is the complex $  MC ( a) $
 +
defined as follows:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411072.png" /></td> </tr></table>
+
$$
 +
MC ( A) _ {n}  = K _ {n} \oplus K _ {n-} 1  ^  \prime
 +
$$
  
 
with
 
with
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411073.png" /></td> </tr></table>
+
$$
 +
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411074.png" /> leads to an exact sequence of complexes
+
The direct sum decomposition of the complex $  MC ( a) $
 +
leads to an exact sequence of complexes
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411075.png" /></td> </tr></table>
+
$$
 +
0  \rightarrow  K  \rightarrow  MC ( a)  \rightarrow  K ^ { \prime } ( - 1 )  \rightarrow  0 ,
 +
$$
  
 
for which the associated long homology sequence is isomorphic to the sequence
 
for which the associated long homology sequence is isomorphic to the sequence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411076.png" /></td> </tr></table>
+
$$
 +
\dots \rightarrow  H _ {n} ( K)  \rightarrow  H _ {n} ( MC ( a) )  \rightarrow \
 +
H _ {n-} 1 ( K ^ { \prime } )  \rightarrow ^ { {H _ n-} 1 ( a) }
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411077.png" /></td> </tr></table>
+
$$
 +
\rightarrow ^ { {H _ n-} 1 ( a) }  H _ {n-} 1 ( K)  \rightarrow  H _ {n-} 1 ( MC ( a) )  \rightarrow \dots .
 +
$$
  
Hence the chain complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411078.png" /> is acyclic if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024110/c02411079.png" /> is an isomorphism. Analogous notions and facts hold for cochain complexes.
+
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====
 
====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>
 
<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>

Revision as of 17:46, 4 June 2020


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).

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

[1] H. Bass, "Algebraic -theory" , Benjamin (1968)
[2] H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956)
[3] P.J. Hilton, U. Stammbach, "A course in homological algebra" , Springer (1971)
[4] S. MacLane, "Homology" , Springer (1963)
How to Cite This Entry:
Complex (in homological algebra). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complex_(in_homological_algebra)&oldid=14272
This article was adapted from an original article by A.V. Mikhalev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article