Namespaces
Variants
Actions

Difference between revisions of "Cohomology"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
 
(2 intermediate revisions by 2 users not shown)
Line 1: Line 1:
 +
<!--
 +
c0230601.png
 +
$#A+1 = 181 n = 0
 +
$#C+1 = 181 : ~/encyclopedia/old_files/data/C023/C.0203060 Cohomology
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
A term used with respect to functors of a homological nature that, in contrast to homology, depend contravariantly, as a rule, on the objects of the basic category on which they are defined. In contrast to homology, connecting homomorphisms in exact cohomology sequences raise the dimension. In typical situations, cohomology occurs simultaneously with the corresponding homology.
 
A term used with respect to functors of a homological nature that, in contrast to homology, depend contravariantly, as a rule, on the objects of the basic category on which they are defined. In contrast to homology, connecting homomorphisms in exact cohomology sequences raise the dimension. In typical situations, cohomology occurs simultaneously with the corresponding homology.
  
Line 6: Line 18:
 
This is a graded group
 
This is a graded group
  
<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/c023/c023060/c0230601.png" /></td> </tr></table>
+
$$
 +
H  ^ {*} ( X , G )  = \
 +
\sum _ {n \geq  0 }
 +
H  ^ {n} ( X , G )
 +
$$
  
associated with a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c0230602.png" /> and an Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c0230603.png" />. The notion of cohomology is dual to that of homology (see [[Homology theory|Homology theory]]; [[Homology group|Homology group]]; [[Aleksandrov–Čech homology and cohomology|Aleksandrov–Čech homology and cohomology]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c0230604.png" /> is a ring, then a natural product is defined in the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c0230605.png" /> (Kolmogorov–Alexander product or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c0230607.png" />-product), converting this group into a graded ring (cohomology ring). In the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c0230608.png" /> is a differentiable manifold, the cohomology ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c0230609.png" /> can be calculated by means of differential forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306010.png" /> (see [[De Rham theorem|de Rham theorem]]).
+
associated with a topological space $  X $
 +
and an Abelian group $  G $.  
 +
The notion of cohomology is dual to that of homology (see [[Homology theory|Homology theory]]; [[Homology group|Homology group]]; [[Aleksandrov–Čech homology and cohomology|Aleksandrov–Čech homology and cohomology]]). If $  G $
 +
is a ring, then a natural product is defined in the group $  H  ^ {*} ( X , G ) $(
 +
Kolmogorov–Alexander product or $  \cup $-
 +
product), converting this group into a graded ring (cohomology ring). In the case when $  X $
 +
is a differentiable manifold, the cohomology ring $  H  ^ {*} ( X , \mathbf R ) $
 +
can be calculated by means of differential forms on $  X $(
 +
see [[De Rham theorem|de Rham theorem]]).
  
 
==Cohomology with values in a sheaf of Abelian groups.==
 
==Cohomology with values in a sheaf of Abelian groups.==
 
This is a generalization of ordinary cohomology of a topological space. There are two cohomology theories with values (or coefficients) in sheaves of Abelian groups: Čech cohomology and Grothendieck cohomology.
 
This is a generalization of ordinary cohomology of a topological space. There are two cohomology theories with values (or coefficients) in sheaves of Abelian groups: Čech cohomology and Grothendieck cohomology.
  
Čech cohomology. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306011.png" /> be a topological space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306012.png" /> a sheaf of Abelian groups on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306014.png" /> an open covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306015.png" />. Then by an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306016.png" />-dimensional cochain of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306017.png" /> one means a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306018.png" /> that associates with each ordered set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306019.png" /> such that
+
Čech cohomology. Let $  X $
 +
be a topological space, $  {\mathcal F} $
 +
a sheaf of Abelian groups on $  X $
 +
and $  \mathfrak U = \{ U _ {i} \} _ {i \in I }  $
 +
an open covering of $  X $.  
 +
Then by an $  N $-
 +
dimensional cochain of $  \mathfrak U $
 +
one means a mapping $  f $
 +
that associates with each ordered set $  i _ {0} \dots i _ {n} \in I $
 +
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/c023/c023060/c02306020.png" /></td> </tr></table>
+
$$
 +
U _ {i _ {0}  \dots i _ {n} } \
 +
= U _ {i _ {0}  } \cap \dots
 +
\cap U _ {i _ {n}  }
 +
\neq  \emptyset ,
 +
$$
  
a section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306021.png" /> of the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306022.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306023.png" />. The set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306024.png" />-dimensional cochains, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306025.png" />, is an Abelian group (with respect to addition). The coboundary operator
+
a section $  f _ {i _ {0}  \dots i _ {n} } $
 +
of the sheaf $  {\mathcal F} $
 +
over $  U _ {i _ {0}  \dots i _ {n} } $.  
 +
The set of all $  n $-
 +
dimensional cochains, $  C  ^ {n} ( \mathfrak U , {\mathcal F} ) $,  
 +
is an Abelian group (with respect to addition). The coboundary operator
  
<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/c023/c023060/c02306026.png" /></td> </tr></table>
+
$$
 +
\delta _ {n} : C  ^ {n} ( \mathfrak U , {\mathcal F} ) \
 +
\rightarrow  C  ^ {n+1} ( \mathfrak U , {\mathcal F} )
 +
$$
  
 
is defined as follows:
 
is 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/c023/c023060/c02306027.png" /></td> </tr></table>
+
$$
 +
( \delta _ {n} f ) _ {i _ {0}  \dots i _ {n+1} } \
 +
= \sum _ {j=0} ^ { n+1}
 +
( - 1 )  ^ {j}
 +
f _ {i _ {0}  \dots \widehat{i}  _ {j} \dots i _ {n+1} } ,
 +
$$
  
where the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306028.png" /> denotes that the corresponding index should be omitted.
+
where the symbol $  \widehat{ {}}  $
 +
denotes that the corresponding index should be omitted.
  
 
The sequence
 
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/c023/c023060/c02306029.png" /></td> </tr></table>
+
$$
 +
C  ^ {*} ( \mathfrak U , {\mathcal F} ) :\
 +
C  ^ {0} ( \mathfrak U , {\mathcal F} )
 +
\rightarrow ^ { {\delta _ 1} } \
 +
C _ {1} ( \mathfrak U , {\mathcal F} )
 +
\rightarrow ^ { {\delta _ 2} } \dots
 +
$$
  
is a complex (the Čech complex). The cohomology of this complex is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306030.png" /> and is called the Čech cohomology of the covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306031.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306032.png" />. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306033.png" /> is the same as the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306034.png" /> of sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306035.png" />. In calculating the cohomology, the Čech complex can be replaced by its subcomplex consisting of the alternating cochains, that is, cochains that change sign on permutation of two indices and are equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306036.png" /> when two indices are equal.
+
is a complex (the Čech complex). The cohomology of this complex is denoted by $  H  ^ {n} ( \mathfrak U , {\mathcal F} ) $
 +
and is called the Čech cohomology of the covering $  \mathfrak U $
 +
with values in $  {\mathcal F} $.  
 +
The group $  H  ^ {0} ( \mathfrak U , {\mathcal F} ) $
 +
is the same as the group $  \Gamma ( X , {\mathcal F} ) $
 +
of sections of $  {\mathcal F} $.  
 +
In calculating the cohomology, the Čech complex can be replaced by its subcomplex consisting of the alternating cochains, that is, cochains that change sign on permutation of two indices and are equal to 0 $
 +
when two indices are equal.
  
If the covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306037.png" /> is a refinement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306038.png" />, that is, for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306039.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306040.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306041.png" />, then a canonical homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306042.png" /> is defined which does not depend on the refinement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306043.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306044.png" />-dimensional Čech cohomology group of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306045.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306046.png" /> is now defined by the formula:
+
If the covering $  \mathfrak U $
 +
is a refinement of $  \mathfrak V = \{ V _ {j} \} $,  
 +
that is, for each $  i \in I $
 +
there exists a $  \tau ( i) \in J $
 +
such that $  U _ {i} \subseteq V _ {\tau ( i) }  $,  
 +
then a canonical homomorphism $  H  ^ {n} ( \mathfrak V , {\mathcal F} ) \rightarrow H  ^ {n} ( \mathfrak U , {\mathcal F} ) $
 +
is defined which does not depend on the refinement $  \tau $.  
 +
The $  n $-
 +
dimensional Čech cohomology group of the space $  X $
 +
with values in $  {\mathcal F} $
 +
is now defined by the formula:
  
<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/c023/c023060/c02306047.png" /></td> </tr></table>
+
$$
 +
\check{H}  {}  ^ {n} ( X , F  )  = \
 +
\lim\limits _  \rightarrow  H  ^ {n}
 +
( \mathfrak U , {\mathcal F} ) ,
 +
$$
  
 
where the inductive limit is taken over the directed (with respect to refinement) set of equivalence classes of open coverings (two coverings being equivalent if and only if each is a refinement of the other). The definition of Čech cohomology is also applicable to pre-sheaves.
 
where the inductive limit is taken over the directed (with respect to refinement) set of equivalence classes of open coverings (two coverings being equivalent if and only if each is a refinement of the other). The definition of Čech cohomology is also applicable to pre-sheaves.
  
A disadvantage of Čech cohomology is that (for non-paracompact spaces) it does not form a cohomology functor (see [[Homology functor|Homology functor]]). In the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306048.png" /> is the constant sheaf corresponding to the Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306049.png" />, the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306050.png" /> are the same as the Aleksandrov–Čech cohomology groups with coefficients in the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306051.png" />.
+
A disadvantage of Čech cohomology is that (for non-paracompact spaces) it does not form a cohomology functor (see [[Homology functor|Homology functor]]). In the case when $  {\mathcal F} $
 +
is the constant sheaf corresponding to the Abelian group $  {\mathcal F} $,  
 +
the groups $  \check{H}  {}  ^ {n} ( X , {\mathcal F} ) $
 +
are the same as the Aleksandrov–Čech cohomology groups with coefficients in the group $  {\mathcal F} $.
  
Grothendieck cohomology. One considers the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306052.png" /> from the category of sheaves of Abelian groups on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306053.png" /> to the category of Abelian groups. The right derived functors (cf. [[Derived functor|Derived functor]]) of this functor are called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306055.png" />-dimensional Grothendieck cohomology groups with values in the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306056.png" /> and are denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306057.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306058.png" />. Corresponding to an exact sequence of sheaves of Abelian groups
+
Grothendieck cohomology. One considers the functor $  {\mathcal F} \rightarrow \Gamma ( X , {\mathcal F} ) $
 +
from the category of sheaves of Abelian groups on $  X $
 +
to the category of Abelian groups. The right derived functors (cf. [[Derived functor|Derived functor]]) of this functor are called the $  n $-
 +
dimensional Grothendieck cohomology groups with values in the sheaf $  {\mathcal F} $
 +
and are denoted by $  H  ^ {n} ( X , {\mathcal F} ) $,
 +
$  n = 0 , 1 ,\dots $.  
 +
Corresponding to an exact sequence of sheaves of Abelian groups
  
<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/c023/c023060/c02306059.png" /></td> </tr></table>
+
$$
 +
0 \rightarrow  {\mathcal F} _ {1}  \rightarrow  {\mathcal F} _ {2}  \rightarrow  {\mathcal F} _ {3}  \rightarrow  0
 +
$$
  
 
there is an exact sequence
 
there is an exact 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/c023/c023060/c02306060.png" /></td> </tr></table>
+
$$
 +
\dots \rightarrow  H  ^ {n-1} ( X , {\mathcal F} _ {3} )  \rightarrow \
 +
H  ^ {n} ( X , {\mathcal F} _ {1} )  \rightarrow  H  ^ {n} ( X , {\mathcal F} _ {2} ) \rightarrow
 +
$$
  
<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/c023/c023060/c02306061.png" /></td> </tr></table>
+
$$
 +
\rightarrow \
 +
H  ^ {n} ( X , {\mathcal F} _ {3} )  \rightarrow  H  ^ {n+1} ( X , {\mathcal F} _ {1} )  \rightarrow \dots ,
 +
$$
  
that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306062.png" /> forms a cohomology functor. Furthermore, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306063.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306064.png" /> is a [[Flabby sheaf|flabby sheaf]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306065.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306066.png" />). These three properties of Grothendieck cohomology characterize the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306067.png" /> uniquely up to an isomorphism.
+
that is, $  \{ H  ^ {n} ( X , {\mathcal F} ) \} _ {n = 0 , 1 ,\dots }  $
 +
forms a cohomology functor. Furthermore, $  H  ^ {0} ( X , {\mathcal F} ) = \Gamma ( X , {\mathcal F}) $.  
 +
If $  {\mathcal F} $
 +
is a [[Flabby sheaf|flabby sheaf]], $  H  ^ {n} ( X , {\mathcal F} ) = 0 $(
 +
$  n > 0 $).  
 +
These three properties of Grothendieck cohomology characterize the functor $  {\mathcal F} \mapsto \{ H  ^ {n} ( X , {\mathcal F} ) \} _ {n = 0 , 1 ,\dots }  $
 +
uniquely up to an isomorphism.
  
For the calculation of the Grothendieck cohomology of the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306068.png" /> one can use the left resolution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306069.png" /> consisting of sheaves the Grothendieck cohomology of which vanishes in positive dimensions. For example, on arbitrary topological spaces one can take the resolution by flabby sheaves, and on paracompact spaces, the resolution by the soft or fine sheaves (cf. [[Fine sheaf|Fine sheaf]]; [[Soft sheaf|Soft sheaf]]).
+
For the calculation of the Grothendieck cohomology of the sheaf $  {\mathcal F} $
 +
one can use the left resolution of $  {\mathcal F} $
 +
consisting of sheaves the Grothendieck cohomology of which vanishes in positive dimensions. For example, on arbitrary topological spaces one can take the resolution by flabby sheaves, and on paracompact spaces, the resolution by the soft or fine sheaves (cf. [[Fine sheaf|Fine sheaf]]; [[Soft sheaf|Soft sheaf]]).
  
Grothendieck cohomology is related to cohomology of coverings in the following way. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306070.png" /> be an open covering of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306071.png" />. Then there exists a spectral sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306072.png" /> converging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306073.png" /> and such that
+
Grothendieck cohomology is related to cohomology of coverings in the following way. Let $  \mathfrak U = \{ U _ {i} \} _ {i \in I }  $
 +
be an open covering of the space $  X $.  
 +
Then there exists a spectral sequence $  \{ E _ {r}  ^ {p,q} \} $
 +
converging to $  \{ H  ^ {n} ( X , {\mathcal F} ) \} $
 +
and 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/c023/c023060/c02306074.png" /></td> </tr></table>
+
$$
 +
E _ {2}  ^ {p,q}  = \
 +
H  ^ {p} ( \mathfrak U , {\mathcal H}  ^ {q} ( X , {\mathcal F} ) ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306075.png" /> is the pre-sheaf associating the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306076.png" /> with the open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306077.png" />. If the cohomology of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306078.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306079.png" /> vanishes in positive dimensions, then the sequence is degenerate and
+
where $  {\mathcal H}  ^ {q} ( X , {\mathcal F} ) $
 +
is the pre-sheaf associating the group $  H  ^ {q} ( V , {\mathcal F} ) $
 +
with the open set $  V \subset  X $.  
 +
If the cohomology of all $  U _ {i _ {0}  \dots i _ {n} } $
 +
with values in $  {\mathcal F} $
 +
vanishes in positive dimensions, then the sequence is degenerate and
  
<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/c023/c023060/c02306080.png" /></td> </tr></table>
+
$$
 +
H  ^ {n} ( \mathfrak U , {\mathcal F} )  \simeq  H  ^ {n} ( X , {\mathcal F} ) ,\ \
 +
n = 0 , 1 ,\dots
 +
$$
  
 
(Leray's theorem). In the general case the spectral sequence defines a functorial homomorphism
 
(Leray's theorem). In the general case the spectral sequence defines a functorial homomorphism
  
<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/c023/c023060/c02306081.png" /></td> </tr></table>
+
$$
 +
H  ^ {n} ( \mathfrak U , {\mathcal F} )  \rightarrow  H  ^ {n} ( X , {\mathcal F} )
 +
$$
  
 
and, on passing to the limit, a functorial homomorphism
 
and, on passing to the limit, a functorial homomorphism
  
<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/c023/c023060/c02306082.png" /></td> </tr></table>
+
$$
 +
\check{H}  {}  ^ {n} ( X , {\mathcal F} )  \rightarrow  H  ^ {n} ( X , {\mathcal F} ) .
 +
$$
  
The latter homomorphism is bijective for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306083.png" />, injective (but not, in general, surjective) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306084.png" /> and, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306085.png" /> is paracompact, bijective for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306086.png" />. Thus, for a paracompact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306087.png" />,
+
The latter homomorphism is bijective for $  n = 0 , 1 $,
 +
injective (but not, in general, surjective) for $  n = 2 $
 +
and, when $  X $
 +
is paracompact, bijective for all $  n $.  
 +
Thus, for a paracompact space $  X $,
  
<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/c023/c023060/c02306088.png" /></td> </tr></table>
+
$$
 +
\check{H}  {}  ^ {n} ( X , {\mathcal F} )  \simeq  H  ^ {n} ( X , {\mathcal F} ) ,\ \
 +
n = 0 , 1 ,\dots .
 +
$$
  
A generalization of the cohomology groups defined above are the cohomology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306089.png" /> with supports in a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306090.png" />. A family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306091.png" /> of closed subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306092.png" /> is called a family of supports if: 1) any closed subset of a member of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306093.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306094.png" />; and 2) the union of any two members of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306095.png" /> is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306096.png" />. The groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306097.png" /> are defined as the right derived functors of the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306098.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c02306099.png" /> is the group of sections of the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060100.png" /> with supports in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060101.png" />. They form a cohomology functor. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060102.png" /> is the family of all closed sets, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060103.png" />. Another important special case: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060104.png" />, the family of all compact subsets. The groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060105.png" /> are called the cohomology groups with compact supports.
+
A generalization of the cohomology groups defined above are the cohomology groups $  H _  \Phi  ^ {n} ( X , {\mathcal F} ) $
 +
with supports in a family $  \Phi $.  
 +
A family $  \Phi $
 +
of closed subsets of $  X $
 +
is called a family of supports if: 1) any closed subset of a member of $  \Phi $
 +
belongs to $  \Phi $;  
 +
and 2) the union of any two members of $  \Phi $
 +
is in $  \Phi $.  
 +
The groups $  H _  \Phi  ^ {n} ( X , {\mathcal F} ) $
 +
are defined as the right derived functors of the functor $  {\mathcal F} \mapsto \Gamma _  \Phi  ( X , {\mathcal F} ) $,  
 +
where $  \Gamma _  \Phi  ( X , {\mathcal F} ) = H  ^ {0} ( X , {\mathcal F} ) $
 +
is the group of sections of the sheaf $  {\mathcal F} $
 +
with supports in $  \Phi $.  
 +
They form a cohomology functor. If $  \Phi $
 +
is the family of all closed sets, then $  H _  \Phi  ^ {n} ( X , {\mathcal F} ) = H  ^ {n} ( X , {\mathcal F} ) $.  
 +
Another important special case: $  \Phi = c $,  
 +
the family of all compact subsets. The groups $  H _ {c}  ^ {n} ( X , {\mathcal F} ) $
 +
are called the cohomology groups with compact supports.
  
In the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060106.png" /> is a sheaf of rings, the group
+
In the case when $  {\mathcal F} $
 +
is a sheaf of rings, the group
  
<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/c023/c023060/c023060107.png" /></td> </tr></table>
+
$$
 +
H  ^ {*} ( X , {\mathcal F} )  = \
 +
\sum _ {n \geq  0 }
 +
H  ^ {n} ( X , {\mathcal F} )
 +
$$
  
has a naturally defined multiplication, converting it into a graded ring (a cohomology ring). Here, associativity in the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060108.png" /> implies associativity of multiplication in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060109.png" />, while a sheaf of commutative rings or Lie rings gives rise to a graded commutative or Lie cohomology ring, respectively. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060110.png" /> is a sheaf of modules over a sheaf of rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060111.png" />, then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060112.png" /> are modules over the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060113.png" />.
+
has a naturally defined multiplication, converting it into a graded ring (a cohomology ring). Here, associativity in the sheaf $  {\mathcal F} $
 +
implies associativity of multiplication in $  H  ^ {*} ( X , {\mathcal F} ) $,  
 +
while a sheaf of commutative rings or Lie rings gives rise to a graded commutative or Lie cohomology ring, respectively. If $  {\mathcal F} $
 +
is a sheaf of modules over a sheaf of rings $  {\mathcal A} $,  
 +
then the $  H  ^ {n} ( X , {\mathcal F} ) $
 +
are modules over the ring $  \Gamma ( X , {\mathcal A} ) $.
  
 
Concerning cohomology with values in a sheaf of non-Abelian groups see [[Non-Abelian cohomology|Non-Abelian cohomology]].
 
Concerning cohomology with values in a sheaf of non-Abelian groups see [[Non-Abelian cohomology|Non-Abelian cohomology]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Grothendieck,   "Sur quelques points d'algèbre homologique"  ''Tôhoku Math. J.'' , '''9'''  (1957)  pp. 119–221</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Godement,   "Topologie algébrique et théorie des faisceaux" , Hermann  (1958)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.-P. Serre,   "Faiseaux algébriques cohérentes"  ''Ann. of Math. (2)'' , '''61''' :  2  (1955)  pp. 197–278</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  A. Grothendieck, "Sur quelques points d'algèbre homologique"  ''Tôhoku Math. J.'' , '''9'''  (1957)  pp. 119–221</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  R. Godement, "Topologie algébrique et théorie des faisceaux" , Hermann  (1958)</TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top">  J.-P. Serre, "Faisceaux algébriques cohérents"  ''Ann. of Math. (2)'' , '''61''' :  2  (1955)  pp. 197–278</TD></TR>
 +
</table>
  
 
''D.A. Ponomarev''
 
''D.A. Ponomarev''
  
 
====Comments====
 
====Comments====
See [[Singular homology|Singular homology]] for a description of singular homology.
+
See [[Singular homology]] for a description of singular homology.
  
 
====References====
 
====References====
Line 97: Line 258:
  
 
==Cohomology of spaces with operators.==
 
==Cohomology of spaces with operators.==
Cohomological invariants of a topological space with a group action defined on it. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060114.png" /> be a group acting on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060115.png" />, where for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060116.png" /> the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060117.png" /> is a homeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060118.png" />. Then by a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060119.png" />-sheaf of Abelian groups on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060120.png" /> one means a sheaf of Abelian groups on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060121.png" /> together with an action of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060122.png" /> which is continuous, compatible with the action on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060123.png" /> and which maps stalks of the sheaf isomorphically onto one another. A natural <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060124.png" />-module structure is defined on the group of sections of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060125.png" />-sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060126.png" /> (and more generally on the cohomology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060127.png" />). The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060128.png" />-sheaves of Abelian groups on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060129.png" /> form an Abelian category, each object of which admits an imbedding into an injective object. The functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060130.png" /> from this category into the category of Abelian groups, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060131.png" /> is the group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060132.png" />-invariant sections of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060133.png" />-sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060134.png" />, has right derived functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060135.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060136.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060137.png" />, which constitute a cohomology functor. The groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060138.png" /> play a fundamental role in the study of the connection between the cohomology of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060139.png" />, the quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060140.png" /> and the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060141.png" />. There exists a spectral sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060142.png" /> with second term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060143.png" /> and converging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060144.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060145.png" /> be the sheaf of invariants of the direct image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060146.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060147.png" /> being the natural projection) regarded as a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060148.png" />-sheaf on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060149.png" /> on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060150.png" /> acts trivially. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060151.png" /> acts properly discontinuously and freely on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060152.png" /> (see [[Discrete group of transformations|Discrete group of transformations]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060153.png" /> (see [[#References|[1]]]). In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060154.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060155.png" />-module, then the constant sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060156.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060157.png" /> has a natural <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060159.png" />-sheaf structure and the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060160.png" /> is locally constant on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060161.png" />. In this case the spectral sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060162.png" /> satisfies the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060163.png" /> and converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060164.png" /> (spectral sequence of a covering). If, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060165.png" /> is connected and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060166.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060167.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060168.png" />, which gives a topological interpretation of the cohomology of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060169.png" /> [[#References|[2]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060170.png" /> is properly discontinuous and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060171.png" /> is paracompact, then the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060172.png" /> can be calculated in the same way as Čech cohomology, by means of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060173.png" />-invariant coverings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060174.png" /> (see [[#References|[1]]]).
+
Cohomological invariants of a topological space with a group action defined on it. Let $  G $
 +
be a group acting on the space $  X $,  
 +
where for each $  g \in G $
 +
the mapping $  x \mapsto g x $
 +
is a homeomorphism $  X \rightarrow X $.  
 +
Then by a $  G $-
 +
sheaf of Abelian groups on $  X $
 +
one means a sheaf of Abelian groups on $  X $
 +
together with an action of the group $  G $
 +
which is continuous, compatible with the action on $  X $
 +
and which maps stalks of the sheaf isomorphically onto one another. A natural $  G $-
 +
module structure is defined on the group of sections of a $  G $-
 +
sheaf $  {\mathcal F} $(
 +
and more generally on the cohomology groups $  H  ^ {n} ( X , {\mathcal F} ) $).  
 +
The $  G $-
 +
sheaves of Abelian groups on $  X $
 +
form an Abelian category, each object of which admits an imbedding into an injective object. The functor $  {\mathcal F} \mapsto \Gamma ( X , {\mathcal F} )  ^ {G} $
 +
from this category into the category of Abelian groups, where $  \Gamma ( X , {\mathcal F} )  ^ {G} $
 +
is the group of $  G $-
 +
invariant sections of the $  G $-
 +
sheaf $  {\mathcal F} $,  
 +
has right derived functors $  {\mathcal F} \mapsto H  ^ {n} ( X , G , {\mathcal F} ) $
 +
$  ( n = 0 , 1 ,\dots ) $,
 +
where $  H  ^ {0} ( X , G , {\mathcal F} ) = \Gamma ( X , {\mathcal F} )  ^ {G} $,  
 +
which constitute a cohomology functor. The groups $  H  ^ {n} ( X , G , {\mathcal F} ) $
 +
play a fundamental role in the study of the connection between the cohomology of the space $  X $,  
 +
the quotient space $  Y = X / G $
 +
and the group $  G $.  
 +
There exists a spectral sequence $  \{ E _ {r} \} $
 +
with second term $  E _ {2}  ^ {p,q} = H  ^ {p} ( G , H  ^ {q} ( X , {\mathcal F} ) ) $
 +
and converging to $  H  ^ {*} ( X , G , {\mathcal F} ) $.  
 +
Let $  {\mathcal F}  ^ {G} $
 +
be the sheaf of invariants of the direct image $  f _ {*} {\mathcal F} $(
 +
$  f : X \rightarrow Y $
 +
being the natural projection) regarded as a $  G $-
 +
sheaf on the space $  Y $
 +
on which $  G $
 +
acts trivially. If $  G $
 +
acts properly discontinuously and freely on $  X $(
 +
see [[Discrete group of transformations|Discrete group of transformations]]), then $  H  ^ {*} ( X , G , {\mathcal F} ) \simeq H  ^ {*} ( Y , {\mathcal F} )  ^ {G} $(
 +
see [[#References|[1]]]). In particular, if $  A $
 +
is a $  G $-
 +
module, then the constant sheaf $  {\mathcal F} = A $
 +
on $  X $
 +
has a natural $  G $-
 +
sheaf structure and the sheaf $  {\mathcal F}  ^ {G} $
 +
is locally constant on $  Y $.  
 +
In this case the spectral sequence $  \{ E _ {r} \} $
 +
satisfies the condition $  E _ {2}  ^ {p,q} = H  ^ {p} ( G , H  ^ {q} ( X , A )) $
 +
and converges to $  H  ^ {*} ( Y , {\mathcal F}  ^ {G} ) $(
 +
spectral sequence of a covering). If, moreover, $  X $
 +
is connected and $  H  ^ {q} ( X , A ) = 0 $
 +
for  $  q > 0 $,  
 +
then $  H  ^ {p} ( G , A ) \simeq H  ^ {p} ( Y , {\mathcal F}  ^ {G} ) $,  
 +
which gives a topological interpretation of the cohomology of the group $  G $[[#References|[2]]]. If $  G $
 +
is properly discontinuous and $  Y $
 +
is paracompact, then the groups $  H  ^ {n} ( X , G , {\mathcal F} ) $
 +
can be calculated in the same way as Čech cohomology, by means of $  G $-
 +
invariant coverings of $  X $(
 +
see [[#References|[1]]]).
  
In the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060175.png" /> is a Lie group acting freely and differentiably on a differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060176.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060177.png" /> is a differentiable manifold, the analogue <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060178.png" /> of the spectral sequence of the covering is well-known [[#References|[3]]]. The sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060179.png" /> converges to the cohomology of the complex of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060180.png" />-invariant differential forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060181.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060182.png" />, where the cohomology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060183.png" /> is calculated by means of cochains of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023060/c023060184.png" />.
+
In the case when $  G $
 +
is a Lie group acting freely and differentiably on a differentiable manifold $  X $,  
 +
where $  X / G $
 +
is a differentiable manifold, the analogue $  \{ \widetilde{E}  _ {r} \} $
 +
of the spectral sequence of the covering is well-known [[#References|[3]]]. The sequence $  \{ \widetilde{E}  _ {r} \} $
 +
converges to the cohomology of the complex of $  G $-
 +
invariant differential forms on $  X $
 +
and $  \widetilde{E}  {} _ {2}  ^ {p,q} = H  ^ {p} ( G , H  ^ {q} ( X , \mathbf R ) ) $,  
 +
where the cohomology of $  G $
 +
is calculated by means of cochains of class $  C  ^  \infty  $.
  
 
See also [[Cohomology of groups|Cohomology of groups]]; [[Equivariant cohomology|Equivariant cohomology]].
 
See also [[Cohomology of groups|Cohomology of groups]]; [[Equivariant cohomology|Equivariant cohomology]].

Latest revision as of 06:20, 21 July 2024


A term used with respect to functors of a homological nature that, in contrast to homology, depend contravariantly, as a rule, on the objects of the basic category on which they are defined. In contrast to homology, connecting homomorphisms in exact cohomology sequences raise the dimension. In typical situations, cohomology occurs simultaneously with the corresponding homology.

E.G. Sklyarenko

Cohomology of a topological space.

This is a graded group

$$ H ^ {*} ( X , G ) = \ \sum _ {n \geq 0 } H ^ {n} ( X , G ) $$

associated with a topological space $ X $ and an Abelian group $ G $. The notion of cohomology is dual to that of homology (see Homology theory; Homology group; Aleksandrov–Čech homology and cohomology). If $ G $ is a ring, then a natural product is defined in the group $ H ^ {*} ( X , G ) $( Kolmogorov–Alexander product or $ \cup $- product), converting this group into a graded ring (cohomology ring). In the case when $ X $ is a differentiable manifold, the cohomology ring $ H ^ {*} ( X , \mathbf R ) $ can be calculated by means of differential forms on $ X $( see de Rham theorem).

Cohomology with values in a sheaf of Abelian groups.

This is a generalization of ordinary cohomology of a topological space. There are two cohomology theories with values (or coefficients) in sheaves of Abelian groups: Čech cohomology and Grothendieck cohomology.

Čech cohomology. Let $ X $ be a topological space, $ {\mathcal F} $ a sheaf of Abelian groups on $ X $ and $ \mathfrak U = \{ U _ {i} \} _ {i \in I } $ an open covering of $ X $. Then by an $ N $- dimensional cochain of $ \mathfrak U $ one means a mapping $ f $ that associates with each ordered set $ i _ {0} \dots i _ {n} \in I $ such that

$$ U _ {i _ {0} \dots i _ {n} } \ = U _ {i _ {0} } \cap \dots \cap U _ {i _ {n} } \neq \emptyset , $$

a section $ f _ {i _ {0} \dots i _ {n} } $ of the sheaf $ {\mathcal F} $ over $ U _ {i _ {0} \dots i _ {n} } $. The set of all $ n $- dimensional cochains, $ C ^ {n} ( \mathfrak U , {\mathcal F} ) $, is an Abelian group (with respect to addition). The coboundary operator

$$ \delta _ {n} : C ^ {n} ( \mathfrak U , {\mathcal F} ) \ \rightarrow C ^ {n+1} ( \mathfrak U , {\mathcal F} ) $$

is defined as follows:

$$ ( \delta _ {n} f ) _ {i _ {0} \dots i _ {n+1} } \ = \sum _ {j=0} ^ { n+1} ( - 1 ) ^ {j} f _ {i _ {0} \dots \widehat{i} _ {j} \dots i _ {n+1} } , $$

where the symbol $ \widehat{ {}} $ denotes that the corresponding index should be omitted.

The sequence

$$ C ^ {*} ( \mathfrak U , {\mathcal F} ) :\ C ^ {0} ( \mathfrak U , {\mathcal F} ) \rightarrow ^ { {\delta _ 1} } \ C _ {1} ( \mathfrak U , {\mathcal F} ) \rightarrow ^ { {\delta _ 2} } \dots $$

is a complex (the Čech complex). The cohomology of this complex is denoted by $ H ^ {n} ( \mathfrak U , {\mathcal F} ) $ and is called the Čech cohomology of the covering $ \mathfrak U $ with values in $ {\mathcal F} $. The group $ H ^ {0} ( \mathfrak U , {\mathcal F} ) $ is the same as the group $ \Gamma ( X , {\mathcal F} ) $ of sections of $ {\mathcal F} $. In calculating the cohomology, the Čech complex can be replaced by its subcomplex consisting of the alternating cochains, that is, cochains that change sign on permutation of two indices and are equal to $ 0 $ when two indices are equal.

If the covering $ \mathfrak U $ is a refinement of $ \mathfrak V = \{ V _ {j} \} $, that is, for each $ i \in I $ there exists a $ \tau ( i) \in J $ such that $ U _ {i} \subseteq V _ {\tau ( i) } $, then a canonical homomorphism $ H ^ {n} ( \mathfrak V , {\mathcal F} ) \rightarrow H ^ {n} ( \mathfrak U , {\mathcal F} ) $ is defined which does not depend on the refinement $ \tau $. The $ n $- dimensional Čech cohomology group of the space $ X $ with values in $ {\mathcal F} $ is now defined by the formula:

$$ \check{H} {} ^ {n} ( X , F ) = \ \lim\limits _ \rightarrow H ^ {n} ( \mathfrak U , {\mathcal F} ) , $$

where the inductive limit is taken over the directed (with respect to refinement) set of equivalence classes of open coverings (two coverings being equivalent if and only if each is a refinement of the other). The definition of Čech cohomology is also applicable to pre-sheaves.

A disadvantage of Čech cohomology is that (for non-paracompact spaces) it does not form a cohomology functor (see Homology functor). In the case when $ {\mathcal F} $ is the constant sheaf corresponding to the Abelian group $ {\mathcal F} $, the groups $ \check{H} {} ^ {n} ( X , {\mathcal F} ) $ are the same as the Aleksandrov–Čech cohomology groups with coefficients in the group $ {\mathcal F} $.

Grothendieck cohomology. One considers the functor $ {\mathcal F} \rightarrow \Gamma ( X , {\mathcal F} ) $ from the category of sheaves of Abelian groups on $ X $ to the category of Abelian groups. The right derived functors (cf. Derived functor) of this functor are called the $ n $- dimensional Grothendieck cohomology groups with values in the sheaf $ {\mathcal F} $ and are denoted by $ H ^ {n} ( X , {\mathcal F} ) $, $ n = 0 , 1 ,\dots $. Corresponding to an exact sequence of sheaves of Abelian groups

$$ 0 \rightarrow {\mathcal F} _ {1} \rightarrow {\mathcal F} _ {2} \rightarrow {\mathcal F} _ {3} \rightarrow 0 $$

there is an exact sequence

$$ \dots \rightarrow H ^ {n-1} ( X , {\mathcal F} _ {3} ) \rightarrow \ H ^ {n} ( X , {\mathcal F} _ {1} ) \rightarrow H ^ {n} ( X , {\mathcal F} _ {2} ) \rightarrow $$

$$ \rightarrow \ H ^ {n} ( X , {\mathcal F} _ {3} ) \rightarrow H ^ {n+1} ( X , {\mathcal F} _ {1} ) \rightarrow \dots , $$

that is, $ \{ H ^ {n} ( X , {\mathcal F} ) \} _ {n = 0 , 1 ,\dots } $ forms a cohomology functor. Furthermore, $ H ^ {0} ( X , {\mathcal F} ) = \Gamma ( X , {\mathcal F}) $. If $ {\mathcal F} $ is a flabby sheaf, $ H ^ {n} ( X , {\mathcal F} ) = 0 $( $ n > 0 $). These three properties of Grothendieck cohomology characterize the functor $ {\mathcal F} \mapsto \{ H ^ {n} ( X , {\mathcal F} ) \} _ {n = 0 , 1 ,\dots } $ uniquely up to an isomorphism.

For the calculation of the Grothendieck cohomology of the sheaf $ {\mathcal F} $ one can use the left resolution of $ {\mathcal F} $ consisting of sheaves the Grothendieck cohomology of which vanishes in positive dimensions. For example, on arbitrary topological spaces one can take the resolution by flabby sheaves, and on paracompact spaces, the resolution by the soft or fine sheaves (cf. Fine sheaf; Soft sheaf).

Grothendieck cohomology is related to cohomology of coverings in the following way. Let $ \mathfrak U = \{ U _ {i} \} _ {i \in I } $ be an open covering of the space $ X $. Then there exists a spectral sequence $ \{ E _ {r} ^ {p,q} \} $ converging to $ \{ H ^ {n} ( X , {\mathcal F} ) \} $ and such that

$$ E _ {2} ^ {p,q} = \ H ^ {p} ( \mathfrak U , {\mathcal H} ^ {q} ( X , {\mathcal F} ) ) , $$

where $ {\mathcal H} ^ {q} ( X , {\mathcal F} ) $ is the pre-sheaf associating the group $ H ^ {q} ( V , {\mathcal F} ) $ with the open set $ V \subset X $. If the cohomology of all $ U _ {i _ {0} \dots i _ {n} } $ with values in $ {\mathcal F} $ vanishes in positive dimensions, then the sequence is degenerate and

$$ H ^ {n} ( \mathfrak U , {\mathcal F} ) \simeq H ^ {n} ( X , {\mathcal F} ) ,\ \ n = 0 , 1 ,\dots $$

(Leray's theorem). In the general case the spectral sequence defines a functorial homomorphism

$$ H ^ {n} ( \mathfrak U , {\mathcal F} ) \rightarrow H ^ {n} ( X , {\mathcal F} ) $$

and, on passing to the limit, a functorial homomorphism

$$ \check{H} {} ^ {n} ( X , {\mathcal F} ) \rightarrow H ^ {n} ( X , {\mathcal F} ) . $$

The latter homomorphism is bijective for $ n = 0 , 1 $, injective (but not, in general, surjective) for $ n = 2 $ and, when $ X $ is paracompact, bijective for all $ n $. Thus, for a paracompact space $ X $,

$$ \check{H} {} ^ {n} ( X , {\mathcal F} ) \simeq H ^ {n} ( X , {\mathcal F} ) ,\ \ n = 0 , 1 ,\dots . $$

A generalization of the cohomology groups defined above are the cohomology groups $ H _ \Phi ^ {n} ( X , {\mathcal F} ) $ with supports in a family $ \Phi $. A family $ \Phi $ of closed subsets of $ X $ is called a family of supports if: 1) any closed subset of a member of $ \Phi $ belongs to $ \Phi $; and 2) the union of any two members of $ \Phi $ is in $ \Phi $. The groups $ H _ \Phi ^ {n} ( X , {\mathcal F} ) $ are defined as the right derived functors of the functor $ {\mathcal F} \mapsto \Gamma _ \Phi ( X , {\mathcal F} ) $, where $ \Gamma _ \Phi ( X , {\mathcal F} ) = H ^ {0} ( X , {\mathcal F} ) $ is the group of sections of the sheaf $ {\mathcal F} $ with supports in $ \Phi $. They form a cohomology functor. If $ \Phi $ is the family of all closed sets, then $ H _ \Phi ^ {n} ( X , {\mathcal F} ) = H ^ {n} ( X , {\mathcal F} ) $. Another important special case: $ \Phi = c $, the family of all compact subsets. The groups $ H _ {c} ^ {n} ( X , {\mathcal F} ) $ are called the cohomology groups with compact supports.

In the case when $ {\mathcal F} $ is a sheaf of rings, the group

$$ H ^ {*} ( X , {\mathcal F} ) = \ \sum _ {n \geq 0 } H ^ {n} ( X , {\mathcal F} ) $$

has a naturally defined multiplication, converting it into a graded ring (a cohomology ring). Here, associativity in the sheaf $ {\mathcal F} $ implies associativity of multiplication in $ H ^ {*} ( X , {\mathcal F} ) $, while a sheaf of commutative rings or Lie rings gives rise to a graded commutative or Lie cohomology ring, respectively. If $ {\mathcal F} $ is a sheaf of modules over a sheaf of rings $ {\mathcal A} $, then the $ H ^ {n} ( X , {\mathcal F} ) $ are modules over the ring $ \Gamma ( X , {\mathcal A} ) $.

Concerning cohomology with values in a sheaf of non-Abelian groups see Non-Abelian cohomology.

References

[1] A. Grothendieck, "Sur quelques points d'algèbre homologique" Tôhoku Math. J. , 9 (1957) pp. 119–221
[2] R. Godement, "Topologie algébrique et théorie des faisceaux" , Hermann (1958)
[3] J.-P. Serre, "Faisceaux algébriques cohérents" Ann. of Math. (2) , 61 : 2 (1955) pp. 197–278

D.A. Ponomarev

Comments

See Singular homology for a description of singular homology.

References

[a1] J.-P. Serre, "Homologie singulière des espaces fibrés. Applications" Ann. of Math. , 54 (1951) pp. 425–505
[a2] N.E. Steenrod, S. Eilenberg, "Foundations of algebraic topology" , Princeton Univ. Press (1966)
[a3] E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) pp. Chapts. 4; 5
[a4] A. Dold, "Lectures on algebraic topology" , Springer (1980)

Cohomology of spaces with operators.

Cohomological invariants of a topological space with a group action defined on it. Let $ G $ be a group acting on the space $ X $, where for each $ g \in G $ the mapping $ x \mapsto g x $ is a homeomorphism $ X \rightarrow X $. Then by a $ G $- sheaf of Abelian groups on $ X $ one means a sheaf of Abelian groups on $ X $ together with an action of the group $ G $ which is continuous, compatible with the action on $ X $ and which maps stalks of the sheaf isomorphically onto one another. A natural $ G $- module structure is defined on the group of sections of a $ G $- sheaf $ {\mathcal F} $( and more generally on the cohomology groups $ H ^ {n} ( X , {\mathcal F} ) $). The $ G $- sheaves of Abelian groups on $ X $ form an Abelian category, each object of which admits an imbedding into an injective object. The functor $ {\mathcal F} \mapsto \Gamma ( X , {\mathcal F} ) ^ {G} $ from this category into the category of Abelian groups, where $ \Gamma ( X , {\mathcal F} ) ^ {G} $ is the group of $ G $- invariant sections of the $ G $- sheaf $ {\mathcal F} $, has right derived functors $ {\mathcal F} \mapsto H ^ {n} ( X , G , {\mathcal F} ) $ $ ( n = 0 , 1 ,\dots ) $, where $ H ^ {0} ( X , G , {\mathcal F} ) = \Gamma ( X , {\mathcal F} ) ^ {G} $, which constitute a cohomology functor. The groups $ H ^ {n} ( X , G , {\mathcal F} ) $ play a fundamental role in the study of the connection between the cohomology of the space $ X $, the quotient space $ Y = X / G $ and the group $ G $. There exists a spectral sequence $ \{ E _ {r} \} $ with second term $ E _ {2} ^ {p,q} = H ^ {p} ( G , H ^ {q} ( X , {\mathcal F} ) ) $ and converging to $ H ^ {*} ( X , G , {\mathcal F} ) $. Let $ {\mathcal F} ^ {G} $ be the sheaf of invariants of the direct image $ f _ {*} {\mathcal F} $( $ f : X \rightarrow Y $ being the natural projection) regarded as a $ G $- sheaf on the space $ Y $ on which $ G $ acts trivially. If $ G $ acts properly discontinuously and freely on $ X $( see Discrete group of transformations), then $ H ^ {*} ( X , G , {\mathcal F} ) \simeq H ^ {*} ( Y , {\mathcal F} ) ^ {G} $( see [1]). In particular, if $ A $ is a $ G $- module, then the constant sheaf $ {\mathcal F} = A $ on $ X $ has a natural $ G $- sheaf structure and the sheaf $ {\mathcal F} ^ {G} $ is locally constant on $ Y $. In this case the spectral sequence $ \{ E _ {r} \} $ satisfies the condition $ E _ {2} ^ {p,q} = H ^ {p} ( G , H ^ {q} ( X , A )) $ and converges to $ H ^ {*} ( Y , {\mathcal F} ^ {G} ) $( spectral sequence of a covering). If, moreover, $ X $ is connected and $ H ^ {q} ( X , A ) = 0 $ for $ q > 0 $, then $ H ^ {p} ( G , A ) \simeq H ^ {p} ( Y , {\mathcal F} ^ {G} ) $, which gives a topological interpretation of the cohomology of the group $ G $[2]. If $ G $ is properly discontinuous and $ Y $ is paracompact, then the groups $ H ^ {n} ( X , G , {\mathcal F} ) $ can be calculated in the same way as Čech cohomology, by means of $ G $- invariant coverings of $ X $( see [1]).

In the case when $ G $ is a Lie group acting freely and differentiably on a differentiable manifold $ X $, where $ X / G $ is a differentiable manifold, the analogue $ \{ \widetilde{E} _ {r} \} $ of the spectral sequence of the covering is well-known [3]. The sequence $ \{ \widetilde{E} _ {r} \} $ converges to the cohomology of the complex of $ G $- invariant differential forms on $ X $ and $ \widetilde{E} {} _ {2} ^ {p,q} = H ^ {p} ( G , H ^ {q} ( X , \mathbf R ) ) $, where the cohomology of $ G $ is calculated by means of cochains of class $ C ^ \infty $.

See also Cohomology of groups; Equivariant cohomology.

References

[1] A. Grothendieck, "Sur quelques points d'algèbre homologique" Tôhoku Math. J. , 9 (1957) pp. 119–221
[2] H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956)
[3] W.T. van Est, "A generalization of the Cartan–Leray spectral sequence I, II" Proc. Nederl. Akad. Wetensch. Ser. A , 61 (1958) pp. 399–413

A.L. OnishchikD.A. Ponomarev

How to Cite This Entry:
Cohomology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cohomology&oldid=12935
This article was adapted from an original article by E.G. Sklyarenko, D.A. Ponomarev, A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article