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The groups
 
The 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/c023110/c0231101.png" /></td> </tr></table>
+
$$
 +
H  ^ {n} ( R, A)  = \
 +
\mathrm{Ext} _ {R}  ^ {n}
 +
( K, A),\ \
 +
n \geq  0
 +
$$
  
(see [[Functor|Functor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c0231102.png" />), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c0231103.png" /> is an associative algebra over a commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c0231104.png" /> with a fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c0231105.png" />-algebra homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c0231106.png" /> (augmentation) enabling one to regard <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c0231107.png" /> as an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c0231108.png" />-module, and where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c0231109.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311010.png" />-module. This definition encompasses many cohomology theories of certain types of (universal) algebras.
+
(see [[Functor|Functor]] $  \mathop{\rm Ext} $),  
 +
where $  R $
 +
is an associative algebra over a commutative ring $  K $
 +
with a fixed $  K $-algebra homomorphism $  \epsilon : R \rightarrow K $ (augmentation) enabling one to regard $  K $
 +
as an $  R $-module, and where $  A $
 +
is an $  R $-module. This definition encompasses many cohomology theories of certain types of (universal) algebras.
  
 
Cohomology groups of groups in all dimensions were introduced in the 1940s first by S. Eilenberg and S. MacLane [[#References|[3]]] in connection with topological investigations, and by D.K. Faddeev [[#References|[5]]] from a purely algebraic point of view — as groups of classes of generalized quotient systems. Cohomology groups in small dimensions were studied earlier in one form or another (see [[#References|[1]]], [[#References|[2]]], [[#References|[4]]]).
 
Cohomology groups of groups in all dimensions were introduced in the 1940s first by S. Eilenberg and S. MacLane [[#References|[3]]] in connection with topological investigations, and by D.K. Faddeev [[#References|[5]]] from a purely algebraic point of view — as groups of classes of generalized quotient systems. Cohomology groups in small dimensions were studied earlier in one form or another (see [[#References|[1]]], [[#References|[2]]], [[#References|[4]]]).
Line 9: Line 31:
 
==Examples of cohomology groups.==
 
==Examples of cohomology groups.==
  
 +
1) If  $  K = \mathbf Z $
 +
is the ring of integers,  $  G $
 +
is a group,  $  R = \mathbf Z G $
 +
is the group ring of  $  G $
 +
over  $  \mathbf Z $,
 +
and
  
1) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311011.png" /> is the ring of integers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311012.png" /> is a group, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311013.png" /> is the group ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311014.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311015.png" />, and
+
$$
 +
\epsilon \left (
 +
\sum n _ {i} g _ {i} \right ) = \
 +
\sum n _ {i} ,\ \
 +
n _ {i} \in \mathbf Z ,\ \
 +
g _ {i} \in G,
 +
$$
  
<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/c023110/c02311016.png" /></td> </tr></table>
+
then the groups  $  H  ^ {n} ( R, A) $
 +
are called the cohomology groups of the group  $  G $
 +
with coefficients (or values) in the  $  R $-module  $  A $;  
 +
they are denoted by  $  H  ^ {n} ( G, A) $.  
 +
Instead of a group  $  G $
 +
one can consider a monoid  $  G $,
 +
and obtain the analogous cohomology groups  $  H  ^ {n} ( G, A) $
 +
of the monoid  $  G $.
  
then the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311017.png" /> are called the cohomology groups of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311018.png" /> with coefficients (or values) in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311019.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311020.png" />; they are denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311021.png" />. Instead of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311022.png" /> one can consider a monoid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311023.png" />, and obtain the analogous cohomology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311024.png" /> of the monoid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311025.png" />.
+
2) If  $  S $
 +
is an associative  $  K $-algebra, $  S  ^ {0} $
 +
is the [[opposite algebra|opposite]]  $  K $-algebra and
  
2) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311026.png" /> is an associative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311027.png" />-algebra, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311028.png" /> is the opposite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311029.png" />-algebra and
+
$$
 +
= S \otimes _ {K} S  ^ {0} ,\ \
 +
\epsilon \left (
 +
\sum s _ {i} \otimes t _ {i} \right )  = \
 +
\sum s _ {i} t _ {i} ,
 +
$$
  
<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/c023110/c02311030.png" /></td> </tr></table>
+
then the groups  $  \mathrm{Ext} _ {R}  ^ {n} ( S, A) $
 +
are called the cohomology groups of the associative algebra  $  S $
 +
with coefficients in the  $  S $-bimodule  $  A $ (that is, in the  $  R $-module  $  A $);  
 +
they are denoted by  $  H  ^ {n} ( S, A) $.  
 +
If  $  K $
 +
is a field, then the groups  $  H  ^ {n} ( S, A) $
 +
are called the Hochschild cohomology groups of the  $  K $-algebra  $  S $.
  
then the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311031.png" /> are called the cohomology groups of the associative algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311032.png" /> with coefficients in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311033.png" />-bimodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311034.png" /> (that is, in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311035.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311036.png" />); they are denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311037.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311038.png" /> is a field, then the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311039.png" /> are called the Hochschild cohomology groups of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311040.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311041.png" />.
+
3) If  $  S $
 +
is a Lie algebra over a field  $  K $
 +
and  $  R = U _ {S} $
 +
is its [[Universal enveloping algebra|universal enveloping algebra]] with augmentation  $  \epsilon :  R \rightarrow K $,
 +
then the groups $  H  ^ {n} ( R, A) $
 +
are called the cohomology groups of the Lie algebra $  S $
 +
with coefficients in the $  U _ {S} $-module  $  A $ (that is, in the Lie  $  S $-module $  A $);  
 +
they are denoted by $  H  ^ {n} ( S, A) $.
  
3) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311042.png" /> is a Lie algebra over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311044.png" /> is its [[Universal enveloping algebra|universal enveloping algebra]] with augmentation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311045.png" />, then the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311046.png" /> are called the cohomology groups of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311047.png" /> with coefficients in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311048.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311049.png" /> (that is, in the Lie <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311050.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311051.png" />); they are denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311052.png" />.
+
The cohomology groups for  $  n = 0, 1 $
 +
and 2 have, in a number of cases, simple interpretations.
  
The cohomology groups for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311053.png" /> and 2 have, in a number of cases, simple interpretations.
+
a) If  $  G $
 +
is a group, then  $  H  ^ {0} ( G, A) $
 +
is isomorphic to the group
  
a) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311054.png" /> is a group, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311055.png" /> is isomorphic to the group
+
$$
 +
A  ^ {G}  = \
 +
\{ {a \in A } : {ga = a \textrm{ for }  \textrm{ all }  g \in G } \}
 +
$$
  
<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/c023110/c02311056.png" /></td> </tr></table>
+
of fixed elements; $  H  ^ {1} ( G, A) $
 
+
is isomorphic to the quotient group $  \mathop{\rm Der} ( G, A)/ \mathop{\rm Ider} ( G, A) $,  
of fixed elements; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311057.png" /> is isomorphic to the quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311058.png" />, where
+
where
  
<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/c023110/c02311059.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Der} ( G, A)  = \
 +
\{ {f : G \rightarrow A } : {f ( xy) = xf ( y) + f ( x)
 +
\textrm{ for }  \textrm{ all }  x, y \in G } \}
 +
$$
  
 
is the group of derivations (or crossed homomorphisms),
 
is the group of derivations (or crossed homomorphisms),
  
<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/c023110/c02311060.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Ider} ( G, A)  = \
 +
\{ {f: G \rightarrow A } : {\exists a \in A  ( f ( x) = xa - a
 +
\textrm{ for }  \textrm{ all }  x \in G ) } \}
 +
$$
  
 
is the group of inner derivations (or principal crossed homomorphisms); here, the sequence
 
is the group of inner derivations (or principal crossed homomorphisms); here, 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/c023110/c02311061.png" /></td> </tr></table>
+
$$
 +
0 \rightarrow  A  ^ {G}  \rightarrow  A  \rightarrow  \mathop{\rm Der} ( G, A)  \rightarrow \
 +
H  ^ {1} ( G, A)  \rightarrow  0
 +
$$
  
is exact; for an Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311063.png" /> is isomorphic to the group of extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311064.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311065.png" /> (see [[Baer multiplication|Baer multiplication]]); the third cohomology group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311066.png" /> is connected with obstructions to extensions (see [[#References|[9]]], Chapt. IV).
+
is exact; for an Abelian group $  G $,  
 +
$  H  ^ {2} ( G, A) $
 +
is isomorphic to the group of extensions of $  A $
 +
by $  G $ (see [[Baer multiplication|Baer multiplication]]); the third cohomology group of $  G $
 +
is connected with obstructions to extensions (see [[#References|[9]]], Chapt. IV).
  
b) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311067.png" /> is an associative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311068.png" />-algebra, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311069.png" /> is isomorphic to the group
+
b) If $  S $
 +
is an associative $  K $-algebra, then $  H  ^ {0} ( S, A) $
 +
is isomorphic to 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/c023110/c02311070.png" /></td> </tr></table>
+
$$
 +
\{ {a \in A } : {xa = ax  \textrm{ for }  \textrm{ all }  x \in S } \}
 +
;
 +
$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311071.png" /> is isomorphic to the quotient group
+
$  H  ^ {1} ( S, A) $
 +
is isomorphic to the quotient 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/c023110/c02311072.png" /></td> </tr></table>
+
$$
 +
{ \mathop{\rm Der} ( S, A) } / { \mathop{\rm Ider} ( S, A) } ,
 +
$$
  
 
where
 
where
  
<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/c023110/c02311073.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Der} ( S, A)  = \
 +
\{ {f: S \rightarrow A } : {f  \textrm{ is }  K-\textrm{linear} \
 +
\textrm{ and }  f ( xy) = xf ( y) + f ( x) y  \textrm{ for }  x, y \in S } \} ,
 +
$$
  
<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/c023110/c02311074.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Ider} ( S, A)  = \{ {f:  S \rightarrow A } : {\exists a \in A  ( f ( x) = xa - ax \
 +
\textrm{ for }  \textrm{ all }  x \in S ) } \} ;
 +
$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311075.png" /> describes the extensions of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311076.png" />-bimodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311077.png" /> by the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311078.png" /> (see [[#References|[14]]]).
+
$  H  ^ {2} ( S, A) $
 +
describes the extensions of the $  S $-bimodule $  A $
 +
by the ring $  S $ (see [[#References|[14]]]).
  
c) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311079.png" /> is a Lie algebra, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311080.png" /> is isomorphic to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311081.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311082.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311083.png" /> is isomorphic to the quotient group
+
c) If $  S $
 +
is a Lie algebra, then $  H  ^ {0} ( S, A) $
 +
is isomorphic to the $  K $-module $  \{ {a \in A } : {xa = 0 \textrm{ for  all  }  x \in S } \} $;  
 +
$  H  ^ {1} ( S, A) $
 +
is isomorphic to the quotient 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/c023110/c02311084.png" /></td> </tr></table>
+
$$
 +
{ \mathop{\rm Der} ( S, A) } / { \mathop{\rm Ider} ( S, A) } ,
 +
$$
  
 
where
 
where
  
<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/c023110/c02311085.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Der} ( S, A)  = \
 +
\{ {f:  S \rightarrow A } : {f ([ x, y]) = xf ( y) - yf ( x)
 +
\textrm{ for }  \textrm{ all }  x, y \in S } \} ,
 +
$$
  
<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/c023110/c02311086.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Ider} ( S, A)  = \{ {f: S \rightarrow A } : {\exists a \in A  ( f ( x) = xa
 +
\textrm{ for }  \textrm{ all }  x \in S ) }\} ;
 +
$$
  
the second cohomology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311087.png" /> of a Lie algebra corresponds to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311088.png" />-split extensions of Lie algebras (see [[#References|[6]]], Chapt. XIV); in certain cases the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311089.png" /> are obstructions in the extension problem.
+
the second cohomology group $  H  ^ {2} ( S, A) $
 +
of a Lie algebra corresponds to the $  K $-split extensions of Lie algebras (see [[#References|[6]]], Chapt. XIV); in certain cases the elements of $  H  ^ {3} ( S, A) $
 +
are obstructions in the extension problem.
  
Cohomology groups find extensive application in various branches of algebra. E.g. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311090.png" /> is a group and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311091.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311092.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311093.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311094.png" /> is free (Stalling's theorem, see [[Homological dimension|Homological dimension]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311095.png" /> is a finite group and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311096.png" /> is the multiplicative group of the complex field, then the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311097.png" /> is called the Schur multiplier of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311098.png" />. It plays an important role in the study of central extensions of groups and in the theory of projective representations of finite groups [[#References|[1]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c02311099.png" /> is a group, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110100.png" /> a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110101.png" />-module and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110102.png" /> for a prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110103.png" />, then
+
Cohomology groups find extensive application in various branches of algebra. E.g. if $  G $
 +
is a group and $  H  ^ {2} ( G, A) = 0 $
 +
for all $  \mathbf Z G $-modules $  A $,  
 +
then $  G $
 +
is free (Stalling's theorem, see [[Homological dimension|Homological dimension]]). If $  G $
 +
is a finite group and $  \mathbf C  ^ {*} $
 +
is the multiplicative group of the complex field, then the group $  M ( G) = H  ^ {2} ( G, \mathbf C  ^ {*} ) $
 +
is called the Schur multiplier of $  G $.  
 +
It plays an important role in the study of central extensions of groups and in the theory of projective representations of finite groups [[#References|[1]]]. If $  G $
 +
is a group, $  A $
 +
a $  \mathbf Z G $-module and $  pA = 0 $
 +
for a prime number $  p $,  
 +
then
  
<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/c023110/c023110104.png" /></td> </tr></table>
+
$$
 +
\mathrm{Ext} _ {\mathbf Z G }  ^ {n}
 +
( \mathbf Z , A)  \cong \
 +
\mathrm{Ext} _ {kG}  ^ {n} ( k, A),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110105.png" /> is the field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110106.png" /> elements. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110107.png" /> is a finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110108.png" />-group, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110109.png" /> is the minimum number of generators of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110110.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110111.png" /> is the minimum number of defining relations for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110112.png" /> considered as a pro-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110113.png" />-group; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110114.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110115.png" /> is the minimum number of defining relations of the discrete group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110116.png" />. The fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110117.png" /> tends to infinity as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110118.png" /> leads to a negative solution of the class field tower problem (cf. [[Class field theory|Class field theory]]), the Kurosh problem on nil algebras (cf. [[Nil algebra|Nil algebra]]) and the unrestricted [[Burnside problem|Burnside problem]] [[#References|[10]]].
+
where $  k = \mathop{\rm GF} ( p) $
 +
is the field of $  p $
 +
elements. If $  G $
 +
is a finite $  p $-group, then $  d ( G) = \mathrm{dim} _ {k}  H  ^ {1} ( G, k) $
 +
is the minimum number of generators of $  G $,  
 +
and $  r ( G) = \mathrm{dim} _ {k}  H  ^ {2} ( G, k) $
 +
is the minimum number of defining relations for $  G $
 +
considered as a pro- $  p $-group; $  r ( G) \leq  R ( G) $,  
 +
where $  R ( G) $
 +
is the minimum number of defining relations of the discrete group $  G $.  
 +
The fact that $  r ( G) - d ( G) $
 +
tends to infinity as $  d ( G) \rightarrow \infty $
 +
leads to a negative solution of the class field tower problem (cf. [[Class field theory|Class field theory]]), the Kurosh problem on nil algebras (cf. [[Nil algebra|Nil algebra]]) and the unrestricted [[Burnside problem|Burnside problem]] [[#References|[10]]].
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110119.png" /> is a profinite group and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110120.png" /> is the family of its open normal subgroups, then the group
+
If $  G $
 +
is a profinite group and $  \{ {U _ {i} } : {i \in I } \} $
 +
is the family of its open normal subgroups, then 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/c023110/c023110121.png" /></td> </tr></table>
+
$$
 +
\lim\limits _  \rightarrow  H  ^ {n} ( G / U _ {i} , A ^ {U _ {i} } )
 +
$$
  
is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110122.png" />-th cohomology group of the profinite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110123.png" /> with coefficients in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110124.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110125.png" />; it is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110126.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110127.png" /> is a Galois extension of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110128.png" /> with Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110129.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110130.png" /> is profinite; in this case the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110131.png" /> are called Galois cohomology groups. An important role is played by the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110132.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110133.png" /> is the multiplicative group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110134.png" />. E.g. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110135.png" />, and a corollary of this fact is Hilbert's theorem 90 (on cyclic extensions). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110136.png" /> is the separable closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110137.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110138.png" /> is called the Brauer group of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110139.png" /> (see [[Brauer group|Brauer group]]). At present (1987) a Galois theory of commutative rings is being developed in which an essential part is played by the Galois cohomology of commutative rings and by the Brauer group.
+
is called the $  n $-th cohomology group of the profinite group $  G $
 +
with coefficients in the $  \mathbf Z G $-module $  A $;  
 +
it is denoted by $  H  ^ {n} ( G, A) $.  
 +
If $  E $
 +
is a Galois extension of a field $  L $
 +
with Galois group $  G = G ( E/L) $,  
 +
then $  G $
 +
is profinite; in this case the groups $  H  ^ {n} ( G, A) $
 +
are called Galois cohomology groups. An important role is played by the groups $  H  ^ {q} ( G, E  ^ {*} ) $
 +
where $  E  ^ {*} $
 +
is the multiplicative group of $  E $.  
 +
E.g. $  H  ^ {1} ( G, E  ^ {*} ) = 0 $,  
 +
and a corollary of this fact is Hilbert's theorem 90 (on cyclic extensions). If $  E $
 +
is the separable closure of $  L $,  
 +
then $  H  ^ {2} ( G ( E/L), E  ^ {*} ) $
 +
is called the Brauer group of the field $  L $ (see [[Brauer group|Brauer group]]). At present (1987) a Galois theory of commutative rings is being developed in which an essential part is played by the Galois cohomology of commutative rings and by the Brauer group.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110140.png" /> is an associative algebra, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110141.png" /> is rigid if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110142.png" /> (see [[Deformation|Deformation]] of an algebra).
+
If $  S $
 +
is an associative algebra, then $  S $
 +
is rigid if $  H  ^ {2} ( S, S) = 0 $ (see [[Deformation|Deformation]] of an algebra).
  
In a sense, the cohomology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110143.png" /> are dual to the homology groups
+
In a sense, the cohomology groups $  H  ^ {n} ( R, A) $
 +
are dual to the homology 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/c023110/c023110144.png" /></td> </tr></table>
+
$$
 +
H _ {n} ( R, A)  = \
 +
\mathrm{Tor} _ {n}  ^ {R} ( A, K)
 +
$$
  
of the associative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110145.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110146.png" /> with coefficients in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110147.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110148.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110149.png" /> is a group, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110150.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110151.png" />, then the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110152.png" /> are called the homology groups of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110153.png" /> with coefficients in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110154.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110155.png" />; they are denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110156.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110157.png" /> is an associative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110158.png" />-algebra and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110159.png" />, then the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110160.png" /> are called the homology groups of the associative algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110161.png" /> with coefficients in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110162.png" />-bimodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110163.png" />; they are denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110164.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110165.png" /> is a Lie algebra and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110166.png" /> is its universal enveloping algebra, the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110167.png" /> are called the homology groups of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110168.png" /> with coefficients in the Lie <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110169.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110170.png" />; they are denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110171.png" />. In a number of cases, the homology groups in small dimensions have a simple interpretation. Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110172.png" /> is a group, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110173.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110174.png" />.
+
of the associative $  K $-algebra $  R $
 +
with coefficients in an $  R $-module $  A $.  
 +
If $  G $
 +
is a group, $  R = \mathbf Z G $
 +
and $  K = \mathbf Z $,  
 +
then the groups $  H _ {n} ( R, A) $
 +
are called the homology groups of the group $  G $
 +
with coefficients in the $  R $-module $  A $;  
 +
they are denoted by $  H _ {n} ( G, A) $.  
 +
If $  S $
 +
is an associative $  K $-algebra and $  R = S \otimes _ {K} S  ^ {0} $,  
 +
then the groups $  \mathrm{Tor} _ {n}  ^ {R} ( S, A) $
 +
are called the homology groups of the associative algebra $  S $
 +
with coefficients in the $  S $-bimodule $  A $;  
 +
they are denoted by $  H _ {n} ( S, A) $.  
 +
If $  S $
 +
is a Lie algebra and $  R = U _ {S} $
 +
is its universal enveloping algebra, the groups $  H _ {n} ( R, A) $
 +
are called the homology groups of the Lie algebra $  S $
 +
with coefficients in the Lie $  S $-module $  A $;  
 +
they are denoted by $  H _ {n} ( S, A) $.  
 +
In a number of cases, the homology groups in small dimensions have a simple interpretation. Thus, if $  G $
 +
is a group, then $  H _ {0} ( G, \mathbf Z ) \cong \mathbf Z $
 +
and $  H _ {1} ( G, \mathbf Z ) \cong G/[ G, G] $.
  
If in an Abelian category the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110175.png" /> has derived functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110176.png" />, and the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110177.png" /> together with its derived functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110178.png" /> are also defined, then the above scheme defines a cohomology and homology theory in this category. A very general approach to the construction of cohomology theories can be developed using co-triples [[#References|[11]]]. The concept of a (co-)triple arose in the analysis of the minimal tools that are necessary for the construction of simplicial resolutions. A triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110179.png" /> in a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110180.png" /> is a functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110181.png" /> together with two natural transformations of functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110182.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110183.png" />, subject to the conditions
+
If in an Abelian category the functor $  \mathop{\rm Hom} $
 +
has derived functor $  \mathop{\rm Ext} $,  
 +
and the functor $  \otimes $
 +
together with its derived functor $  \mathop{\rm Tor} $
 +
are also defined, then the above scheme defines a cohomology and homology theory in this category. A very general approach to the construction of cohomology theories can be developed using co-triples [[#References|[11]]]. The concept of a (co-)triple arose in the analysis of the minimal tools that are necessary for the construction of simplicial resolutions. A triple $  T = ( T, k, p) $
 +
in a category $  \mathfrak A $
 +
is a functor $  T: \mathfrak A \rightarrow \mathfrak A $
 +
together with two natural transformations of functors $  k: 1 _ {\mathfrak A} \rightarrow T $,  
 +
$  p: T  ^ {2} \rightarrow T $,  
 +
subject to the conditions
  
<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/c023110/c023110184.png" /></td> </tr></table>
+
$$
 +
p \circ Tk  = \
 +
p \circ kT  = 1,\ \
 +
p \circ Tp  = \
 +
p \circ pT.
 +
$$
  
The concept of a co-triple is dual to this, that is, it is obtained by reversing arrows. If an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110185.png" /> and a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110186.png" /> are such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110187.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110188.png" />, then the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110189.png" /> is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110190.png" />-algebra. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110191.png" /> be the category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110192.png" />-algebras. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110193.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110194.png" />. This defines a functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110195.png" /> (in a sense, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110196.png" /> is a free object over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110197.png" />). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110198.png" /> be the functor that forgets the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110199.png" />-structure. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110200.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110201.png" /> are adjoint functors (cf. [[Adjoint functor|Adjoint functor]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110202.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110203.png" />, together with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110204.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110205.png" />, defines a co-triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110206.png" /> and a complex
+
The concept of a co-triple is dual to this, that is, it is obtained by reversing arrows. If an object $  X \in \mathfrak A $
 +
and a morphism $  q: T ( X) \rightarrow X $
 +
are such that $  q \circ k ( X) = 1 _ {X} : X \rightarrow X $
 +
and $  q \circ T ( q) = q \circ p ( X): T  ^ {2} ( X) \rightarrow X $,  
 +
then the pair $  ( X, q) $
 +
is called a $  T $-algebra. Let $  \mathfrak A  ^ {T} $
 +
be the category of $  T $-algebras. If $  X \in \mathfrak A $,  
 +
then $  F ( X) = ( T ( X), p ( X)) \in \mathfrak A  ^ {T} $.  
 +
This defines a functor $  F: \mathfrak A \rightarrow \mathfrak A  ^ {T} $ (in a sense, $  F ( X) $
 +
is a free object over $  X $).  
 +
Let $  U: \mathfrak A  ^ {T} \rightarrow \mathfrak A $
 +
be the functor that forgets the $  T $-structure. Then $  F $
 +
and $  U $
 +
are adjoint functors (cf. [[Adjoint functor|Adjoint functor]]), $  UF = T $,  
 +
and $  G = FU: \mathfrak A  ^ {T} \rightarrow \mathfrak A  ^ {T} $,  
 +
together with $  l: G \rightarrow 1 _ {\mathfrak A  ^ {T}  } $,  
 +
$  q: G \rightarrow G  ^ {2} $,  
 +
defines a co-triple $  ( G, l, q) $
 +
and a complex
  
<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/c023110/c023110207.png" /></td> </tr></table>
+
$$
 +
X  \leftarrow ^ { {d _ 0} } \
 +
G ( X)  \leftarrow ^ { {d _ 1} }
 +
G  ^ {2} ( X)  \leftarrow ^ { {d _ 2} } \
 +
G  ^ {3} ( X)  \leftarrow \dots
 +
$$
  
with differentiation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110208.png" /> (this complex is an analogue of the canonical free resolution of the object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110209.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110210.png" /> is an Abelian category and the complex so obtained is acyclic, the standard application of the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110211.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110212.png" />) gives rise to the construction of the cohomology groups (or homology groups) of the object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110213.png" />. In general it is necessary to construct a new Abelian category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110214.png" />-modules over the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110215.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110216.png" />, on which there is a natural co-triple structure enabling one to construct groups, which are then called the cohomology groups of the original category (analogous to the construction of cohomology groups for the categories of groups, associative algebras and Lie algebras). This scheme embraces the cohomology of groups, associative algebras and Lie algebras, as well as a number of other cohomology theories (Harrison cohomology of commutative algebras, André–Quillen cohomology, Amitsur cohomology, etc; see [[#References|[8]]]).
+
with differentiation $  d _ {n} = \sum _ {i = 0 }  ^ {n} (- 1)  ^ {i} G ^ {n - i } lG  ^ {i} $ (this complex is an analogue of the canonical free resolution of the object $  X $).  
 +
If $  \mathfrak A  ^ {T} $
 +
is an Abelian category and the complex so obtained is acyclic, the standard application of the functor $  \mathop{\rm Hom} $ (or $  \otimes $)  
 +
gives rise to the construction of the cohomology groups (or homology groups) of the object $  X $.  
 +
In general it is necessary to construct a new Abelian category of $  ( X, q) $-modules over the $  T $-algebra $  ( X, q) $,  
 +
on which there is a natural co-triple structure enabling one to construct groups, which are then called the cohomology groups of the original category (analogous to the construction of cohomology groups for the categories of groups, associative algebras and Lie algebras). This scheme embraces the cohomology of groups, associative algebras and Lie algebras, as well as a number of other cohomology theories (Harrison cohomology of commutative algebras, André–Quillen cohomology, Amitsur cohomology, etc; see [[#References|[8]]]).
  
All the constructions specified here relate to some Abelian category. At the same time, a number of mathematical disciplines (for example, the theory of group extensions) require the construction of cohomology theories with coefficients in a non-Abelian category (for example, in a non-Abelian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110217.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110218.png" /> in the case of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110219.png" />) (see [[#References|[8]]], [[#References|[11]]]). The starting-point for the construction of various non-Abelian cohomology theories of algebras is the interpretation of cohomology in dimension 0 and 1, but certain aspects of the classical theory have to be relinquished (group structures on cohomology, etc.). Cohomology of topological algebraic structures has been considered (for example, the cohomology of topological groups [[#References|[5]]], Banach algebras, etc.).
+
All the constructions specified here relate to some Abelian category. At the same time, a number of mathematical disciplines (for example, the theory of group extensions) require the construction of cohomology theories with coefficients in a non-Abelian category (for example, in a non-Abelian $  G $-module $  A $
 +
in the case of a group $  G $)  
 +
(see [[#References|[8]]], [[#References|[11]]]). The starting-point for the construction of various non-Abelian cohomology theories of algebras is the interpretation of cohomology in dimension 0 and 1, but certain aspects of the classical theory have to be relinquished (group structures on cohomology, etc.). Cohomology of topological algebraic structures has been considered (for example, the cohomology of topological groups [[#References|[5]]], Banach algebras, etc.).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> J. Schur, "Untersuchungen über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen" ''J. Reine Angew. Math.'' , '''132''' (1907) pp. 85–137 {{MR|}} {{ZBL|38.0174.02}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> R. Baer, "Erweiterung von Gruppen und ihren Isomorphismen" ''Math. Z.'' , '''38''' (1934) pp. 374–416 {{MR|1545456}} {{ZBL|0009.01101}} {{ZBL|60.0079.04}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S. Eilenberg, S. MacLane, "Relations between homology and homotopy groups" ''Proc. Nat. Acad. Sci. USA'' , '''29''' (1943) pp. 155–158 {{MR|0007982}} {{ZBL|0061.40701}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> H. Hopf, "Ueber die Bettischen Gruppen, die zu einer beliebigen Gruppe gehören" ''Comment. Math. Helv.'' , '''17''' (1944–1945) pp. 39–79 {{MR|}} {{ZBL|0061.40703}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> D.K. Faddeev, "On quotient systems in Abelian groups with operators" ''Dokl. Akad. Nauk SSSR'' , '''58''' : 3 (1947) pp. 361–364 (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956) {{MR|0077480}} {{ZBL|0075.24305}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> A. Grothendieck, "Sur quelques points d'algèbre homologique" ''Tohôku Math. J.'' , '''9''' (1957) pp. 119–221 {{MR|0102537}} {{ZBL|}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> ''Itogi Nauk. Algebra 1964'' (1966) pp. 203–235</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> S. MacLane, "Homology" , Springer (1963) {{MR|}} {{ZBL|0818.18001}} {{ZBL|0328.18009}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> J.-P. Serre, "Cohomologie Galoisienne" , Springer (1964) {{MR|0180551}} {{ZBL|0128.26303}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> B. Eckmann (ed.) , ''Seminar on triples and categorical homology theory Zürich, 1966–1967'' , ''Lect. notes in math.'' , '''80''' , Springer (1969)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> K.W. Gruenberg, "Cohomological topics in group theory" , Springer (1970) {{MR|0279200}} {{ZBL|0205.32701}} </TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> U. Stammbach, "Homology in group theory" , Springer (1973) {{MR|0382477}} {{ZBL|0272.20049}} </TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top"> R. Fossum, P.A. Griffith, I. Reiten, "Trivial extensions of Abelian categories. Homological algebra of trivial extensions of Abelian categories with applications to ring theory" , Springer (1975) {{MR|0389981}} {{ZBL|0303.18006}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> J. Schur, "Untersuchungen über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen" ''J. Reine Angew. Math.'' , '''132''' (1907) pp. 85–137 {{MR|}} {{ZBL|38.0174.02}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> R. Baer, "Erweiterung von Gruppen und ihren Isomorphismen" ''Math. Z.'' , '''38''' (1934) pp. 374–416 {{MR|1545456}} {{ZBL|0009.01101}} {{ZBL|60.0079.04}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S. Eilenberg, S. MacLane, "Relations between homology and homotopy groups" ''Proc. Nat. Acad. Sci. USA'' , '''29''' (1943) pp. 155–158 {{MR|0007982}} {{ZBL|0061.40701}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> H. Hopf, "Ueber die Bettischen Gruppen, die zu einer beliebigen Gruppe gehören" ''Comment. Math. Helv.'' , '''17''' (1944–1945) pp. 39–79 {{MR|}} {{ZBL|0061.40703}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> D.K. Faddeev, "On quotient systems in Abelian groups with operators" ''Dokl. Akad. Nauk SSSR'' , '''58''' : 3 (1947) pp. 361–364 (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956) {{MR|0077480}} {{ZBL|0075.24305}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> A. Grothendieck, "Sur quelques points d'algèbre homologique" ''Tohôku Math. J.'' , '''9''' (1957) pp. 119–221 {{MR|0102537}} {{ZBL|}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> ''Itogi Nauk. Algebra 1964'' (1966) pp. 203–235</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> S. MacLane, "Homology" , Springer (1963) {{MR|}} {{ZBL|0818.18001}} {{ZBL|0328.18009}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> J.-P. Serre, "Cohomologie Galoisienne" , Springer (1964) {{MR|0180551}} {{ZBL|0128.26303}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> B. Eckmann (ed.) , ''Seminar on triples and categorical homology theory Zürich, 1966–1967'' , ''Lect. notes in math.'' , '''80''' , Springer (1969)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> K.W. Gruenberg, "Cohomological topics in group theory" , Springer (1970) {{MR|0279200}} {{ZBL|0205.32701}} </TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> U. Stammbach, "Homology in group theory" , Springer (1973) {{MR|0382477}} {{ZBL|0272.20049}} </TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top"> R. Fossum, P.A. Griffith, I. Reiten, "Trivial extensions of Abelian categories. Homological algebra of trivial extensions of Abelian categories with applications to ring theory" , Springer (1975) {{MR|0389981}} {{ZBL|0303.18006}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
In the definition of Schur multiplier above, the group need not be finite. However, there is no definite agreement on terminology in the case of infinite groups.
 
In the definition of Schur multiplier above, the group need not be finite. However, there is no definite agreement on terminology in the case of infinite groups.
  
In the early 1960s S.U. Chase, D.K. Harrison and A. Rosenberg [[#References|[a1]]] developed a Galois theory of commutative rings. In particular, they set up a seven-term exact sequence incorporating Hilbert's theorem 90 and the Brauer group, using Amitsur cohomology as an appropriate generalization of Galois cohomology. In 1982, A.S. Merkurev and A.A. Suslin [[#References|[a3]]] showed that for a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110220.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110221.png" /> there is an isomorphism between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110222.png" /> and a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110223.png" /> from [[Algebraic K-theory|algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110224.png" />-theory]]. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110225.png" /> is the group (scheme) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110226.png" />-th roots of unity. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110227.png" /> contains a primitive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110228.png" />-th root of unity this gives an explicit computation of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110229.png" />-torsion of the Brauer group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110230.png" />.
+
In the early 1960s S.U. Chase, D.K. Harrison and A. Rosenberg [[#References|[a1]]] developed a Galois theory of commutative rings. In particular, they set up a seven-term exact sequence incorporating Hilbert's theorem 90 and the Brauer group, using Amitsur cohomology as an appropriate generalization of Galois cohomology. In 1982, A.S. Merkurev and A.A. Suslin [[#References|[a3]]] showed that for a field $  F $
 +
and $  n \neq  \mathop{\rm char}  F $
 +
there is an isomorphism between $  H  ^ {2} ( F, \mu _ {n} \otimes \mu _ {n} ) $
 +
and a group $  K _ {2} ( F  ) / nK _ {2} ( F  ) $
 +
from [[Algebraic K-theory|algebraic $  K $-theory]]. Here $  \mu _ {n} $
 +
is the group (scheme) of $  n $-th roots of unity. If $  F $
 +
contains a primitive $  n $-th root of unity this gives an explicit computation of the $  n $-torsion of the Brauer group of $  F $.
  
 
A comprehensive text on (co)homology of Banach algebras is [[#References|[a4]]]. For non-Abelian cohomology cf. [[#References|[a2]]].
 
A comprehensive text on (co)homology of Banach algebras is [[#References|[a4]]]. For non-Abelian cohomology cf. [[#References|[a2]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S.U. Chase, D.K. Harrison, A. Rosenberg, "Galois theory and cohomology of commutative rings" , ''Mem. Amer. Math. Soc.'' , '''52''' , Amer. Math. Soc. (1965) {{MR|0195922}} {{ZBL|0143.05902}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J. Giraud, "Cohomologie non abélienne" , Springer (1971) {{MR|0344253}} {{ZBL|0226.14011}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A.S. Merkurev, A.A. Suslin, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023110/c023110231.png" />-cohomology of Severi-Brauer varieties and the norm residue homomorphism" ''Math. USSR-Izv.'' , '''21''' (1983) pp. 307–340 ''Izv. Akad. Nauk SSSR'' , '''46''' (1982) pp. 1011–1046 {{MR|0675529}} {{MR|0659762}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> A.Ya. [A.Ya. Khelemskii] Helemskii, "Cohomology of Banach and topological spaces" , Reidel (Forthcoming) (Translated from Russian)</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S.U. Chase, D.K. Harrison, A. Rosenberg, "Galois theory and cohomology of commutative rings" , ''Mem. Amer. Math. Soc.'' , '''52''' , Amer. Math. Soc. (1965) {{MR|0195922}} {{ZBL|0143.05902}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J. Giraud, "Cohomologie non abélienne" , Springer (1971) {{MR|0344253}} {{ZBL|0226.14011}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A.S. Merkurev, A.A. Suslin, "K-cohomology of Severi-Brauer varieties and the norm residue homomorphism" ''Math. USSR-Izv.'' , '''21''' (1983) pp. 307–340 ''Izv. Akad. Nauk SSSR'' , '''46''' (1982) pp. 1011–1046 {{MR|0675529}} {{MR|0659762}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> A.Ya. [A.Ya. Khelemskii] Helemskii, "Cohomology of Banach and topological spaces" , Reidel (Forthcoming) (Translated from Russian)</TD></TR></table>

Latest revision as of 09:48, 26 March 2023


The groups

$$ H ^ {n} ( R, A) = \ \mathrm{Ext} _ {R} ^ {n} ( K, A),\ \ n \geq 0 $$

(see Functor $ \mathop{\rm Ext} $), where $ R $ is an associative algebra over a commutative ring $ K $ with a fixed $ K $-algebra homomorphism $ \epsilon : R \rightarrow K $ (augmentation) enabling one to regard $ K $ as an $ R $-module, and where $ A $ is an $ R $-module. This definition encompasses many cohomology theories of certain types of (universal) algebras.

Cohomology groups of groups in all dimensions were introduced in the 1940s first by S. Eilenberg and S. MacLane [3] in connection with topological investigations, and by D.K. Faddeev [5] from a purely algebraic point of view — as groups of classes of generalized quotient systems. Cohomology groups in small dimensions were studied earlier in one form or another (see [1], [2], [4]).

Examples of cohomology groups.

1) If $ K = \mathbf Z $ is the ring of integers, $ G $ is a group, $ R = \mathbf Z G $ is the group ring of $ G $ over $ \mathbf Z $, and

$$ \epsilon \left ( \sum n _ {i} g _ {i} \right ) = \ \sum n _ {i} ,\ \ n _ {i} \in \mathbf Z ,\ \ g _ {i} \in G, $$

then the groups $ H ^ {n} ( R, A) $ are called the cohomology groups of the group $ G $ with coefficients (or values) in the $ R $-module $ A $; they are denoted by $ H ^ {n} ( G, A) $. Instead of a group $ G $ one can consider a monoid $ G $, and obtain the analogous cohomology groups $ H ^ {n} ( G, A) $ of the monoid $ G $.

2) If $ S $ is an associative $ K $-algebra, $ S ^ {0} $ is the opposite $ K $-algebra and

$$ R = S \otimes _ {K} S ^ {0} ,\ \ \epsilon \left ( \sum s _ {i} \otimes t _ {i} \right ) = \ \sum s _ {i} t _ {i} , $$

then the groups $ \mathrm{Ext} _ {R} ^ {n} ( S, A) $ are called the cohomology groups of the associative algebra $ S $ with coefficients in the $ S $-bimodule $ A $ (that is, in the $ R $-module $ A $); they are denoted by $ H ^ {n} ( S, A) $. If $ K $ is a field, then the groups $ H ^ {n} ( S, A) $ are called the Hochschild cohomology groups of the $ K $-algebra $ S $.

3) If $ S $ is a Lie algebra over a field $ K $ and $ R = U _ {S} $ is its universal enveloping algebra with augmentation $ \epsilon : R \rightarrow K $, then the groups $ H ^ {n} ( R, A) $ are called the cohomology groups of the Lie algebra $ S $ with coefficients in the $ U _ {S} $-module $ A $ (that is, in the Lie $ S $-module $ A $); they are denoted by $ H ^ {n} ( S, A) $.

The cohomology groups for $ n = 0, 1 $ and 2 have, in a number of cases, simple interpretations.

a) If $ G $ is a group, then $ H ^ {0} ( G, A) $ is isomorphic to the group

$$ A ^ {G} = \ \{ {a \in A } : {ga = a \textrm{ for } \textrm{ all } g \in G } \} $$

of fixed elements; $ H ^ {1} ( G, A) $ is isomorphic to the quotient group $ \mathop{\rm Der} ( G, A)/ \mathop{\rm Ider} ( G, A) $, where

$$ \mathop{\rm Der} ( G, A) = \ \{ {f : G \rightarrow A } : {f ( xy) = xf ( y) + f ( x) \textrm{ for } \textrm{ all } x, y \in G } \} $$

is the group of derivations (or crossed homomorphisms),

$$ \mathop{\rm Ider} ( G, A) = \ \{ {f: G \rightarrow A } : {\exists a \in A ( f ( x) = xa - a \textrm{ for } \textrm{ all } x \in G ) } \} $$

is the group of inner derivations (or principal crossed homomorphisms); here, the sequence

$$ 0 \rightarrow A ^ {G} \rightarrow A \rightarrow \mathop{\rm Der} ( G, A) \rightarrow \ H ^ {1} ( G, A) \rightarrow 0 $$

is exact; for an Abelian group $ G $, $ H ^ {2} ( G, A) $ is isomorphic to the group of extensions of $ A $ by $ G $ (see Baer multiplication); the third cohomology group of $ G $ is connected with obstructions to extensions (see [9], Chapt. IV).

b) If $ S $ is an associative $ K $-algebra, then $ H ^ {0} ( S, A) $ is isomorphic to the group

$$ \{ {a \in A } : {xa = ax \textrm{ for } \textrm{ all } x \in S } \} ; $$

$ H ^ {1} ( S, A) $ is isomorphic to the quotient group

$$ { \mathop{\rm Der} ( S, A) } / { \mathop{\rm Ider} ( S, A) } , $$

where

$$ \mathop{\rm Der} ( S, A) = \ \{ {f: S \rightarrow A } : {f \textrm{ is } K-\textrm{linear} \ \textrm{ and } f ( xy) = xf ( y) + f ( x) y \textrm{ for } x, y \in S } \} , $$

$$ \mathop{\rm Ider} ( S, A) = \{ {f: S \rightarrow A } : {\exists a \in A ( f ( x) = xa - ax \ \textrm{ for } \textrm{ all } x \in S ) } \} ; $$

$ H ^ {2} ( S, A) $ describes the extensions of the $ S $-bimodule $ A $ by the ring $ S $ (see [14]).

c) If $ S $ is a Lie algebra, then $ H ^ {0} ( S, A) $ is isomorphic to the $ K $-module $ \{ {a \in A } : {xa = 0 \textrm{ for all } x \in S } \} $; $ H ^ {1} ( S, A) $ is isomorphic to the quotient group

$$ { \mathop{\rm Der} ( S, A) } / { \mathop{\rm Ider} ( S, A) } , $$

where

$$ \mathop{\rm Der} ( S, A) = \ \{ {f: S \rightarrow A } : {f ([ x, y]) = xf ( y) - yf ( x) \textrm{ for } \textrm{ all } x, y \in S } \} , $$

$$ \mathop{\rm Ider} ( S, A) = \{ {f: S \rightarrow A } : {\exists a \in A ( f ( x) = xa \textrm{ for } \textrm{ all } x \in S ) }\} ; $$

the second cohomology group $ H ^ {2} ( S, A) $ of a Lie algebra corresponds to the $ K $-split extensions of Lie algebras (see [6], Chapt. XIV); in certain cases the elements of $ H ^ {3} ( S, A) $ are obstructions in the extension problem.

Cohomology groups find extensive application in various branches of algebra. E.g. if $ G $ is a group and $ H ^ {2} ( G, A) = 0 $ for all $ \mathbf Z G $-modules $ A $, then $ G $ is free (Stalling's theorem, see Homological dimension). If $ G $ is a finite group and $ \mathbf C ^ {*} $ is the multiplicative group of the complex field, then the group $ M ( G) = H ^ {2} ( G, \mathbf C ^ {*} ) $ is called the Schur multiplier of $ G $. It plays an important role in the study of central extensions of groups and in the theory of projective representations of finite groups [1]. If $ G $ is a group, $ A $ a $ \mathbf Z G $-module and $ pA = 0 $ for a prime number $ p $, then

$$ \mathrm{Ext} _ {\mathbf Z G } ^ {n} ( \mathbf Z , A) \cong \ \mathrm{Ext} _ {kG} ^ {n} ( k, A), $$

where $ k = \mathop{\rm GF} ( p) $ is the field of $ p $ elements. If $ G $ is a finite $ p $-group, then $ d ( G) = \mathrm{dim} _ {k} H ^ {1} ( G, k) $ is the minimum number of generators of $ G $, and $ r ( G) = \mathrm{dim} _ {k} H ^ {2} ( G, k) $ is the minimum number of defining relations for $ G $ considered as a pro- $ p $-group; $ r ( G) \leq R ( G) $, where $ R ( G) $ is the minimum number of defining relations of the discrete group $ G $. The fact that $ r ( G) - d ( G) $ tends to infinity as $ d ( G) \rightarrow \infty $ leads to a negative solution of the class field tower problem (cf. Class field theory), the Kurosh problem on nil algebras (cf. Nil algebra) and the unrestricted Burnside problem [10].

If $ G $ is a profinite group and $ \{ {U _ {i} } : {i \in I } \} $ is the family of its open normal subgroups, then the group

$$ \lim\limits _ \rightarrow H ^ {n} ( G / U _ {i} , A ^ {U _ {i} } ) $$

is called the $ n $-th cohomology group of the profinite group $ G $ with coefficients in the $ \mathbf Z G $-module $ A $; it is denoted by $ H ^ {n} ( G, A) $. If $ E $ is a Galois extension of a field $ L $ with Galois group $ G = G ( E/L) $, then $ G $ is profinite; in this case the groups $ H ^ {n} ( G, A) $ are called Galois cohomology groups. An important role is played by the groups $ H ^ {q} ( G, E ^ {*} ) $ where $ E ^ {*} $ is the multiplicative group of $ E $. E.g. $ H ^ {1} ( G, E ^ {*} ) = 0 $, and a corollary of this fact is Hilbert's theorem 90 (on cyclic extensions). If $ E $ is the separable closure of $ L $, then $ H ^ {2} ( G ( E/L), E ^ {*} ) $ is called the Brauer group of the field $ L $ (see Brauer group). At present (1987) a Galois theory of commutative rings is being developed in which an essential part is played by the Galois cohomology of commutative rings and by the Brauer group.

If $ S $ is an associative algebra, then $ S $ is rigid if $ H ^ {2} ( S, S) = 0 $ (see Deformation of an algebra).

In a sense, the cohomology groups $ H ^ {n} ( R, A) $ are dual to the homology groups

$$ H _ {n} ( R, A) = \ \mathrm{Tor} _ {n} ^ {R} ( A, K) $$

of the associative $ K $-algebra $ R $ with coefficients in an $ R $-module $ A $. If $ G $ is a group, $ R = \mathbf Z G $ and $ K = \mathbf Z $, then the groups $ H _ {n} ( R, A) $ are called the homology groups of the group $ G $ with coefficients in the $ R $-module $ A $; they are denoted by $ H _ {n} ( G, A) $. If $ S $ is an associative $ K $-algebra and $ R = S \otimes _ {K} S ^ {0} $, then the groups $ \mathrm{Tor} _ {n} ^ {R} ( S, A) $ are called the homology groups of the associative algebra $ S $ with coefficients in the $ S $-bimodule $ A $; they are denoted by $ H _ {n} ( S, A) $. If $ S $ is a Lie algebra and $ R = U _ {S} $ is its universal enveloping algebra, the groups $ H _ {n} ( R, A) $ are called the homology groups of the Lie algebra $ S $ with coefficients in the Lie $ S $-module $ A $; they are denoted by $ H _ {n} ( S, A) $. In a number of cases, the homology groups in small dimensions have a simple interpretation. Thus, if $ G $ is a group, then $ H _ {0} ( G, \mathbf Z ) \cong \mathbf Z $ and $ H _ {1} ( G, \mathbf Z ) \cong G/[ G, G] $.

If in an Abelian category the functor $ \mathop{\rm Hom} $ has derived functor $ \mathop{\rm Ext} $, and the functor $ \otimes $ together with its derived functor $ \mathop{\rm Tor} $ are also defined, then the above scheme defines a cohomology and homology theory in this category. A very general approach to the construction of cohomology theories can be developed using co-triples [11]. The concept of a (co-)triple arose in the analysis of the minimal tools that are necessary for the construction of simplicial resolutions. A triple $ T = ( T, k, p) $ in a category $ \mathfrak A $ is a functor $ T: \mathfrak A \rightarrow \mathfrak A $ together with two natural transformations of functors $ k: 1 _ {\mathfrak A} \rightarrow T $, $ p: T ^ {2} \rightarrow T $, subject to the conditions

$$ p \circ Tk = \ p \circ kT = 1,\ \ p \circ Tp = \ p \circ pT. $$

The concept of a co-triple is dual to this, that is, it is obtained by reversing arrows. If an object $ X \in \mathfrak A $ and a morphism $ q: T ( X) \rightarrow X $ are such that $ q \circ k ( X) = 1 _ {X} : X \rightarrow X $ and $ q \circ T ( q) = q \circ p ( X): T ^ {2} ( X) \rightarrow X $, then the pair $ ( X, q) $ is called a $ T $-algebra. Let $ \mathfrak A ^ {T} $ be the category of $ T $-algebras. If $ X \in \mathfrak A $, then $ F ( X) = ( T ( X), p ( X)) \in \mathfrak A ^ {T} $. This defines a functor $ F: \mathfrak A \rightarrow \mathfrak A ^ {T} $ (in a sense, $ F ( X) $ is a free object over $ X $). Let $ U: \mathfrak A ^ {T} \rightarrow \mathfrak A $ be the functor that forgets the $ T $-structure. Then $ F $ and $ U $ are adjoint functors (cf. Adjoint functor), $ UF = T $, and $ G = FU: \mathfrak A ^ {T} \rightarrow \mathfrak A ^ {T} $, together with $ l: G \rightarrow 1 _ {\mathfrak A ^ {T} } $, $ q: G \rightarrow G ^ {2} $, defines a co-triple $ ( G, l, q) $ and a complex

$$ X \leftarrow ^ { {d _ 0} } \ G ( X) \leftarrow ^ { {d _ 1} } G ^ {2} ( X) \leftarrow ^ { {d _ 2} } \ G ^ {3} ( X) \leftarrow \dots $$

with differentiation $ d _ {n} = \sum _ {i = 0 } ^ {n} (- 1) ^ {i} G ^ {n - i } lG ^ {i} $ (this complex is an analogue of the canonical free resolution of the object $ X $). If $ \mathfrak A ^ {T} $ is an Abelian category and the complex so obtained is acyclic, the standard application of the functor $ \mathop{\rm Hom} $ (or $ \otimes $) gives rise to the construction of the cohomology groups (or homology groups) of the object $ X $. In general it is necessary to construct a new Abelian category of $ ( X, q) $-modules over the $ T $-algebra $ ( X, q) $, on which there is a natural co-triple structure enabling one to construct groups, which are then called the cohomology groups of the original category (analogous to the construction of cohomology groups for the categories of groups, associative algebras and Lie algebras). This scheme embraces the cohomology of groups, associative algebras and Lie algebras, as well as a number of other cohomology theories (Harrison cohomology of commutative algebras, André–Quillen cohomology, Amitsur cohomology, etc; see [8]).

All the constructions specified here relate to some Abelian category. At the same time, a number of mathematical disciplines (for example, the theory of group extensions) require the construction of cohomology theories with coefficients in a non-Abelian category (for example, in a non-Abelian $ G $-module $ A $ in the case of a group $ G $) (see [8], [11]). The starting-point for the construction of various non-Abelian cohomology theories of algebras is the interpretation of cohomology in dimension 0 and 1, but certain aspects of the classical theory have to be relinquished (group structures on cohomology, etc.). Cohomology of topological algebraic structures has been considered (for example, the cohomology of topological groups [5], Banach algebras, etc.).

References

[1] J. Schur, "Untersuchungen über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen" J. Reine Angew. Math. , 132 (1907) pp. 85–137 Zbl 38.0174.02
[2] R. Baer, "Erweiterung von Gruppen und ihren Isomorphismen" Math. Z. , 38 (1934) pp. 374–416 MR1545456 Zbl 0009.01101 Zbl 60.0079.04
[3] S. Eilenberg, S. MacLane, "Relations between homology and homotopy groups" Proc. Nat. Acad. Sci. USA , 29 (1943) pp. 155–158 MR0007982 Zbl 0061.40701
[4] H. Hopf, "Ueber die Bettischen Gruppen, die zu einer beliebigen Gruppe gehören" Comment. Math. Helv. , 17 (1944–1945) pp. 39–79 Zbl 0061.40703
[5] D.K. Faddeev, "On quotient systems in Abelian groups with operators" Dokl. Akad. Nauk SSSR , 58 : 3 (1947) pp. 361–364 (In Russian)
[6] H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956) MR0077480 Zbl 0075.24305
[7] A. Grothendieck, "Sur quelques points d'algèbre homologique" Tohôku Math. J. , 9 (1957) pp. 119–221 MR0102537
[8] Itogi Nauk. Algebra 1964 (1966) pp. 203–235
[9] S. MacLane, "Homology" , Springer (1963) Zbl 0818.18001 Zbl 0328.18009
[10] J.-P. Serre, "Cohomologie Galoisienne" , Springer (1964) MR0180551 Zbl 0128.26303
[11] B. Eckmann (ed.) , Seminar on triples and categorical homology theory Zürich, 1966–1967 , Lect. notes in math. , 80 , Springer (1969)
[12] K.W. Gruenberg, "Cohomological topics in group theory" , Springer (1970) MR0279200 Zbl 0205.32701
[13] U. Stammbach, "Homology in group theory" , Springer (1973) MR0382477 Zbl 0272.20049
[14] R. Fossum, P.A. Griffith, I. Reiten, "Trivial extensions of Abelian categories. Homological algebra of trivial extensions of Abelian categories with applications to ring theory" , Springer (1975) MR0389981 Zbl 0303.18006

Comments

In the definition of Schur multiplier above, the group need not be finite. However, there is no definite agreement on terminology in the case of infinite groups.

In the early 1960s S.U. Chase, D.K. Harrison and A. Rosenberg [a1] developed a Galois theory of commutative rings. In particular, they set up a seven-term exact sequence incorporating Hilbert's theorem 90 and the Brauer group, using Amitsur cohomology as an appropriate generalization of Galois cohomology. In 1982, A.S. Merkurev and A.A. Suslin [a3] showed that for a field $ F $ and $ n \neq \mathop{\rm char} F $ there is an isomorphism between $ H ^ {2} ( F, \mu _ {n} \otimes \mu _ {n} ) $ and a group $ K _ {2} ( F ) / nK _ {2} ( F ) $ from algebraic $ K $-theory. Here $ \mu _ {n} $ is the group (scheme) of $ n $-th roots of unity. If $ F $ contains a primitive $ n $-th root of unity this gives an explicit computation of the $ n $-torsion of the Brauer group of $ F $.

A comprehensive text on (co)homology of Banach algebras is [a4]. For non-Abelian cohomology cf. [a2].

References

[a1] S.U. Chase, D.K. Harrison, A. Rosenberg, "Galois theory and cohomology of commutative rings" , Mem. Amer. Math. Soc. , 52 , Amer. Math. Soc. (1965) MR0195922 Zbl 0143.05902
[a2] J. Giraud, "Cohomologie non abélienne" , Springer (1971) MR0344253 Zbl 0226.14011
[a3] A.S. Merkurev, A.A. Suslin, "K-cohomology of Severi-Brauer varieties and the norm residue homomorphism" Math. USSR-Izv. , 21 (1983) pp. 307–340 Izv. Akad. Nauk SSSR , 46 (1982) pp. 1011–1046 MR0675529 MR0659762
[a4] A.Ya. [A.Ya. Khelemskii] Helemskii, "Cohomology of Banach and topological spaces" , Reidel (Forthcoming) (Translated from Russian)
How to Cite This Entry:
Cohomology of algebras. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cohomology_of_algebras&oldid=24147
This article was adapted from an original article by V.E. GovorovA.V. Mikhalev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article