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Cohomology of a [[Galois group|Galois group]]. Let <math>
+
Cohomology of a [[Galois group|Galois group]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g0431001.png" /> be an Abelian group, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g0431002.png" /> be the Galois group of an extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g0431003.png" /> and suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g0431004.png" /> acts on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g0431005.png" />; the Galois cohomology groups will then be the cohomology groups
M
 
</math> be an Abelian group, let <math>
 
G(K/k)
 
</math> be the
 
Galois group of an extension <math>
 
K/k
 
</math> and suppose
 
<math>
 
G(K/k)
 
</math> acts on
 
<math>
 
M
 
</math>; the Galois
 
cohomology groups will then be the cohomology groups
 
<br /><center><math>
 
H^n(K/k,M) = H^n(G(K/k),M), n \ge 0,
 
</math></center><br />
 
defined by the complex <math>
 
(C^n,d)
 
</math>, where
 
<math>
 
C^n
 
</math> consists of all mappings <math>
 
G(K/k)^n \to M
 
</math> and
 
<math>
 
d
 
</math> is the coboundary
 
operator (cf. [[Cohomology of groups|Cohomology of groups]]). If
 
<math>
 
K/k
 
</math> is an extension of infinite degree, an additional requirement is that the [[Galois topological group|Galois topological group]] acts continuously on the
 
discrete group <math>
 
M
 
</math>, and continuous
 
mappings are taken for the cochains in <math>
 
C^n
 
</math>.
 
<br />
 
Usually, only zero-dimensional <math>
 
(H^0)
 
</math> and
 
one-dimensional <math>
 
(H^1)
 
</math> cohomology are
 
defined for a non-Abelian group <math>
 
M
 
</math>. Namely,
 
<math>
 
H^0(K/k,M) = M^{G(K/k)}
 
</math> is the set of
 
fixed points under the group <math>
 
G(K/k)
 
</math> in <math>
 
M
 
</math>, while <math>
 
H^1(K/k,M)
 
</math> is the quotient
 
set of the set of one-dimensional cocycles, i.e. continuous mappings
 
<math>
 
z:G(K/k) \to M
 
</math> that satisfy the relation
 
<br /><center><math>
 
z(g_1g_2) = z(g_1)\;^{g_1}z(g_2)
 
</math></center><br />
 
for all <math>
 
g_1, g_2 \in G(K/k)
 
</math>, by the
 
equivalence relation <math>
 
\sim
 
</math>, where
 
<math>
 
z_1 \sim z_2
 
</math> if and only if
 
<math>
 
z_1(g) = m^{-1} \; z_2(g)\; ^gm
 
</math> for some <math>
 
m \in M
 
</math> and all <math>
 
g \in G(K/k)
 
</math>. In the non-Abelian case <math>
 
H^1(K/k,M)
 
</math> is a set with a
 
distinguished point corresponding to the trivial cocycle
 
<math>
 
G(K/k) \to (e)
 
</math>, where
 
<math>
 
e
 
</math> is the unit of
 
<math>
 
M
 
</math>, and usually has
 
no group structure. Nevertheless, a standard cohomology formalism can
 
be developed for such cohomology as well (cf. [[Non-Abelian
 
cohomology|Non-Abelian cohomology]]).
 
  
If <math>
+
<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/g/g043/g043100/g0431006.png" /></td> </tr></table>
K=k_s
 
</math> is the separable closure of a field <math>
 
k
 
</math>, it is customary
 
to denote the group <math>
 
G(k_s/k)
 
</math> by
 
<math>
 
G_k
 
</math>, and to write
 
<math>
 
H^n(k,M)
 
</math> for
 
<math>
 
H^n(k_s/k,M)
 
</math>.
 
  
Galois cohomology groups were implicitly present in the work of
+
defined by the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g0431007.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g0431008.png" /> consists of all mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g0431009.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310010.png" /> is the coboundary operator (cf. [[Cohomology of groups|Cohomology of groups]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310011.png" /> is an extension of infinite degree, an additional requirement is that the [[Galois topological group|Galois topological group]] acts continuously on the discrete group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310012.png" />, and continuous mappings are taken for the cochains in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310013.png" />.
D. Hilbert, E. Artin, R. Brauer, H. Hasse, and C. Chevalley on class
 
field theory, finite-dimensional simple algebras and quadratic
 
forms. The development of the ideas and methods of homological algebra
 
resulted in the introduction of Galois cohomology groups of finite
 
extensions with values in an Abelian group by E. Artin, A. Weil,
 
G. Hochschild, and J. Tate in the 1950s, in connection with class
 
field theory. The general theory of Abelian Galois cohomology groups
 
was then developed by Tate and J.-P. Serre [[#References|[1]]],
 
[[#References|[3]]], [[#References|[6]]].
 
  
Tate used Galois cohomology to introduce the concept of the
+
Usually, only zero-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310014.png" /> and one-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310015.png" /> cohomology are defined for a non-Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310016.png" />. Namely, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310017.png" /> is the set of fixed points under the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310018.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310019.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310020.png" /> is the quotient set of the set of one-dimensional cocycles, i.e. continuous mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310021.png" /> that satisfy the relation
cohomological dimension of the Galois group <math>
 
G_k
 
</math> of a
 
field <math>
 
k
 
</math> (denoted by
 
<math>
 
{\rm cd}\; G_k
 
</math>). It is defined in
 
terms of the cohomological <math>
 
p
 
</math>-dimension
 
<math>
 
{\rm cd}_p\; G_k
 
</math>, which is the
 
smallest integer <math>
 
n
 
</math> such that for any
 
torsion <math>
 
G_k
 
</math>-module
 
<math>
 
A
 
</math> and any integer
 
<math>
 
q > n
 
</math> the
 
<math>
 
p
 
</math>-primary component
 
of the group <math>
 
H^q(G_k,A)
 
</math> is zero. The
 
cohomological dimension <math>
 
{\rm cd}\; G_k
 
</math> is
 
<br /><center><math>
 
\underset{p}{\rm sup}\; {\rm cd}_p\; G_k
 
</math></center><br />
 
  
For any algebraically closed field <math>
+
<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/g/g043/g043100/g04310022.png" /></td> </tr></table>
k
 
</math> one has
 
<math>
 
{\rm cd}\; G_k = 0
 
</math>; for all fields
 
<math>
 
k
 
</math> such that the
 
[[Brauer group|Brauer group]] <math>
 
B(K)
 
</math> of an arbitrary
 
extension <math>
 
K/k
 
</math> is trivial,
 
<math>
 
{\rm cd}\; G_k \le 1
 
</math>; for the
 
<math>
 
p
 
</math>-adic field, the
 
field of algebraic functions of one variable over a finite field of
 
constants and for a totally-complex number field,
 
<math>
 
{\rm cd}\; G_k = 1
 
</math>
 
[[#References|[1]]]. Fields <math>
 
k
 
</math> whose Galois group
 
has cohomological dimension <math>
 
\le 1
 
</math> and whose Brauer
 
group <math>
 
B(k) = 0
 
</math> are called fields
 
of dimension <math>
 
\le 1
 
</math>; this is denoted
 
by <math>
 
{\rm dim}\; k \le 1
 
</math>. Such fields
 
include all finite fields, maximal unramified extensions of
 
<math>
 
p
 
</math>-adic fields, and
 
the field of rational functions in one variable over an algebraically
 
closed field of constants. If a Galois group <math>
 
G(K/k)
 
</math> is a
 
[[Pro-p group|pro-<math>
 
p
 
</math>-group]], i.e. is
 
the projective limit of finite <math>
 
p
 
</math>-groups, the
 
dimension of <math>
 
H^1(G(K/k),{\Bbb Z}/p{\Bbb Z}
 
</math> over
 
<math>
 
{\Bbb Z}/p{\Bbb Z}
 
</math> is equal to the
 
minimal number of topological generators of <math>
 
G(K/k)
 
</math>, while
 
the dimension of <math>
 
H^2(G(K/k),{\Bbb Z}/p{\Bbb Z}
 
</math> is the number of
 
defining relations between these generators. If <math>
 
{\rm cd}\; G(K/k) = 1
 
</math>, then
 
<math>
 
G(K/k)
 
</math> is a free
 
pro-<math>
 
p
 
</math>-group.
 
  
Non-Abelian Galois cohomology appeared in the late 1950s, but
+
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310023.png" />, by the equivalence relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310024.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310025.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310026.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310027.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310028.png" />. In the non-Abelian case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310029.png" /> is a set with a distinguished point corresponding to the trivial cocycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310030.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310031.png" /> is the unit of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310032.png" />, and usually has no group structure. Nevertheless, a standard cohomology formalism can be developed for such cohomology as well (cf. [[Non-Abelian cohomology|Non-Abelian cohomology]]).
systematic research began only in the 1960s, mainly in response to the
 
need for the classification of algebraic groups over not algebraically
 
closed fields. One of the principal problems which stimulated the
 
development of non-Abelian Galois cohomology is the task of
 
classifying principal homogeneous spaces of group schemes. Galois
 
cohomology groups proved to be specially effective in the problem of
 
classifying types of algebraic varieties.
 
  
These problems led to the problem of computing the Galois cohomology
+
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310033.png" /> is the separable closure of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310034.png" />, it is customary to denote the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310035.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310036.png" />, and to write <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310037.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310038.png" />.
groups of algebraic groups. The general theorems on the structure of
 
algebraic groups essentially reduce the study of Galois cohomology
 
groups to a separate consideration of the Galois cohomology groups of
 
finite groups, unipotent groups, tori, semi-simple groups, and Abelian
 
varieties.
 
  
The Galois cohomology groups of a connected unipotent group
+
Galois cohomology groups were implicitly present in the work of D. Hilbert, E. Artin, R. Brauer, H. Hasse, and C. Chevalley on class field theory, finite-dimensional simple algebras and quadratic forms. The development of the ideas and methods of homological algebra resulted in the introduction of Galois cohomology groups of finite extensions with values in an Abelian group by E. Artin, A. Weil, G. Hochschild, and J. Tate in the 1950s, in connection with class field theory. The general theory of Abelian Galois cohomology groups was then developed by Tate and J.-P. Serre [[#References|[1]]], [[#References|[3]]], [[#References|[6]]].
<math>
 
U
 
</math> are trivial if
 
<math>
 
U
 
</math> is defined over a perfect field <math>
 
k
 
</math>,
 
i.e. <math>
 
H^1(k,U) = 0
 
</math> for an arbitrary
 
unipotent group <math>
 
U
 
</math>, and
 
<math>
 
H^n(k,U) = 0
 
</math> for all
 
<math>
 
n \ge 1
 
</math> if
 
<math>
 
U
 
</math> is an Abelian
 
group. In particular, for the additive group <math>
 
G_\alpha
 
</math> of an
 
arbitrary field one always has <math>
 
H^1(k,G_\alpha)
 
</math>. For an imperfect field <math>
 
k
 
</math>, in general
 
<math>
 
H^1(k,G_\alpha) \ne 0
 
</math>.
 
  
One of the first significant facts about Galois cohomology groups was
+
Tate used Galois cohomology to introduce the concept of the cohomological dimension of the Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310039.png" /> of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310040.png" /> (denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310041.png" />). It is defined in terms of the cohomological <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310043.png" />-dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310044.png" />, which is the smallest integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310045.png" /> such that for any torsion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310046.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310047.png" /> and any integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310048.png" /> the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310049.png" />-primary component of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310050.png" /> is zero. The cohomological dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310051.png" /> is
Hilbert's "Theorem 90" , one formulation of which states that
 
<math>
 
H^1(k,G_m) = 0
 
</math> (where
 
<math>
 
G_m
 
</math> is the
 
multiplicative group). Moreover, for any <math>
 
k
 
</math>-split
 
algebraic torus <math>
 
T
 
</math> one has
 
<math>
 
H^1(k,T) = 0
 
</math>. The computation of <math>
 
H^1(k,T)
 
</math> for an arbitrary
 
<math>
 
k
 
</math>-defined torus
 
<math>
 
T
 
</math> can be reduced, in
 
the general case, to the computation of <math>
 
H^1(K/k,T)
 
</math> where
 
<math>
 
K
 
</math> is a Galois
 
splitting field of <math>
 
T
 
</math>; so far (1989)
 
this has only been accomplished for special fields. The case when
 
<math>
 
k
 
</math> is an algebraic
 
number field is especially important in practical
 
applications. Duality theorems, with various applications, have been
 
developed for this case.
 
  
Let <math>
+
<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/g/g043/g043100/g04310052.png" /></td> </tr></table>
K/k
 
</math> be a [[Galois
 
extension|Galois extension]] of finite degree, let
 
<math>
 
C((K)
 
</math> be the group of
 
adèles (cf. [[Adèle|Adèle]]) of a multiplicative
 
<math>
 
K
 
</math>-group
 
<math>
 
G_m
 
</math>, and let
 
<math>
 
\hat T = {\rm Hom}_k(T,G_m)
 
</math> be the group of
 
characters of a torus. The duality theorem states that the cup-product
 
<br /><center><math>
 
H^{2-r}(K/k),\hat T)\times H^r(K/k,{\rm Hom}(\hat T,C(K))) \to H^2(K/k,C(K))
 
</math></center><br />
 
defines non-degenerate pairing for <math>
 
r = 0,1,2
 
</math>. This theorem was
 
used to find the formula for expressing the Tamagawa numbers
 
(cf. [[Tamagawa number|Tamagawa number]]) of the torus
 
<math>
 
T
 
</math> by invariants
 
connected with its Galois cohomology groups. Other important duality
 
theorems for Galois cohomology groups also exist [[#References|[1]]].
 
  
It has been proved [[#References|[11]]] that the groups
+
For any algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310053.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310054.png" />; for all fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310055.png" /> such that the [[Brauer group|Brauer group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310056.png" /> of an arbitrary extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310057.png" /> is trivial, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310058.png" />; for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310059.png" />-adic field, the field of algebraic functions of one variable over a finite field of constants and for a totally-complex number field, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310060.png" /> [[#References|[1]]]. Fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310061.png" /> whose Galois group has cohomological dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310062.png" /> and whose Brauer group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310063.png" /> are called fields of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310065.png" />; this is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310066.png" />. Such fields include all finite fields, maximal unramified extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310067.png" />-adic fields, and the field of rational functions in one variable over an algebraically closed field of constants. If a Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310068.png" /> is a [[Pro-p group|pro-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310069.png" />-group]], i.e. is the projective limit of finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310070.png" />-groups, the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310071.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310072.png" /> is equal to the minimal number of topological generators of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310073.png" />, while the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310074.png" /> is the number of defining relations between these generators. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310075.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310076.png" /> is a free pro-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310077.png" />-group.
<math>
 
H^1(k,G)
 
</math> over fields
 
<math>
 
k
 
</math> of dimension
 
<math>
 
\le 1
 
</math> are trivial. A
 
natural class of fields has been distinguished with only a finite
 
number of extensions of a given degree (the so-called type
 
<math>
 
(F)
 
</math> fields); these
 
include, for example, the <math>
 
p
 
</math>-adic number
 
fields. It was proved [[#References|[1]]] that for any algebraic group
 
<math>
 
G
 
</math> over a field
 
<math>
 
k
 
</math> of type
 
<math>
 
(F)
 
</math> the cohomology
 
group <math>
 
H^1(k,G)
 
</math> is a finite set.
 
  
The theory of Galois cohomology of semi-simple algebraic groups has
+
Non-Abelian Galois cohomology appeared in the late 1950s, but systematic research began only in the 1960s, mainly in response to the need for the classification of algebraic groups over not algebraically closed fields. One of the principal problems which stimulated the development of non-Abelian Galois cohomology is the task of classifying principal homogeneous spaces of group schemes. Galois cohomology groups proved to be specially effective in the problem of classifying types of algebraic varieties.
far-reaching arithmetical and analytical applications. The
 
Kneser–Bruhat–Tits theorem states that <math>
 
H^1(k,G) = 0
 
</math> for
 
simply-connected semi-simple algebraic groups <math>
 
G
 
</math> over
 
local fields <math>
 
k
 
</math> whose residue
 
field has cohomological dimension <math>
 
\le 1
 
</math>. This theorem was
 
first proved for <math>
 
p
 
</math>-adic number
 
fields , after which a proof was obtained for the general case. It was
 
proved that <math>
 
H^1(k,G)
 
</math> is trivial for a
 
field of algebraic functions in one variable over a finite field of
 
constants. In all these cases the cohomological dimension
 
<math>
 
{\rm cd}\; G_k \le 2
 
</math>, which confirms
 
the general conjecture of Serre to the effect that
 
<math>
 
H^1(k,G)
 
</math> is trivial for
 
simply-connected semi-simple <math>
 
G
 
</math> over fields
 
<math>
 
k
 
</math> with <math>
 
{\rm cd}\; G_k \le 2
 
</math>.
 
  
Let <math>
+
These problems led to the problem of computing the Galois cohomology groups of algebraic groups. The general theorems on the structure of algebraic groups essentially reduce the study of Galois cohomology groups to a separate consideration of the Galois cohomology groups of finite groups, unipotent groups, tori, semi-simple groups, and Abelian varieties.
K
+
 
</math> be a global
+
The Galois cohomology groups of a connected unipotent group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310078.png" /> are trivial if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310079.png" /> is defined over a perfect field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310080.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310081.png" /> for an arbitrary unipotent group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310082.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310083.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310084.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310085.png" /> is an Abelian group. In particular, for the additive group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310086.png" /> of an arbitrary field one always has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310087.png" />. For an imperfect field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310088.png" />, in general <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310089.png" />.
field, let <math>
+
 
V
+
One of the first significant facts about Galois cohomology groups was Hilbert's  "Theorem 90" , one formulation of which states that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310090.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310091.png" /> is the multiplicative group). Moreover, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310092.png" />-split algebraic torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310093.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310094.png" />. The computation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310095.png" /> for an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310096.png" />-defined torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310097.png" /> can be reduced, in the general case, to the computation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310098.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g04310099.png" /> is a Galois splitting field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100100.png" />; so far (1989) this has only been accomplished for special fields. The case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100101.png" /> is an algebraic number field is especially important in practical applications. Duality theorems, with various applications, have been developed for this case.
</math> be the set of all
+
 
non-equivalent valuations of <math>
+
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100102.png" /> be a [[Galois extension|Galois extension]] of finite degree, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100103.png" /> be the group of adèles (cf. [[Adèle|Adèle]]) of a multiplicative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100104.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100105.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100106.png" /> be the group of characters of a torus. The duality theorem states that the cup-product
k
+
 
</math>, let
+
<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/g/g043/g043100/g043100107.png" /></td> </tr></table>
<math>
+
 
k_\nu
+
<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/g/g043/g043100/g043100108.png" /></td> </tr></table>
</math> be the completion
+
 
of <math>
+
defines non-degenerate pairing for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100109.png" />. This theorem was used to find the formula for expressing the Tamagawa numbers (cf. [[Tamagawa number|Tamagawa number]]) of the torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100110.png" /> by invariants connected with its Galois cohomology groups. Other important duality theorems for Galois cohomology groups also exist [[#References|[1]]].
k
+
 
</math>. The imbeddings
+
It has been proved [[#References|[11]]] that the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100111.png" /> over fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100112.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100113.png" /> are trivial. A natural class of fields has been distinguished with only a finite number of extensions of a given degree (the so-called type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100115.png" /> fields); these include, for example, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100116.png" />-adic number fields. It was proved [[#References|[1]]] that for any algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100117.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100118.png" /> of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100119.png" /> the cohomology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100120.png" /> is a finite set.
<math>
+
 
k\to k_\nu
+
The theory of Galois cohomology of semi-simple algebraic groups has far-reaching arithmetical and analytical applications. The Kneser–Bruhat–Tits theorem states that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100121.png" /> for simply-connected semi-simple algebraic groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100122.png" /> over local fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100123.png" /> whose residue field has cohomological dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100124.png" />. This theorem was first proved for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100125.png" />-adic number fields , after which a proof was obtained for the general case. It was proved
</math> induce a natural mapping
+
 
<br /><center><math>
+
that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100126.png" /> is trivial for a field of algebraic functions in one variable over a finite field of constants. In all these cases the cohomological dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100127.png" />, which confirms the general conjecture of Serre to the effect that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100128.png" /> is trivial for simply-connected semi-simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100129.png" /> over fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100130.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100131.png" />.
i : H^1(k,G) \to \displaystyle\prod_{\nu\in V} H^1(k_\nu,G)
+
 
</math></center><br />
+
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100132.png" /> be a global field, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100133.png" /> be the set of all non-equivalent valuations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100134.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100135.png" /> be the completion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100136.png" />. The imbeddings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100137.png" /> induce a natural mapping
for an arbitrary algebraic group <math>
+
 
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/g/g043/g043100/g043100138.png" /></td> </tr></table>
</math> defined over <math>
+
 
k
+
for an arbitrary algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100139.png" /> defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100140.png" />, the kernel of which is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100141.png" /> and, in the case of Abelian varieties, is called the Tate–Shafarevich group. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100142.png" /> measures the extent to which the Galois cohomology groups over a global field are described by Galois cohomology groups over localizations. The principal result on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100143.png" /> for linear algebraic groups is due to A. Borel, who proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100144.png" /> is finite. There exists a conjecture according to which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100145.png" /> is finite in the case of Abelian varieties as well. The situation in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100146.png" />, i.e. the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100147.png" /> is injective, is a special case. One then says that the [[Hasse principle|Hasse principle]] applies to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100148.png" />. This terminology is explained by the fact that for an orthogonal group the injectivity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100149.png" /> is equivalent to the classical theorem of Minkowski–Hasse on quadratic forms, and in the case of a projective group it is equivalent to the Brauer–Hasse–Noether theorem on the splitting of simple algebras. According to a conjecture of Serre one always has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100150.png" /> for a simply-connected or adjoint semi-simple group. This conjecture was proved for most simply-connected semi-simple groups over global number fields (except for groups with simple components of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100151.png" />) , and also for arbitrary simply-connected algebraic groups over global function fields.
</math>, the kernel of
 
which is denoted by <math>
 
{\rm Shaf}\;(G)
 
</math> and, in the case
 
of Abelian varieties, is called the Tate–Shafarevich group. The group
 
<math>
 
{\rm Shaf}\; (G)
 
</math> measures the
 
extent to which the Galois cohomology groups over a global field are
 
described by Galois cohomology groups over localizations. The
 
principal result on <math>
 
{\rm Shaf}\;(G)
 
</math> for linear
 
algebraic groups is due to A. Borel, who proved that
 
<math>
 
{\rm Shaf}\;(G)
 
</math> is finite. There
 
exists a conjecture according to which <math>
 
{\rm Shaf}\;(G)
 
</math> is
 
finite in the case of Abelian varieties as well. The situation in
 
which <math>
 
{\rm Shaf}\;(G) = 0
 
</math>, i.e. the mapping
 
<math>
 
i
 
</math> is injective, is
 
a special case. One then says that the [[Hasse principle|Hasse
 
principle]] applies to <math>
 
G
 
</math>. This terminology
 
is explained by the fact that for an orthogonal group the injectivity
 
of <math>
 
i
 
</math> is equivalent to
 
the classical theorem of Minkowski–Hasse on quadratic forms, and in
 
the case of a projective group it is equivalent to the
 
Brauer–Hasse–Noether theorem on the splitting of simple
 
algebras. According to a conjecture of Serre one always has
 
<math>
 
{\rm Shaf}\;(G) = 0
 
</math> for a
 
simply-connected or adjoint semi-simple group. This conjecture was
 
proved for most simply-connected semi-simple groups over global number
 
fields (except for groups with simple components of type
 
<math>
 
E_8
 
</math>), and also for
 
arbitrary simply-connected algebraic groups over global function
 
fields.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> J.-P. Serre,
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> J.-P. Serre,   "Cohomologie Galoisienne" , Springer (1964)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.-P. Serre,   "Groupes algébrique et corps des classes" , Hermann (1959)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J.W.S. Cassels (ed.)  A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press (1986)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> H. Koch,   "Galoissche Theorie der <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100152.png" />-Erweiterungen" , Deutsch. Verlag Wissenschaft. (1970)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> E. Artin,   J. Tate,   "Class field theory" , Benjamin (1967)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> J.-P. Serre,   "Local fields" , Springer (1979) (Translated from French)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> A. Borel,   J.-P. Serre,   "Théorèmes de finitude en cohomologie Galoisienne" ''Comment Math. Helv.'' , '''39''' (1964) pp. 111–164</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> "Théorie des toposes et cohomologie étale des schémas" A. Grothendieck (ed.)  J.-L. Verdier (ed.)  E. Artin (ed.) , ''Sem. Geom. Alg. 4'' , '''1–3''' , Springer (1972)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> F. Bruhat,   J. Tits,   "Groupes réductifs sur un corps local I. Données radicielles valuées" ''Publ. Math. IHES'' : 41 (1972) pp. 5–252</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> A. Borel,   "Some finiteness properties of adèle groups over number fields" ''Publ. Math. IHES'' : 16 (1963) pp. 5–30</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> R. Steinberg,   "Regular elements of semisimple algebraic groups" ''Publ. Math. IHES'' : 25 (1965) pp. 49–80</TD></TR><TR><TD valign="top">[12a]</TD> <TD valign="top"> M. Kneser,   "Galois-Kohomologie halbeinfacher algebraischer Gruppen über <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100153.png" />-adische Körpern I" ''Math. Z.'' , '''88''' (1965) pp. 40–47</TD></TR><TR><TD valign="top">[12b]</TD> <TD valign="top"> M. Kneser,   "Galois-Kohomologie halbeinfacher algebraischer Gruppen über <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100154.png" />-adische Körpern II" ''Math. Z.'' , '''89''' (1965) pp. 250–272</TD></TR><TR><TD valign="top">[13a]</TD> <TD valign="top"> G. Harder,   "Ueber die Galoiskohomologie halbeinfacher Matrizengruppen I" ''Math. Z.'' , '''90''' (1965) pp. 404–428</TD></TR><TR><TD valign="top">[13b]</TD> <TD valign="top"> G. Harder,   "Ueber die Galoiskohomologie halbeinfacher Matrizengruppen II" ''Math. Z.'' , '''92''' (1966) pp. 396–415</TD></TR></table>
"Cohomologie Galoisienne" , Springer
 
(1964)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">
 
J.-P. Serre, "Groupes algébrique et corps des classes" , Hermann
 
(1959)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">
 
J.W.S. Cassels (ed.)  A. Fröhlich (ed.) , ''Algebraic number theory''
 
, Acad. Press
 
(1986)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">
 
H. Koch, "Galoissche Theorie der <math>
 
p
 
</math>-Erweiterungen" ,
 
Deutsch. Verlag Wissenschaft.
 
(1970)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">
 
E. Artin, J. Tate, "Class field theory" , Benjamin
 
(1967)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">
 
J.-P. Serre, "Local fields" , Springer (1979) (Translated from
 
French)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">
 
A. Borel, J.-P. Serre, "Théorèmes de finitude en cohomologie
 
Galoisienne" ''Comment Math. Helv.'' , '''39''' (1964)
 
pp. 111–164</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">
 
"Théorie des toposes et cohomologie étale des schémas" A. Grothendieck
 
(ed.)  J.-L. Verdier (ed.)  E. Artin (ed.) , ''Sem. Geom. Alg. 4'' ,
 
'''1–3''' , Springer
 
(1972)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">
 
F. Bruhat, J. Tits, "Groupes réductifs sur un corps local I. Données
 
radicielles valuées" ''Publ. Math. IHES'' : 41 (1972)
 
pp. 5–252</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">
 
A. Borel, "Some finiteness properties of adèle groups over number
 
fields" ''Publ. Math. IHES'' : 16 (1963)
 
pp. 5–30</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">
 
R. Steinberg, "Regular elements of semisimple algebraic groups"
 
''Publ. Math. IHES'' : 25 (1965)
 
pp. 49–80</TD></TR><TR><TD valign="top">[12a]</TD> <TD valign="top">
 
M. Kneser, "Galois-Kohomologie halbeinfacher algebraischer Gruppen
 
über <math>
 
p
 
</math>-adische Körpern
 
I" ''Math. Z.'' , '''88''' (1965)
 
pp. 40–47</TD></TR><TR><TD valign="top">[12b]</TD> <TD valign="top">
 
M. Kneser, "Galois-Kohomologie halbeinfacher algebraischer Gruppen
 
über <math>
 
p
 
</math>-adische Körpern
 
II" ''Math. Z.'' , '''89''' (1965)
 
pp. 250–272</TD></TR><TR><TD valign="top">[13a]</TD> <TD valign="top">
 
G. Harder, "Ueber die Galoiskohomologie halbeinfacher Matrizengruppen
 
I" ''Math. Z.'' , '''90''' (1965)
 
pp. 404–428</TD></TR><TR><TD valign="top">[13b]</TD> <TD valign="top">
 
G. Harder, "Ueber die Galoiskohomologie halbeinfacher Matrizengruppen
 
II" ''Math. Z.'' , '''92''' (1966) pp. 396–415</TD></TR></table>
 
  
  
  
====Comments==== Let <math>
+
====Comments====
G
+
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100155.png" /> be a finite (or pro-finite) group, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100156.png" /> a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100157.png" />-group, i.e. a group together with an action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100158.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100159.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100160.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100161.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100162.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100163.png" />-set, i.e. there is an action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100164.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100165.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100166.png" /> acts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100168.png" />-equivariantly on the right on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100169.png" /> if there is given a right action <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100170.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100171.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100172.png" />. Such a right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100173.png" />-set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100174.png" /> is a principal homogeneous space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100175.png" /> if the action makes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100176.png" /> an affine space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100177.png" /> (an affine version of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100178.png" />), i.e. if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100179.png" /> there is a unique <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100180.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100181.png" />. (This is precisely the situation of a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100182.png" /> and its corresponding affine space.) There is a natural bijective correspondence between isomorphism classes of principal homogeneous spaces over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100183.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100184.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100185.png" /> is a principal homogeneous space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100186.png" />, choose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100187.png" /> and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100188.png" /> define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100189.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100190.png" />. This defines the corresponding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100191.png" />-cocycle.
</math> be a finite (or
 
pro-finite) group, <math>
 
A
 
</math> a
 
<math>
 
G
 
</math>-group, i.e. a
 
group together with an action of <math>
 
G
 
</math> on
 
<math>
 
A
 
</math>,
 
<math>
 
(g,a) \mapsto g(a)
 
</math>, such that
 
<math>
 
g(ab) = g(a)g(b)
 
</math>, and let
 
<math>
 
E
 
</math> be a
 
<math>
 
G
 
</math>-set, i.e. there
 
is an action of <math>
 
G
 
</math> on
 
<math>
 
E
 
</math>. <math>
 
A
 
</math> acts
 
<math>
 
G
 
</math>-equivariantly on
 
the right on <math>
 
E
 
</math> if there is given
 
a right action <math>
 
E\times A \to E
 
</math>,
 
<math>
 
(x,a) \mapsto x.a
 
</math>, such that
 
<math>
 
g(x.a) = g(x)g(a)
 
</math>. Such a right
 
<math>
 
A
 
</math>-set
 
<math>
 
E
 
</math> is a principal
 
homogeneous space over <math>
 
A
 
</math> if the action
 
makes <math>
 
E
 
</math> an affine space
 
over <math>
 
A
 
</math> (an affine
 
version of <math>
 
A
 
</math>), i.e. if for all
 
<math>
 
x,y
 
</math> there is a unique
 
<math>
 
a \in A
 
</math> such that
 
<math>
 
x = y.a
 
</math>. (This is
 
precisely the situation of a vector space <math>
 
V
 
</math> and
 
its corresponding affine space.) There is a natural bijective
 
correspondence between isomorphism classes of principal homogeneous
 
spaces over <math>
 
A
 
</math> and
 
<math>
 
H^1(G,A)
 
</math>. If
 
<math>
 
E
 
</math> is a principal
 
homogeneous space over <math>
 
A
 
</math>, choose
 
<math>
 
x\in E
 
</math> and for
 
<math>
 
g\in G
 
</math> define
 
<math>
 
a_g
 
</math> by
 
<math>
 
g(x) = x.a_g
 
</math>. This defines the
 
corresponding <math>
 
1
 
</math>-cocycle.
 
  
Let <math>
+
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100192.png" /> be a cyclic Galois extension of (commutative) fields of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100193.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100194.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100195.png" /> be an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100196.png" />. Let the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100197.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100198.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100199.png" /> be constructed as follows: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100200.png" /> for some symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100201.png" />, with the multiplication rules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100202.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100203.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100204.png" />. This defines an associative non-commutative algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100205.png" />. Such an algebra is called a cyclic algebra. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100206.png" /> it is a central simple algebra with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100207.png" />. The Brauer–Hasse–Noether theorem, [[#References|[a8]]], now says that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100208.png" /> is a finite-dimensional division algebra over its centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100209.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100210.png" /> is an algebraic number field, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100211.png" /> is a cyclic algebra. The same conclusion holds if instead <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100212.png" /> is a finite extension of one of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100213.png" />-adic fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100214.png" />, [[#References|[a7]]].
K/F
 
</math> be a cyclic
 
Galois extension of (commutative) fields of degree
 
<math>
 
m
 
</math>. Let
 
<math>
 
Gal(K/F) = \{\sigma,\sigma^2,\dots,\sigma^m =1 \}
 
</math>. Let
 
<math>
 
b
 
</math> be an element of
 
<math>
 
K
 
</math>. Let the algebra
 
<math>
 
A
 
</math> of dimension
 
<math>
 
m
 
</math> over
 
<math>
 
K
 
</math> be constructed as
 
follows: <math>
 
A = K + yK + \cdots + y^{m-1}K
 
</math> for some symbol
 
<math>
 
y
 
</math>, with the
 
multiplication rules <math>
 
y^m = b,\; \alpha y = y \sigma(\alpha)
 
</math>, for all
 
<math>
 
\alpha \in K
 
</math>. This defines an
 
associative non-commutative algebra over <math>
 
F
 
</math>. Such
 
an algebra is called a cyclic algebra. If <math>
 
b\ne 0
 
</math> it is
 
a central simple algebra with centre <math>
 
F
 
</math>. The
 
Brauer–Hasse–Noether theorem, [[#References|[a8]]], now says that if
 
<math>
 
D
 
</math> is a
 
finite-dimensional division algebra over its centre
 
<math>
 
F
 
</math> and
 
<math>
 
F
 
</math> is an algebraic
 
number field, then <math>
 
D
 
</math> is a cyclic
 
algebra. The same conclusion holds if instead <math>
 
F
 
</math> is a
 
finite extension of one of the <math>
 
p
 
</math>-adic fields
 
<math>
 
{\Bbb Q}_p
 
</math>,
 
[[#References|[a7]]].
 
  
For the Minkowski–Hasse theorem on quadratic forms see [[Quadratic
+
For the Minkowski–Hasse theorem on quadratic forms see [[Quadratic form|Quadratic form]].
form|Quadratic form]].
 
  
Cohomology of Galois groups is also used in the birational
+
Cohomology of Galois groups is also used in the birational classification of rational varieties over not algebraically closed fields (cf. also [[Rational variety|Rational variety]]). An important birational invariant is the cohomology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100215.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100216.png" /> is the [[Picard group|Picard group]] of the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100217.png" /> which is defined over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100218.png" />. As in the case of algebraic groups, Galois cohomology provides important tools in the study of arithmetical properties of rational varieties. The use of Galois cohomology for the study of birational and arithmetical characteristics of rational varieties was initiated by Yu.I. Manin in the 1960s (see [[#References|[a1]]]) and was continued by J.-L. Colliot-Thélène and J.J. Sansuc (see [[#References|[a2]]]), V.E. Voskresenskii ([[#References|[a3]]]), etc.
classification of rational varieties over not algebraically closed
 
fields (cf. also [[Rational variety|Rational variety]]). An important
 
birational invariant is the cohomology group <math>
 
H^1(k,{\rm Pic}\; V)
 
</math>, where
 
<math>
 
{\rm Pic}\; V
 
</math> is the [[Picard
 
group|Picard group]] of the variety <math>
 
V
 
</math> which is defined
 
over a field <math>
 
k
 
</math>. As in the case
 
of algebraic groups, Galois cohomology provides important tools in the
 
study of arithmetical properties of rational varieties. The use of
 
Galois cohomology for the study of birational and arithmetical
 
characteristics of rational varieties was initiated by Yu.I. Manin in
 
the 1960s (see [[#References|[a1]]]) and was continued by
 
J.-L. Colliot-Thélène and J.J. Sansuc (see [[#References|[a2]]]),
 
V.E. Voskresenskii ([[#References|[a3]]]), etc.
 
  
It was proved recently (1988) by V.I. Chernusov [[#References|[a4]]]
+
It was proved recently (1988) by V.I. Chernusov [[#References|[a4]]] that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100219.png" /> for a simple group of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100220.png" /> over a number field. It follows that the [[Hasse principle|Hasse principle]] holds for simply-connected semi-simple algebraic groups over number fields.
that <math>
 
{\rm Shaf}\;(G) = 0
 
</math> for a simple
 
group of type <math>
 
E_8
 
</math> over a number
 
field. It follows that the [[Hasse principle|Hasse principle]] holds
 
for simply-connected semi-simple algebraic groups over number fields.
 
  
For a proof of the general case of the Kneser–Bruhat–Tits theorem see,
+
For a proof of the general case of the Kneser–Bruhat–Tits theorem see, e.g., [[#References|[a5]]].
e.g., [[#References|[a5]]].
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> Yu.I. Manin,
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> Yu.I. Manin,   "Cubic forms. Algebra, geometry, arithmetic" , North-Holland (1974) (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J.-L. Colliot-Thélène,   J.J. Sansuc,   "La descente sur les variétés rationnelles II" ''Duke Math. J.'' , '''54''' (1987) pp. 375–492</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> V.E. Voskresenskii,   "Algebraic tori" , Moscow (1977) (In Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> V.I. Chernusov,   "On the Hasse principle for groups of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043100/g043100221.png" />" (To appear) (In Russian)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> F. Bruhat,   J. Tits,   "Groupes réductifs sur un corps local III. Complements et applications à la cohomologie Galoisiènne" ''J. Fac. Sci. Univ. Tokyo'' , '''34''' (1987) pp. 671–698</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> G. Harder,   "Chevalley groups over function fields and automorphic forms" ''Ann. of Math.'' , '''100''' (1974) pp. 249–306</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> A.A. Albert,   "Structure of algebras" , Amer. Math. Soc.  (1939) pp. 143</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> R. Brauer,   H. Hasse,   E. Noether,   "Beweis eines Haupsatzes in der Theorie der Algebren" ''J. Reine Angew. Math.'' , '''107''' (1931) pp. 399–404</TD></TR></table>
"Cubic forms. Algebra, geometry, arithmetic" , North-Holland (1974)
 
(Translated from
 
Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">
 
J.-L. Colliot-Thélène, J.J. Sansuc, "La descente sur les variétés
 
rationnelles II" ''Duke Math. J.'' , '''54''' (1987)
 
pp. 375–492</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">
 
V.E. Voskresenskii, "Algebraic tori" , Moscow (1977) (In
 
Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">
 
V.I. Chernusov, "On the Hasse principle for groups of type
 
<math>
 
E_8
 
</math>" (To appear) (In
 
Russian)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">
 
F. Bruhat, J. Tits, "Groupes réductifs sur un corps local
 
III. Complements et applications à la cohomologie Galoisiènne"
 
''J. Fac. Sci. Univ. Tokyo'' , '''34''' (1987)
 
pp. 671–698</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">
 
G. Harder, "Chevalley groups over function fields and automorphic
 
forms" ''Ann. of Math.'' , '''100''' (1974)
 
pp. 249–306</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">
 
A.A. Albert, "Structure of algebras" , Amer. Math. Soc.  (1939)
 
pp. 143</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">
 
R. Brauer, H. Hasse, E. Noether, "Beweis eines Haupsatzes in der
 
Theorie der Algebren" ''J. Reine Angew. Math.'' , '''107''' (1931)
 
pp. 399–404</TD></TR></table>
 

Revision as of 23:04, 20 June 2011

Cohomology of a Galois group. Let be an Abelian group, let be the Galois group of an extension and suppose acts on ; the Galois cohomology groups will then be the cohomology groups

defined by the complex , where consists of all mappings and is the coboundary operator (cf. Cohomology of groups). If is an extension of infinite degree, an additional requirement is that the Galois topological group acts continuously on the discrete group , and continuous mappings are taken for the cochains in .

Usually, only zero-dimensional and one-dimensional cohomology are defined for a non-Abelian group . Namely, is the set of fixed points under the group in , while is the quotient set of the set of one-dimensional cocycles, i.e. continuous mappings that satisfy the relation

for all , by the equivalence relation , where if and only if for some and all . In the non-Abelian case is a set with a distinguished point corresponding to the trivial cocycle , where is the unit of , and usually has no group structure. Nevertheless, a standard cohomology formalism can be developed for such cohomology as well (cf. Non-Abelian cohomology).

If is the separable closure of a field , it is customary to denote the group by , and to write for .

Galois cohomology groups were implicitly present in the work of D. Hilbert, E. Artin, R. Brauer, H. Hasse, and C. Chevalley on class field theory, finite-dimensional simple algebras and quadratic forms. The development of the ideas and methods of homological algebra resulted in the introduction of Galois cohomology groups of finite extensions with values in an Abelian group by E. Artin, A. Weil, G. Hochschild, and J. Tate in the 1950s, in connection with class field theory. The general theory of Abelian Galois cohomology groups was then developed by Tate and J.-P. Serre [1], [3], [6].

Tate used Galois cohomology to introduce the concept of the cohomological dimension of the Galois group of a field (denoted by ). It is defined in terms of the cohomological -dimension , which is the smallest integer such that for any torsion -module and any integer the -primary component of the group is zero. The cohomological dimension is

For any algebraically closed field one has ; for all fields such that the Brauer group of an arbitrary extension is trivial, ; for the -adic field, the field of algebraic functions of one variable over a finite field of constants and for a totally-complex number field, [1]. Fields whose Galois group has cohomological dimension and whose Brauer group are called fields of dimension ; this is denoted by . Such fields include all finite fields, maximal unramified extensions of -adic fields, and the field of rational functions in one variable over an algebraically closed field of constants. If a Galois group is a pro--group, i.e. is the projective limit of finite -groups, the dimension of over is equal to the minimal number of topological generators of , while the dimension of is the number of defining relations between these generators. If , then is a free pro--group.

Non-Abelian Galois cohomology appeared in the late 1950s, but systematic research began only in the 1960s, mainly in response to the need for the classification of algebraic groups over not algebraically closed fields. One of the principal problems which stimulated the development of non-Abelian Galois cohomology is the task of classifying principal homogeneous spaces of group schemes. Galois cohomology groups proved to be specially effective in the problem of classifying types of algebraic varieties.

These problems led to the problem of computing the Galois cohomology groups of algebraic groups. The general theorems on the structure of algebraic groups essentially reduce the study of Galois cohomology groups to a separate consideration of the Galois cohomology groups of finite groups, unipotent groups, tori, semi-simple groups, and Abelian varieties.

The Galois cohomology groups of a connected unipotent group are trivial if is defined over a perfect field , i.e. for an arbitrary unipotent group , and for all if is an Abelian group. In particular, for the additive group of an arbitrary field one always has . For an imperfect field , in general .

One of the first significant facts about Galois cohomology groups was Hilbert's "Theorem 90" , one formulation of which states that (where is the multiplicative group). Moreover, for any -split algebraic torus one has . The computation of for an arbitrary -defined torus can be reduced, in the general case, to the computation of where is a Galois splitting field of ; so far (1989) this has only been accomplished for special fields. The case when is an algebraic number field is especially important in practical applications. Duality theorems, with various applications, have been developed for this case.

Let be a Galois extension of finite degree, let be the group of adèles (cf. Adèle) of a multiplicative -group , and let be the group of characters of a torus. The duality theorem states that the cup-product

defines non-degenerate pairing for . This theorem was used to find the formula for expressing the Tamagawa numbers (cf. Tamagawa number) of the torus by invariants connected with its Galois cohomology groups. Other important duality theorems for Galois cohomology groups also exist [1].

It has been proved [11] that the groups over fields of dimension are trivial. A natural class of fields has been distinguished with only a finite number of extensions of a given degree (the so-called type fields); these include, for example, the -adic number fields. It was proved [1] that for any algebraic group over a field of type the cohomology group is a finite set.

The theory of Galois cohomology of semi-simple algebraic groups has far-reaching arithmetical and analytical applications. The Kneser–Bruhat–Tits theorem states that for simply-connected semi-simple algebraic groups over local fields whose residue field has cohomological dimension . This theorem was first proved for -adic number fields , after which a proof was obtained for the general case. It was proved

that is trivial for a field of algebraic functions in one variable over a finite field of constants. In all these cases the cohomological dimension , which confirms the general conjecture of Serre to the effect that is trivial for simply-connected semi-simple over fields with .

Let be a global field, let be the set of all non-equivalent valuations of , let be the completion of . The imbeddings induce a natural mapping

for an arbitrary algebraic group defined over , the kernel of which is denoted by and, in the case of Abelian varieties, is called the Tate–Shafarevich group. The group measures the extent to which the Galois cohomology groups over a global field are described by Galois cohomology groups over localizations. The principal result on for linear algebraic groups is due to A. Borel, who proved that is finite. There exists a conjecture according to which is finite in the case of Abelian varieties as well. The situation in which , i.e. the mapping is injective, is a special case. One then says that the Hasse principle applies to . This terminology is explained by the fact that for an orthogonal group the injectivity of is equivalent to the classical theorem of Minkowski–Hasse on quadratic forms, and in the case of a projective group it is equivalent to the Brauer–Hasse–Noether theorem on the splitting of simple algebras. According to a conjecture of Serre one always has for a simply-connected or adjoint semi-simple group. This conjecture was proved for most simply-connected semi-simple groups over global number fields (except for groups with simple components of type ) , and also for arbitrary simply-connected algebraic groups over global function fields.

References

[1] J.-P. Serre, "Cohomologie Galoisienne" , Springer (1964)
[2] J.-P. Serre, "Groupes algébrique et corps des classes" , Hermann (1959)
[3] J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1986)
[4] H. Koch, "Galoissche Theorie der -Erweiterungen" , Deutsch. Verlag Wissenschaft. (1970)
[5] E. Artin, J. Tate, "Class field theory" , Benjamin (1967)
[6] J.-P. Serre, "Local fields" , Springer (1979) (Translated from French)
[7] A. Borel, J.-P. Serre, "Théorèmes de finitude en cohomologie Galoisienne" Comment Math. Helv. , 39 (1964) pp. 111–164
[8] "Théorie des toposes et cohomologie étale des schémas" A. Grothendieck (ed.) J.-L. Verdier (ed.) E. Artin (ed.) , Sem. Geom. Alg. 4 , 1–3 , Springer (1972)
[9] F. Bruhat, J. Tits, "Groupes réductifs sur un corps local I. Données radicielles valuées" Publ. Math. IHES : 41 (1972) pp. 5–252
[10] A. Borel, "Some finiteness properties of adèle groups over number fields" Publ. Math. IHES : 16 (1963) pp. 5–30
[11] R. Steinberg, "Regular elements of semisimple algebraic groups" Publ. Math. IHES : 25 (1965) pp. 49–80
[12a] M. Kneser, "Galois-Kohomologie halbeinfacher algebraischer Gruppen über -adische Körpern I" Math. Z. , 88 (1965) pp. 40–47
[12b] M. Kneser, "Galois-Kohomologie halbeinfacher algebraischer Gruppen über -adische Körpern II" Math. Z. , 89 (1965) pp. 250–272
[13a] G. Harder, "Ueber die Galoiskohomologie halbeinfacher Matrizengruppen I" Math. Z. , 90 (1965) pp. 404–428
[13b] G. Harder, "Ueber die Galoiskohomologie halbeinfacher Matrizengruppen II" Math. Z. , 92 (1966) pp. 396–415


Comments

Let be a finite (or pro-finite) group, a -group, i.e. a group together with an action of on , , such that , and let be a -set, i.e. there is an action of on . acts -equivariantly on the right on if there is given a right action , , such that . Such a right -set is a principal homogeneous space over if the action makes an affine space over (an affine version of ), i.e. if for all there is a unique such that . (This is precisely the situation of a vector space and its corresponding affine space.) There is a natural bijective correspondence between isomorphism classes of principal homogeneous spaces over and . If is a principal homogeneous space over , choose and for define by . This defines the corresponding -cocycle.

Let be a cyclic Galois extension of (commutative) fields of degree . Let . Let be an element of . Let the algebra of dimension over be constructed as follows: for some symbol , with the multiplication rules , for all . This defines an associative non-commutative algebra over . Such an algebra is called a cyclic algebra. If it is a central simple algebra with centre . The Brauer–Hasse–Noether theorem, [a8], now says that if is a finite-dimensional division algebra over its centre and is an algebraic number field, then is a cyclic algebra. The same conclusion holds if instead is a finite extension of one of the -adic fields , [a7].

For the Minkowski–Hasse theorem on quadratic forms see Quadratic form.

Cohomology of Galois groups is also used in the birational classification of rational varieties over not algebraically closed fields (cf. also Rational variety). An important birational invariant is the cohomology group , where is the Picard group of the variety which is defined over a field . As in the case of algebraic groups, Galois cohomology provides important tools in the study of arithmetical properties of rational varieties. The use of Galois cohomology for the study of birational and arithmetical characteristics of rational varieties was initiated by Yu.I. Manin in the 1960s (see [a1]) and was continued by J.-L. Colliot-Thélène and J.J. Sansuc (see [a2]), V.E. Voskresenskii ([a3]), etc.

It was proved recently (1988) by V.I. Chernusov [a4] that for a simple group of type over a number field. It follows that the Hasse principle holds for simply-connected semi-simple algebraic groups over number fields.

For a proof of the general case of the Kneser–Bruhat–Tits theorem see, e.g., [a5].

References

[a1] Yu.I. Manin, "Cubic forms. Algebra, geometry, arithmetic" , North-Holland (1974) (Translated from Russian)
[a2] J.-L. Colliot-Thélène, J.J. Sansuc, "La descente sur les variétés rationnelles II" Duke Math. J. , 54 (1987) pp. 375–492
[a3] V.E. Voskresenskii, "Algebraic tori" , Moscow (1977) (In Russian)
[a4] V.I. Chernusov, "On the Hasse principle for groups of type " (To appear) (In Russian)
[a5] F. Bruhat, J. Tits, "Groupes réductifs sur un corps local III. Complements et applications à la cohomologie Galoisiènne" J. Fac. Sci. Univ. Tokyo , 34 (1987) pp. 671–698
[a6] G. Harder, "Chevalley groups over function fields and automorphic forms" Ann. of Math. , 100 (1974) pp. 249–306
[a7] A.A. Albert, "Structure of algebras" , Amer. Math. Soc. (1939) pp. 143
[a8] R. Brauer, H. Hasse, E. Noether, "Beweis eines Haupsatzes in der Theorie der Algebren" J. Reine Angew. Math. , 107 (1931) pp. 399–404
How to Cite This Entry:
Galois cohomology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Galois_cohomology&oldid=19460
This article was adapted from an original article by E.A. NisnevichV.P. Platonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article