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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
+
Cohomology of a [[Galois group|Galois group]]. Let <math>
 +
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>
 +
\equiv
 +
</math>, where
 +
<math>
 +
z_1 \equiv 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]]).
  
<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>
+
If <math>
 +
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>.
  
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" />.
+
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]]].
  
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
+
Tate used Galois cohomology to introduce the concept of the
 +
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>
 +
\displaystyle\mathop{{\rm sup}}_p {\rm cd}_p\; G_k
 +
</math></center><br />
  
<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>
+
For any algebraically closed field <math>
 +
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.
  
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]]).
+
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.
  
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" />.
+
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.
  
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]]].
+
The Galois cohomology groups of a connected unipotent group
 +
<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>.
  
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
+
One of the first significant facts about Galois cohomology groups was
 +
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.
  
<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>
+
Let <math>
 +
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]]].
  
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.
+
It has been proved [[#References|[11]]] that the groups
 +
<math>
 +
H^(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.
  
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.
+
The theory of Galois cohomology of semi-simple algebraic groups has
 +
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>.
  
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.
+
Let <math>
 
+
K
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" />.
+
</math> be a global
 
+
field, let <math>
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.
+
V
 
+
</math> be the set of all
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
+
non-equivalent valuations of <math>
 
+
k
<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>, 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/g043100108.png" /></td> </tr></table>
+
k_\nu
 
+
</math> be the completion
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]]].
+
of <math>
 
+
k
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>. The imbeddings
 
+
<math>
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
+
k\to k_\nu
 
+
</math> induce a natural mapping
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" />.
+
<br /><center><math>
 
+
i : H^(k,G) \to \Pi_{\nu\in V} H^1(k_\nu,G)
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
+
</math></center><br />
 
+
for an arbitrary algebraic group <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/g043100138.png" /></td> </tr></table>
+
G
 
+
</math> defined over <math>
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.
+
k
 +
</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,   "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>
+
<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 <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====
+
====Comments==== Let <math>
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.
+
G
 +
</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 <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]]].
+
Let <math>
 +
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 form|Quadratic form]].
+
For the Minkowski–Hasse theorem on quadratic forms see [[Quadratic
 +
form|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|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.
+
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 <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]]] 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.
+
It was proved recently (1988) by V.I. Chernusov [[#References|[a4]]]
 +
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, e.g., [[#References|[a5]]].
+
For a proof of the general case of the Kneser–Bruhat–Tits theorem see,
 +
e.g., [[#References|[a5]]].
  
 
====References====
 
====References====
<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>
+
<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
 +
<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 22:30, 20 June 2011

Cohomology of a Galois group. Let \( M \) be an Abelian group, let \( G(K/k) \) be the Galois group of an extension \( K/k \) and suppose \( G(K/k) \) acts on \( M \); the Galois cohomology groups will then be the cohomology groups


\( H^n(K/k,M) = H^n(G(K/k),M), n \ge 0, \)


defined by the complex \( (C^n,d) \), where \( C^n \) consists of all mappings \( G(K/k)^n \to M \) and \( d \) is the coboundary operator (cf. Cohomology of groups). If \( K/k \) is an extension of infinite degree, an additional requirement is that the Galois topological group acts continuously on the discrete group \( M \), and continuous mappings are taken for the cochains in \( C^n \).
Usually, only zero-dimensional \( (H^0) \) and one-dimensional \( (H^1) \) cohomology are defined for a non-Abelian group \( M \). Namely, \( H^0(K/k,M) = M^{G(K/k)} \) is the set of fixed points under the group \( G(K/k) \) in \( M \), while \( H^1(K/k,M) \) is the quotient set of the set of one-dimensional cocycles, i.e. continuous mappings \( z:G(K/k) \to M \) that satisfy the relation


\( z(g_1g_2) = z(g_1)\;^{g_1}z(g_2) \)


for all \( g_1, g_2 \in G(K/k) \), by the equivalence relation \( \equiv \), where \( z_1 \equiv z_2 \) if and only if \( z_1(g) = m^{-1} \; z_2(g)\; ^gm \) for some \( m \in M \) and all \( g \in G(K/k) \). In the non-Abelian case \( H^1(K/k,M) \) is a set with a distinguished point corresponding to the trivial cocycle \( G(K/k) \to (e) \), where \( e \) is the unit of \( M \), 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 \( K=k_s \) is the separable closure of a field \( k \), it is customary to denote the group \( G(k_s/k) \) by \( G_k \), and to write \( H^n(k,M) \) for \( H^n(k_s/k,M) \).

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 \( G_k \) of a field \( k \) (denoted by \( {\rm cd}\; G_k \)). It is defined in terms of the cohomological \( p \)-dimension \( {\rm cd}_p\; G_k \), which is the smallest integer \( n \) such that for any torsion \( G_k \)-module \( A \) and any integer \( q > n \) the \( p \)-primary component of the group \( H^q(G_k,A) \) is zero. The cohomological dimension \( {\rm cd}\; G_k \) is


\( \displaystyle\mathop[[:Template:\rm sup]]_p {\rm cd}_p\; G_k \)


For any algebraically closed field \( k \) one has \( {\rm cd}\; G_k = 0 \); for all fields \( k \) such that the Brauer group \( B(K) \) of an arbitrary extension \( K/k \) is trivial, \( {\rm cd}\; G_k \le 1 \); for the \( p \)-adic field, the field of algebraic functions of one variable over a finite field of constants and for a totally-complex number field, \( {\rm cd}\; G_k = 1 \) [1]. Fields \( k \) whose Galois group has cohomological dimension \( \le 1 \) and whose Brauer group \( B(k) = 0 \) are called fields of dimension \( \le 1 \); this is denoted by \( {\rm dim}\; k \le 1 \). Such fields include all finite fields, maximal unramified extensions of \( p \)-adic fields, and the field of rational functions in one variable over an algebraically closed field of constants. If a Galois group \( G(K/k) \) is a pro-\( p \)-group, i.e. is the projective limit of finite \( p \)-groups, the dimension of \( H^1(G(K/k),{\Bbb Z}/p{\Bbb Z} \) over \( {\Bbb Z}/p{\Bbb Z} \) is equal to the minimal number of topological generators of \( G(K/k) \), while the dimension of \( H^2(G(K/k),{\Bbb Z}/p{Bbb Z} \) is the number of defining relations between these generators. If \( {\rm cd}\; G(K/k) = 1; \), then \( G(K/k) \) is a free pro-\( p \)-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 \( U \) are trivial if \( U \) is defined over a perfect field \( k \), i.e. \( H^1(k,U) = 0 \) for an arbitrary unipotent group \( U \), and \( H^n(k,U) = 0 \) for all \( n \ge 1 \) if \( U \) is an Abelian group. In particular, for the additive group \( G_\alpha \) of an arbitrary field one always has \( H^1(k,G_\alpha) \). For an imperfect field \( k \), in general \( H^1(k,G\alpha) \ne 0 \).

One of the first significant facts about Galois cohomology groups was Hilbert's "Theorem 90" , one formulation of which states that \( H^1(k,G_m) = 0 \) (where \( G_m \) is the multiplicative group). Moreover, for any \( k \)-split algebraic torus \( T \) one has \( H^1(k,T) = 0 \). The computation of \( H^1(k,T) \) for an arbitrary \( k \)-defined torus \( T \) can be reduced, in the general case, to the computation of \( H^1(K/k,T) \) where \( K \) is a Galois splitting field of \( T \); so far (1989) this has only been accomplished for special fields. The case when \( k \) is an algebraic number field is especially important in practical applications. Duality theorems, with various applications, have been developed for this case.

Let \( K/k \) be a [[Galois extension|Galois extension]] of finite degree, let \( C((K) \) be the group of adèles (cf. Adèle) of a multiplicative \( K \)-group \( G_m \), and let \( \hat T = {\rm Hom}_k(T,G_m) \) be the group of characters of a torus. The duality theorem states that the cup-product


\( H^{2-r}(K/k),\hat T)\times H^r(K/k,{\rm Hom}(\hat T,C(K))) \to H^2(K/k,C(K)) \)


defines non-degenerate pairing for \( r = 0,1,2 \). This theorem was used to find the formula for expressing the Tamagawa numbers (cf. Tamagawa number) of the torus \( T \) 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 \( H^(k,G) \) over fields \( k \) of dimension \( \le 1 \) 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 \( (F) \) fields); these include, for example, the \( p \)-adic number fields. It was proved [1] that for any algebraic group \( G \) over a field \( k \) of type \( (F) \) the cohomology group \( H^1(k,G) \) 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 \( H^1(k,G) = 0 \) for simply-connected semi-simple algebraic groups \( G \) over local fields \( k \) whose residue field has cohomological dimension \( \le 1 \). This theorem was first proved for \( p \)-adic number fields , after which a proof was obtained for the general case. It was proved that \( H^1(k,G) \) is trivial for a field of algebraic functions in one variable over a finite field of constants. In all these cases the cohomological dimension \( {\rm cd}\; G_k \le 2 \), which confirms the general conjecture of Serre to the effect that \( H^1(k,G) \) is trivial for simply-connected semi-simple \( G \) over fields \( k \) with \( {\rm cd}\; G_k \le 2 \).

Let \( K \) be a global field, let \( V \) be the set of all non-equivalent valuations of \( k \), let \( k_\nu \) be the completion of \( k \). The imbeddings \( k\to k_\nu \) induce a natural mapping


\( i : H^(k,G) \to \Pi_{\nu\in V} H^1(k_\nu,G) \)


for an arbitrary algebraic group \( G \) defined over \( k \), the kernel of which is denoted by \( {\rm Shaf}\;(G) \) and, in the case of Abelian varieties, is called the Tate–Shafarevich group. The group \( {\rm Shaf}\; (G) \) 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 \( {\rm Shaf}\;(G) \) for linear algebraic groups is due to A. Borel, who proved that \( {\rm Shaf}\;(G) \) is finite. There exists a conjecture according to which \( {\rm Shaf}\;(G) \) is finite in the case of Abelian varieties as well. The situation in which \( {\rm Shaf}\;(G) = 0 \), i.e. the mapping \( i \) is injective, is a special case. One then says that the Hasse principle applies to \( G \). This terminology is explained by the fact that for an orthogonal group the injectivity of \( i \) 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 \( {\rm Shaf}\;(G) = 0 \) 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 \( E_8 \)), 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 \( p \)-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 \( p \)-adische Körpern I" Math. Z. , 88 (1965)

pp. 40–47
[12b]

M. Kneser, "Galois-Kohomologie halbeinfacher algebraischer Gruppen über \( p \)-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 \( G \) be a finite (or pro-finite) group, \( A \) a \( G \)-group, i.e. a group together with an action of \( G \) on \( A \), \( (g,a) \mapsto g(a) \), such that \( g(ab) = g(a)g(b) \), and let \( E \) be a \( G \)-set, i.e. there is an action of \( G \) on \( E \). \( A \) acts \( G \)-equivariantly on the right on \( E \) if there is given a right action \( E\times A \to E \), \( (x,a) \mapsto x.a \), such that \( g(x.a) = g(x)g(a) \). Such a right \( A \)-set \( E \) is a principal homogeneous space over \( A \) if the action makes \( E \) an affine space over \( A \) (an affine version of \( A \)), i.e. if for all \( x,y \) there is a unique \( a \in A \) such that \( x = y.a \). (This is precisely the situation of a vector space \( V \) and its corresponding affine space.) There is a natural bijective correspondence between isomorphism classes of principal homogeneous spaces over \( A \) and \( H^1(G,A) \). If \( E \) is a principal homogeneous space over \( A \), choose \( x\in E \) and for \( g\in G \) define \( a_g \) by \( g(x) = x.a_g \). This defines the corresponding \( 1 \)-cocycle.

Let \( K/F \) be a cyclic Galois extension of (commutative) fields of degree \( m \). Let \( Gal(K/F) = \{\sigma,\sigma^2,\dots,\sigma^m =1 \} \). Let \( b \) be an element of \( K \). Let the algebra \( A \) of dimension \( m \) over \( K \) be constructed as follows\[ A = K + yK + \cdots + y^{m-1}K \] for some symbol \( y \), with the multiplication rules \( y^m = b,\; \alpha y = y \sigma(\alpha) \), for all \( \alpha \in K \). This defines an associative non-commutative algebra over \( F \). Such an algebra is called a cyclic algebra. If \( b\ne 0 \) it is a central simple algebra with centre \( F \). The Brauer–Hasse–Noether theorem, [a8], now says that if \( D \) is a finite-dimensional division algebra over its centre \( F \) and \( F \) is an algebraic number field, then \( D \) is a cyclic algebra. The same conclusion holds if instead \( F \) is a finite extension of one of the \( p \)-adic fields \( {\Bbb Q}_p \), [a7].

For the Minkowski–Hasse theorem on quadratic forms see [[Quadratic form|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 \( H^1(k,{\rm Pic}\; V) \), where \( {\rm Pic}\; V \) is the [[Picard group|Picard group]] of the variety \( V \) which is defined over a field \( k \). 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 \( {\rm Shaf}\;(G) = 0 \) for a simple group of type \( E_8 \) 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 \( E_8 \)" (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=19459
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