View source for Galois cohomology

{{TEX|done}}
{{MSC|12G05|11-02,11R34}}

Galois cohomology studies the cohomology of a [[Galois group|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|Cohomology of groups]]). If
$K/k$ is an extension of infinite degree, an additional requirement is that the [[Galois topological group|Galois topological group]] acts continuously on the
discrete group $M$, and continuous
mappings are taken for the cochains in $C^n$.
<br />
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 $\sim$, where
$z_1 \sim z_2$ 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 {{Cite|Se}},
{{Cite|CaFr}}, {{Cite|Se3}}.

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
$$\underset{p}{\rm sup}\; {\rm cd}_p\; G_k$$

For any algebraically closed field $k$ one has
${\rm cd}\; G_k = 0$; for all fields
$k$ such that the
[[Brauer group|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,
${\rm cd}\; G_k = 1$
{{Cite|Se}}. 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|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|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|Tamagawa number]]) of the torus
$T$ by invariants
connected with its Galois cohomology groups. Other important duality
theorems for Galois cohomology groups also exist {{Cite|Se}}.

It has been proved [[#References|[11]]] that the groups
$H^1(k,G)$ 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 {{Cite|Se}} 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
$$i : H^1(k,G) \to \displaystyle\prod_{\nu\in V} H^1(k_\nu,G)$$
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|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====
{|
|-
|valign="top"|{{Ref|ArTa}}||valign="top"| E. Artin, J. Tate, "Class field theory", Benjamin (1967)      
|-
|valign="top"|{{Ref|Bo}}||valign="top"| A. Borel, "Some finiteness properties of adèle groups over number fields" ''Publ. Math. IHES'' : 16 (1963) pp. 5–30  {{MR|0202718}}  {{ZBL|0135.08902}}    
|-
|valign="top"|{{Ref|BoSe}}||valign="top"| A. Borel, J.-P. Serre, "Théorèmes de finitude en cohomologie Galoisienne" ''Comment Math. Helv.'', '''39''' (1964) pp. 111–164  {{MR|0181643}}  {{ZBL|0143.05901}}    
|-
|valign="top"|{{Ref|BrTi}}||valign="top"| F. Bruhat, J. Tits, "Groupes réductifs sur un corps local I. Données radicielles valuées" ''Publ. Math. IHES'' : 41 (1972) pp. 5–252      
|-
|valign="top"|{{Ref|CaFr}}||valign="top"| J.W.S. Cassels (ed.)  A. Fröhlich (ed.), ''Algebraic number theory'', Acad. Press (1986)      
|-
|valign="top"|{{Ref|GrVeAr}}||valign="top"| A. Grothendieck (ed.)  J.-L. Verdier (ed.)  E. Artin (ed.), "Théorie des toposes et cohomologie étale des schémas" ''Sem. Geom. Alg. 4'', '''1–3''', Springer (1972)      
|-
|valign="top"|{{Ref|Ha}}||valign="top"| G. Harder, "Über die Galoiskohomologie halbeinfacher Matrizengruppen I" ''Math. Z.'', '''90''' (1965) pp. 404–428    {{MR|0190264}}  {{ZBL|0152.00903}} 
|-
|valign="top"|{{Ref|Ha2}}||valign="top"| G. Harder, "Über die Galoiskohomologie halbeinfacher Matrizengruppen II" ''Math. Z.'', '''92''' (1966) pp. 396–415   {{MR|0202918}}  {{ZBL|0152.01001}} 
|-
|valign="top"|{{Ref|Kn}}||valign="top"| M. Kneser, "Galois-Kohomologie halbeinfacher algebraischer Gruppen über $p$-adische Körpern I" ''Math. Z.'', '''88''' (1965) pp. 40–47  {{MR|0188219}} {{MR|0174559}}     
|-
|valign="top"|{{Ref|Kn2}}||valign="top"| M. Kneser, "Galois-Kohomologie halbeinfacher algebraischer Gruppen über $p$-adische Körpern II" ''Math. Z.'', '''89''' (1965) pp. 250–272  {{MR|0188219}} {{MR|0174559}}     
|-
|valign="top"|{{Ref|Ko}}||valign="top"| H. Koch, "Galoissche Theorie der $p$-Erweiterungen", Deutsch. Verlag Wissenschaft. (1970)  {{MR|0291139}}  {{ZBL|0216.04704}}    
|-
|valign="top"|{{Ref|Se}}||valign="top"| J.-P. Serre, "Cohomologie Galoisienne", Springer (1964)  {{MR|0181643}}  {{ZBL|0143.05901}} {{ZBL|0128.26303}}    
|-
|valign="top"|{{Ref|Se2}}||valign="top"| J.-P. Serre, "Groupes algébrique et corps des classes", Hermann (1959)  {{MR|0103191}}     
|-
|valign="top"|{{Ref|Se3}}||valign="top"| J.-P. Serre, "Local fields", Springer (1979) (Translated from French)  {{MR|0554237}}  {{ZBL|0423.12016}}    
|-
|valign="top"|{{Ref|St}}||valign="top"| R. Steinberg, "Regular elements of semisimple algebraic groups" ''Publ. Math. IHES'' : 25 (1965) pp. 49–80  {{MR|0180554}}  {{ZBL|0136.30002}}    
|-
|}




====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, [[#References|[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
${\Bbb Q}_p$,
[[#References|[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|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 [[#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 ${\rm Shaf}\;(G) = 0$ for a simple
group of type $E_8$ 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]]].

====References====
{|
|-
|valign="top"|{{Ref|Al}}||valign="top"| A.A. Albert, "Structure of algebras", Amer. Math. Soc.  (1939) pp. 143  {{MR|0000595}}  {{ZBL|0023.19901}}    
|-
|valign="top"|{{Ref|BrHaNo}}||valign="top"| R. Brauer, H. Hasse, E. Noether, "Beweis eines Hauptsatzes in der Theorie der Algebren" ''J. Reine Angew. Math.'', '''107''' (1932) pp. 399–404      {{ZBL|0003.24404}} 
|-
|valign="top"|{{Ref|BrTi2}}||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      
|-
|valign="top"|{{Ref|CoSa}}||valign="top"| J.-L. Colliot-Thélène, J.J. Sansuc, "La descente sur les variétés rationnelles II" ''Duke Math. J.'', '''54''' (1987) pp. 375–492   {{MR|0899402}} {{ZBL|0659.14028}}    
|-
|valign="top"|{{Ref|Ha3}}||valign="top"| G. Harder, "Chevalley groups over function fields and automorphic forms" ''Ann. of Math.'', '''100''' (1974) pp. 249–306  {{MR|0563090}}  {{ZBL|0309.14041}}    
|-
|valign="top"|{{Ref|Ma}}||valign="top"| Yu.I. Manin, "Cubic forms. Algebra, geometry, arithmetic", North-Holland (1974) (Translated from Russian)  {{MR|0460349}}  {{ZBL|0277.14014}}    
|-
|valign="top"|{{Ref|Vo}}||valign="top"| V.E. Voskresenskii, "Algebraic tori", Moscow (1977) (In Russian)  {{MR|0506279}} {{MR|0472845}}  {{ZBL|0379.14001}} {{ZBL|0367.14007}}    
|-
|}