Cohomology of a
Galois group.
Let
be an Abelian group, let
be the Galois group of an extension
and suppose
acts on
;
the Galois cohomology groups will then be the cohomology groups
defined by the complex
,
where
consists of all mappings
and
is the coboundary operator (cf.
Cohomology of groups).
If
is an extension of infinite degree, an additional requirement is that the
Galois topological group
acts continuously on the discrete group
,
and continuous mappings are taken for the cochains in
.
Usually, only zero-dimensional
and one-dimensional
cohomology are defined for a non-Abelian group
.
Namely,
is the set of fixed points under the group
in
,
while
is the quotient set of the set of one-dimensional cocycles, i.e. continuous mappings
that satisfy the relation
for all
,
by the equivalence relation
,
where
if and only if
for some
and all
.
In the non-Abelian case
is a set with a distinguished point corresponding to the trivial cocycle
,
where
is the unit of
,
and usually has no group structure. Nevertheless, a
standard cohomology formalism can be developed for such cohomology as well (cf.
Non-Abelian cohomology).
If
is the separable closure of a field
,
it is customary to denote the group
by
,
and to write
for
.
Galois cohomology groups were implicitly present in the work of
D. Hilbert,
E. Artin,
R. Brauer,
H. Hasse,
and
C. Chevalley
on class field theory, finite-dimensional simple algebras and quadratic forms.
The development of the ideas and methods of homological algebra
resulted in the introduction of Galois cohomology groups of finite extensions
with values in an Abelian group by
E. Artin,
A. Weil,
G. Hochschild,
and
J. Tate
in the
1950s,
in connection with class field theory. The general theory of
Abelian Galois cohomology groups was then developed by Tate and
J.-P. Serre
[1],
[3],
[6].
Tate used Galois cohomology to introduce the concept
of the cohomological dimension of the Galois group
of a field
(denoted by
).
It is defined in terms of the
cohomological
-dimension
,
which is the smallest integer
such that for any torsion
-module
and any integer
the
-primary
component of the group
is zero. The
cohomological dimension
is
 |
For any algebraically closed field
one has
;
for all fields
such that the
Brauer group
of an arbitrary extension
is trivial,
;
for the
-adic
field, the field of algebraic functions of one variable over a
finite field of constants and for a totally-complex number field,
[1].
Fields
whose Galois group has cohomological dimension
and whose Brauer group
are called
fields of dimension
;
this is denoted by
.
Such fields include all finite fields, maximal unramified extensions of
-adic
fields, and the field of rational functions in one
variable over an algebraically closed field of constants. If a Galois group
is a
pro-
-group,
i.e. is the projective limit of finite
-groups,
the dimension of
over
is equal to the minimal number of topological generators of
,
while the dimension of
is the number of defining relations between these generators. If
,
then
is a free pro-
-group.
Non-Abelian Galois cohomology appeared in the late
1950s,
but
systematic research began only in the
1960s,
mainly in response
to the need for the classification of algebraic groups over
not algebraically closed fields. One of the principal
problems which stimulated the development of non-Abelian Galois
cohomology is the task of classifying principal homogeneous spaces
of group schemes. Galois cohomology groups proved to be specially
effective in the problem of classifying types of algebraic varieties.
These problems led to the problem of computing the Galois cohomology
groups of algebraic groups. The general theorems on the structure of
algebraic groups essentially reduce the study of Galois cohomology groups
to a separate consideration of the Galois cohomology groups
of finite groups, unipotent groups, tori, semi-simple groups, and Abelian varieties.
The Galois cohomology groups of a connected unipotent group
are trivial if
is defined over a perfect field
,
i.e.
for an arbitrary unipotent group
,
and
for all
if
is an Abelian group. In particular, for the additive group
of an arbitrary field one always has
.
For an imperfect field
,
in general
.
One of the first significant facts about Galois cohomology groups
was Hilbert's
"Theorem 90" ,
one formulation of which states that
(where
is the multiplicative group). Moreover, for any
-split
algebraic torus
one has
.
The computation of
for an arbitrary
-defined
torus
can be reduced, in the general case, to the computation of
where
is a Galois splitting field of
;
so far
(1989)
this has only been accomplished for special fields. The case when
is an algebraic number field is especially important
in practical applications. Duality theorems, with various
applications, have been developed for this case.
Let
be a
Galois extension
of finite degree, let
be the group of adèles (cf.
Adèle)
of a multiplicative
-group
,
and let
be the group of characters of a torus. The duality theorem states that the cup-product
defines non-degenerate pairing for
.
This theorem was used to find the formula for expressing the Tamagawa numbers (cf.
Tamagawa number)
of the torus
by invariants connected with its Galois
cohomology groups. Other important duality theorems
for Galois cohomology groups also exist
[1].
It has been proved
[11]
that the groups
over fields
of dimension
are trivial. A natural class of fields has been distinguished with only
a finite number of extensions of a given degree (the so-called
type
fields);
these include, for example, the
-adic
number fields. It was proved
[1]
that for any algebraic group
over a field
of type
the cohomology group
is a finite set.
The theory of Galois cohomology of semi-simple
algebraic groups has far-reaching arithmetical and analytical applications. The
Kneser–Bruhat–Tits theorem
states that
for simply-connected semi-simple algebraic groups
over local fields
whose residue field has cohomological dimension
.
This theorem was first proved for
-adic
number fields
,
after which a proof was obtained for the general case. It was proved
that
is trivial for a field of algebraic functions in one variable over
a finite field of constants. In all these cases the cohomological dimension
,
which confirms the
general conjecture of Serre
to the effect that
is trivial for simply-connected semi-simple
over fields
with
.
Let
be a global field, let
be the set of all non-equivalent valuations of
,
let
be the completion of
.
The imbeddings
induce a natural mapping
for an arbitrary algebraic group
defined over
,
the kernel of which is denoted by
and, in the case of Abelian varieties, is called the
Tate–Shafarevich group.
The group
measures the extent to which the Galois cohomology groups over a
global field are described by Galois cohomology
groups over localizations. The principal result on
for linear algebraic groups is due to
A. Borel,
who proved that
is finite. There exists a conjecture according to which
is finite in the case of Abelian varieties as well. The situation in which
,
i.e. the mapping
is injective, is a special case. One then says that the
Hasse principle
applies to
.
This terminology is explained by the fact that for an orthogonal group the injectivity of
is equivalent to the classical theorem of Minkowski–Hasse on quadratic forms, and
in the case of a projective group it is
equivalent to the Brauer–Hasse–Noether theorem on the splitting of
simple algebras. According to a conjecture of Serre one always has
for a simply-connected or adjoint semi-simple group.
This conjecture was proved for most simply-connected semi-simple groups
over global number fields (except for groups with simple components of type
)
,
and also for arbitrary simply-connected algebraic groups over global function fields.