Galois cohomology

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

References

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

Comments

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

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

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

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

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

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

References

[a1]	Yu.I. Manin, "Cubic forms. Algebra, geometry, arithmetic" , North-Holland (1974) (Translated from Russian)
[a2]	J.-L. Colliot-Thélène, J.J. Sansuc, "La descente sur les variétés rationnelles II" Duke Math. J. , 54 (1987) pp. 375–492
[a3]	V.E. Voskresenskii, "Algebraic tori" , Moscow (1977) (In Russian)
[a4]	V.I. Chernusov, "On the Hasse principle for groups of type " (To appear) (In Russian)
[a5]	F. Bruhat, J. Tits, "Groupes réductifs sur un corps local III. Complements et applications à la cohomologie Galoisiènne" J. Fac. Sci. Univ. Tokyo , 34 (1987) pp. 671–698
[a6]	G. Harder, "Chevalley groups over function fields and automorphic forms" Ann. of Math. , 100 (1974) pp. 249–306
[a7]	A.A. Albert, "Structure of algebras" , Amer. Math. Soc. (1939) pp. 143
[a8]	R. Brauer, H. Hasse, E. Noether, "Beweis eines Haupsatzes in der Theorie der Algebren" J. Reine Angew. Math. , 107 (1931) pp. 399–404