Galois cohomology

2020 Mathematics Subject Classification: Primary: 12G05 Secondary: 11-0211R34 [MSN][ZBL]

Galois cohomology studies the 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 $\sim$, 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 [Se], [CaFr], [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 $B(K)$ 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, ${\rm cd}\; G_k = 1$ [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, 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 [Se].

It has been proved [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 [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 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

[ArTa]	E. Artin, J. Tate, "Class field theory", Benjamin (1967)
[Bo]	A. Borel, "Some finiteness properties of adèle groups over number fields" Publ. Math. IHES : 16 (1963) pp. 5–30
[BoSe]	A. Borel, J.-P. Serre, "Théorèmes de finitude en cohomologie Galoisienne" Comment Math. Helv., 39 (1964) pp. 111–164
[BrTi]	F. Bruhat, J. Tits, "Groupes réductifs sur un corps local I. Données radicielles valuées" Publ. Math. IHES : 41 (1972) pp. 5–252
[CaFr]	J.W.S. Cassels (ed.) A. Fröhlich (ed.), Algebraic number theory, Acad. Press (1986)
[GrVeAr]	"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)
[Ha]	G. Harder, "Ueber die Galoiskohomologie halbeinfacher Matrizengruppen I" Math. Z., 90 (1965) pp. 404–428
[Ha2]	G. Harder, "Ueber die Galoiskohomologie halbeinfacher Matrizengruppen II" Math. Z., 92 (1966) pp. 396–415
[Kn]	M. Kneser, "Galois-Kohomologie halbeinfacher algebraischer Gruppen über $p$-adische Körpern I" Math. Z., 88 (1965) pp. 40–47
[Kn2]	M. Kneser, "Galois-Kohomologie halbeinfacher algebraischer Gruppen über $p$-adische Körpern II" Math. Z., 89 (1965) pp. 250–272
[Ko]	H. Koch, "Galoissche Theorie der $p$-Erweiterungen", Deutsch. Verlag Wissenschaft. (1970)
[Se]	J.-P. Serre, "Cohomologie Galoisienne", Springer (1964)
[Se2]	J.-P. Serre, "Groupes algébrique et corps des classes", Hermann (1959)
[Se3]	J.-P. Serre, "Local fields", Springer (1979) (Translated from French)
[St]	R. Steinberg, "Regular elements of semisimple algebraic groups" Publ. Math. IHES : 25 (1965) pp. 49–80

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

[Al]	A.A. Albert, "Structure of algebras", Amer. Math. Soc. (1939) pp. 143
[BrHaNo]	R. Brauer, H. Hasse, E. Noether, "Beweis eines Haupsatzes in der Theorie der Algebren" J. Reine Angew. Math., 107 (1931) pp. 399–404
[BrTi2]	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
[Ch]	V.I. Chernusov, "On the Hasse principle for groups of type $E_8$" (To appear) (In Russian)
[CoSa]	J.-L. Colliot-Thélène, J.J. Sansuc, "La descente sur les variétés rationnelles II" Duke Math. J., 54 (1987) pp. 375–492
[Ha3]	G. Harder, "Chevalley groups over function fields and automorphic forms" Ann. of Math., 100 (1974) pp. 249–306
[Ma]	Yu.I. Manin, "Cubic forms. Algebra, geometry, arithmetic", North-Holland (1974) (Translated from Russian)
[Vo]	V.E. Voskresenskii, "Algebraic tori", Moscow (1977) (In Russian)