xvi
Introduction
Throughout its history, evidence for and progress towards establishing conjecture
II has been obtained by either considering special classes of fields or looking at the
implications of the conjecture for the classification of algebraic groups.
From the point of view of groups, the strongest evidence for the validity of the
conjecture is the case of classical groups (and groups of type G 2 and F 4 ) established
in 1995 by Bayer and Parimala [11]; see also Berhuy et al. for the generalization to
imperfect fields [15]. For exceptional groups (trialitarian of type D 4 , type E 6 , E 7
and E 8 ), the general conjecture is still open in spite of some considerable progress
[38, 49, 77].
From the point of view of fields, we know that the conjecture holds in the case
of imaginary number fields (Kneser, Harder, Chernousov; see [149, §6]) and, more
recently, in the case of function fields of complex surfaces. A general proof for all
types of groups was recently given by de Jong et al. [57] using deformation methods.
This result has a clear geometric meaning: if G/C is a semisimple simply connected
group and X a smooth complex surface, then any G-torsor over X (or a G-bundle)
is locally trivial with respect to the Zariski topology.
The preceding examples of fields with separable cohomological dimension ≤ 2
are quite specific since their central simple division algebras of period 2 are
quaternion algebras. This is in contrast to Merkurjev’s construction of a field of
cohomological dimension 2 having a central simple division algebra of period 2
and index 2 n , n ≥ 1 arbitrary (and also anisotropic quadratic forms of dimension
2n − 2) [131]. It is therefore not surprising that the complexity of central simple
algebras plays a crucial role in the statements of the results.
In this monograph, we present our approach to the conjecture based on the study
of cohomology classes arising from finite diagonalizable subgroups. The key result
is the following:
Proposition 5.5.1 Let G/k be a semisimple simply connected group satisfying the
hypothesis of conjecture II. Let f : μ n → G be a k-group homomorphism. Then
the induced map
f ∗ : k
× /k
×n ∼ = H
1 (k, μ n ) → H
1 (k, G)
is trivial.
If μ n is central in G, this follows from the norm principle (Sect. 5.4) and the study
of norm group of varieties of Borel subgroups (Sect. 5.3). The new ingredient is the
generalization to the non-central case, which involves the study of the centralizer
C G (μ n ) (Sect. 3.1).
We then use known techniques (e.g. Harder quadratic trick, Sect. 3.2.4) to cover
all known cases of the conjecture. We start with the quasi-split case (Chap. 6),
continue with the classical groups (Chap. 7) and end with the exceptional groups
(Chap. 8).
In the classical group setting, our proofs are somewhat simpler than the original
ones since we avoid certain results and constructions from algebras with involutions.
Introduction
Throughout its history, evidence for and progress towards establishing conjecture
II has been obtained by either considering special classes of fields or looking at the
implications of the conjecture for the classification of algebraic groups.
From the point of view of groups, the strongest evidence for the validity of the
conjecture is the case of classical groups (and groups of type G 2 and F 4 ) established
in 1995 by Bayer and Parimala [11]; see also Berhuy et al. for the generalization to
imperfect fields [15]. For exceptional groups (trialitarian of type D 4 , type E 6 , E 7
and E 8 ), the general conjecture is still open in spite of some considerable progress
[38, 49, 77].
From the point of view of fields, we know that the conjecture holds in the case
of imaginary number fields (Kneser, Harder, Chernousov; see [149, §6]) and, more
recently, in the case of function fields of complex surfaces. A general proof for all
types of groups was recently given by de Jong et al. [57] using deformation methods.
This result has a clear geometric meaning: if G/C is a semisimple simply connected
group and X a smooth complex surface, then any G-torsor over X (or a G-bundle)
is locally trivial with respect to the Zariski topology.
The preceding examples of fields with separable cohomological dimension ≤ 2
are quite specific since their central simple division algebras of period 2 are
quaternion algebras. This is in contrast to Merkurjev’s construction of a field of
cohomological dimension 2 having a central simple division algebra of period 2
and index 2 n , n ≥ 1 arbitrary (and also anisotropic quadratic forms of dimension
2n − 2) [131]. It is therefore not surprising that the complexity of central simple
algebras plays a crucial role in the statements of the results.
In this monograph, we present our approach to the conjecture based on the study
of cohomology classes arising from finite diagonalizable subgroups. The key result
is the following:
Proposition 5.5.1 Let G/k be a semisimple simply connected group satisfying the
hypothesis of conjecture II. Let f : μ n → G be a k-group homomorphism. Then
the induced map
f ∗ : k
× /k
×n ∼ = H
1 (k, μ n ) → H
1 (k, G)
is trivial.
If μ n is central in G, this follows from the norm principle (Sect. 5.4) and the study
of norm group of varieties of Borel subgroups (Sect. 5.3). The new ingredient is the
generalization to the non-central case, which involves the study of the centralizer
C G (μ n ) (Sect. 3.1).
We then use known techniques (e.g. Harder quadratic trick, Sect. 3.2.4) to cover
all known cases of the conjecture. We start with the quasi-split case (Chap. 6),
continue with the classical groups (Chap. 7) and end with the exceptional groups
(Chap. 8).
In the classical group setting, our proofs are somewhat simpler than the original
ones since we avoid certain results and constructions from algebras with involutions.