Introduction
xv
that over function fields of transcendence degree 1 all almost simple algebraic
groups are quasi-split, i.e. contain a Borel subgroup [168, Th. 11.1].
The next case is that of the function field of a complex algebraic surface. This is
the degree 2 case, an important topic discussed in this book. These fields share an
important invariant with p-adic fields and totally imaginary number fields, namely
their cohomological dimension (which is 2). This dimension is defined by means
of Galois cohomology (Sect. 1.3.1); as a first approach, we can think of it as a
generalization of the transcendence degree of function fields of complex varieties
to arbitrary fields.
One of the main goals of this book is to classify (semi)simple algebraic groups
over a field of cohomological dimension ≤ 2. Let us give the result in the case of
even dimensional quadratic forms q, q with the same discriminant. According to
[15, Cor. 5.3], q and q are similar if and only if C 0 (q) ∼ = C 0 (q ) where C 0 (q)
is the even Clifford algebra of q. It follows that SO(q) ∼ = SO(q ) if and only if
C 0 (q) ∼ = C 0 (q ). This can be interpreted in terms of Galois cohomology as follows:
SO(q) and SO(q ) share the same Tits class in H 2 (k, μ 2 ). In Sect. 9.4, we provide
a general result of this form, conditional on the validity of Serre’s conjecture II, the
main topic of this monograph.
Serre’s original conjecture II (1962) states that the Galois cohomology set
H 1 (k, G) vanishes for a semisimple simply connected algebraic group G defined
over a perfect field k of cohomological dimension cd(k) ≤ 2. In other words,
it predicts that all G-torsors (or principal homogeneous spaces) over Spec(k) are
trivial. Serre extended his conjecture to imperfect fields in 1994. We discuss this
in detail in Sect. 4.8. We need to replace the condition cd(k) ≤ 2 by the condition
scd(k) ≤ 2, where scd stands for the separable cohomological dimension which
agrees with cd for perfect fields.
One of the simplest examples of the conjecture is that of a central simple algebra
A defined over a field k and a non-zero scalar c ∈ k × . The subvariety
X c := {Nrd A (y) = c} ⊂ GL 1 (A)
of elements of reduced norm c is a torsor under the special linear group G =
SL 1 (A), which is semisimple and simply connected. If scd(k) ≤ 2, the conjecture
says that this G-torsor is trivial, i.e. X c (k) = ∅. By considering all scalars c,
we therefore expect that the reduced norm map A × → k × is surjective. For
function fields of complex surfaces, this follows from the Tsen-Lang theorem,
since the reduced norm is a homogeneous form of degree deg(A) in deg(A) 2 -
indeterminates [159, II.4.5]. The general case of the surjectivity of the reduced norm
map was established by Suslin. This fact essentially characterizes fields of separable
cohomological dimension ≤ 2; see Sect. 4.7.
An important method used when dealing with classification problems over fields
of separable cohomological dimension ≤ 2 is the very precise classification of
quadratic forms in terms of multiquaternion division algebras, due to Sivatski.
Although this result can be deduced from Serre’s conjecture II for spin groups, we
present an independent proof of that classification (Sect. 7.1.3).
xv
that over function fields of transcendence degree 1 all almost simple algebraic
groups are quasi-split, i.e. contain a Borel subgroup [168, Th. 11.1].
The next case is that of the function field of a complex algebraic surface. This is
the degree 2 case, an important topic discussed in this book. These fields share an
important invariant with p-adic fields and totally imaginary number fields, namely
their cohomological dimension (which is 2). This dimension is defined by means
of Galois cohomology (Sect. 1.3.1); as a first approach, we can think of it as a
generalization of the transcendence degree of function fields of complex varieties
to arbitrary fields.
One of the main goals of this book is to classify (semi)simple algebraic groups
over a field of cohomological dimension ≤ 2. Let us give the result in the case of
even dimensional quadratic forms q, q with the same discriminant. According to
[15, Cor. 5.3], q and q are similar if and only if C 0 (q) ∼ = C 0 (q ) where C 0 (q)
is the even Clifford algebra of q. It follows that SO(q) ∼ = SO(q ) if and only if
C 0 (q) ∼ = C 0 (q ). This can be interpreted in terms of Galois cohomology as follows:
SO(q) and SO(q ) share the same Tits class in H 2 (k, μ 2 ). In Sect. 9.4, we provide
a general result of this form, conditional on the validity of Serre’s conjecture II, the
main topic of this monograph.
Serre’s original conjecture II (1962) states that the Galois cohomology set
H 1 (k, G) vanishes for a semisimple simply connected algebraic group G defined
over a perfect field k of cohomological dimension cd(k) ≤ 2. In other words,
it predicts that all G-torsors (or principal homogeneous spaces) over Spec(k) are
trivial. Serre extended his conjecture to imperfect fields in 1994. We discuss this
in detail in Sect. 4.8. We need to replace the condition cd(k) ≤ 2 by the condition
scd(k) ≤ 2, where scd stands for the separable cohomological dimension which
agrees with cd for perfect fields.
One of the simplest examples of the conjecture is that of a central simple algebra
A defined over a field k and a non-zero scalar c ∈ k × . The subvariety
X c := {Nrd A (y) = c} ⊂ GL 1 (A)
of elements of reduced norm c is a torsor under the special linear group G =
SL 1 (A), which is semisimple and simply connected. If scd(k) ≤ 2, the conjecture
says that this G-torsor is trivial, i.e. X c (k) = ∅. By considering all scalars c,
we therefore expect that the reduced norm map A × → k × is surjective. For
function fields of complex surfaces, this follows from the Tsen-Lang theorem,
since the reduced norm is a homogeneous form of degree deg(A) in deg(A) 2 -
indeterminates [159, II.4.5]. The general case of the surjectivity of the reduced norm
map was established by Suslin. This fact essentially characterizes fields of separable
cohomological dimension ≤ 2; see Sect. 4.7.
An important method used when dealing with classification problems over fields
of separable cohomological dimension ≤ 2 is the very precise classification of
quadratic forms in terms of multiquaternion division algebras, due to Sivatski.
Although this result can be deduced from Serre’s conjecture II for spin groups, we
present an independent proof of that classification (Sect. 7.1.3).