étale cohomology group Br (X) := Hét(X,
m). This is what is called the cohomo-
logical Brauer group. On the other hand, a rather naive generalization of the defi-
nition for fields yields the concept of the Brauer group. One has Br(X) ⊆ Br (X).
In general, the two are not equal to each other.
In Section III.7, we give a proof for the Theorem of Auslander and Goldman stating
that Br(X) = Br (X) in the case of a smooth surface. This result was originally
shown in [A/G] before the actual invention of schemes. The proof of Auslan-
der and Goldman was formulated in the language of Brauer groups for commuta-
tive rings. However, all the arguments given carry over immediately to the case
of a two-dimensional regular scheme. Although better results are available today,
most notably Gabber’s Theorem [dJo2], we feel that the proof of the Theorem of
Auslander and Goldman gives a good impression of the methods used to compare
Br(X) and Br (X).
The chapter is closed by computations of Brauer groups in particular examples.
In the case of a variety over an algebraically non-closed field, we study the relation-
ship of Br(X) with H1
. We prove Manin’s formula expressing
the latter cohomology group in terms of the Galois operation on a specific set
of divisors. For smooth cubic surfaces, one may work with the classes given by
the 27 lines.
This leads to the result of Sir Peter Swinnerton-Dyer [SD93] that, for a smooth
cubic surface, H1
is one of the groups 0, /2 , /3 , ( /2 )2,
and ( /3
Swinnerton-Dyer’s proof filled the entire article [SD93] and was later
modified by P. K. Corn in his thesis [Cor].
We discovered that Swinnerton-Dyer’s result may be obtained in a manner, which
is rather brute force, but very simple. The Galois group acting on the 27 lines on a
smooth cubic surface is a subgroup of W (E6). There are only conjugacy classes
of subgroups of W (E6). We computed
in each of these cases
using GAP. This took 28 seconds of CPU time.
As an application of Brauer groups, the third chapter is concerned with the Brauer–
Manin obstruction. We recall the notion of an adelic point and define the local and
global evaluation maps. An adelic point x = (xν)ν∈Val(
is “obstructed” from being
approximated by rational points if the global evaluation map ev gives a non-zero
value ev(α, x) for a certain Brauer class α ∈ Br(X).
We then describe a strategy on how the Brauer–Manin obstruction may be explicitly
computed in concrete examples. We carry out this strategy for two special types of
cubic surfacess, which, as we think, are representative but particularly interesting.
The first type is given as follows. Let p0 ≡ 1 (mod 3) be a prime number, and
let K/ be the unique cubic field extension contained in the cyclotomic exten-
sion (ζp0 )/ . Fix the explicit generator θ ∈ K given by
θ := tr
)/K(ζp0 − 1) = −2n +
for n :=
. Then consider the cubic surface X ⊂ P3 , given by
x3(a1x0 + d1x3)(a2x0 + d2x3) =
Here, a1,a2,d1,d2 ∈ . The
denote the three images of θ under Gal(K/ ).