CHAPTER 2
The Brauer group and the Tate-Shafarevich group
We need some basic facts relating elements of the Brauer group to
elements of the Tate-Shafarevich group of an elliptic fibration. We dis-
cuss
(9x-gerbes
and the Brauer groups which classify them in section 1,
then genus-1 fibrations and the Tate-Shafarevich group which classifies
them, in section 2. For an elliptic fibration there is a simple, direct
relation between these two groups. The extension to genus-1 fibrations
though is more delicate, and is defined only when a certain alternating
pairing vanishes. This is discussed in section 3.
1. Brauer groups and
(9x-gerbes
In this section we review the notions of 0x-gerbe and presentation,
and discuss the relationship between (9x-gerbes and elements in the
Brauer group.
1.1. Jf-gerbes. Let Jf7 be a sheaf of abelian groups on a topo-
logical space (or a site) X. The case of main interest for us is when
(X, Ox) is a ringed space and ffl 0\ is the sheaf of invertible el-
ements in the structure sheaf. In fact, most of the time we will have
3tf Ox in either the etale or the analytic topologies on a complex
scheme (or an algebraic or analytic space) X. In chapter 5 we will be
interested also in the case when Jf? is the sheaf of germs of smooth
maps from a C°° manifold X to the circle 5
1
.
An J^f-gevbe on X is a global structure on X which "locally looks
like the quotient of X by the trivial action of Jt?". More precisely
"the quotient of X by the trivial action of Jf?" is the classifying object
BJ^. For example, in case J4? is the sheaf of holomorphic maps from
X to a fixed group if, BJif is the sheaf of sections of X x BH over
X, where BH is the classifying space of H. In the general case, B3tf
can be interpreted either as a topological space over X (defined up to
homotopy), or as a stack in groupoids over X (see [LMBOO, §3] for the
definition). We adopt the second approach and treat BM" as a stack
(=4sheaf of categories'): over any open set V, the objects of BJ4?(V)
are the Jf?-torsors on V and the morphisms are the isomorphisms of
torsors. In particular, the automorphisms of the trivial torsor ly are
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