An exact sequence relating
Br(X), the Brauer group of a regular scheme of dimension ≦ 2, and Amitsur
cohomology (obtained as the cohomology of the sheaf of units on an appropriate
Grothendieck topology) is derived by functorial methods. In order to do this we first
show that any torsion element of H1(Xet,Gm), i.e., Pic(X), and H2(Xet,Gm), i.e.,
Br (X), is split by a finite, faithfully flat covering Y → X. After proving a
divisibility result for Pic (X) under such coverings and some preliminary
investigation of cohomology in the topology defined from such coverings,
the exact sequence which is analogous to that of Chase and Rosenberg is
obtained.
