The classical Brauer group
B(R) is formed from equivalence classes of Azumaya algebras over the ring R.
The bigger Brauer group B(R) is formed in a similar way from equivalence
classes in a larger category of R-algebras. This larger category is defined
through axioms similar to those defining Azumaya algebras but with the
requirement for an identity dropped. In this paper we identify B(R) with the
second étale cohomology of Spec(R) (with Gm as coefficients). The classical
Brauer group consists of the torsion subgroup of this cohomology group.
This result yields a concrete realization of second étale cohomology and
also enables us to settle several questions about the relation of B(R) to
H2(Δ,Z) in the case where R is a Banach algebra with maximal ideal space
Δ.
