  Let S be a commutative ring, Q a group that acts on S, and let R be the
subring of S fixed under Q. A Q-normal S-algebra consists of a central
S-algebra A and a homomorphism s from Q to the group Out(A) of outer
automorphisms of A that lifts the Q-action on S. We associate to a Q-normal
S-algebra (A,s) a crossed 2-fold extension which, in turn, represents a class,
the Teichmueller class of (A,s), in the third cohomology group of Q with
coefficients in the group U(S) of units of S, endowed with the obvious Q-module
structure. Suitable equivalence classes of Q-normal Azumaya S-algebras
constitute an abelian group XB(S,Q), the crossed Brauer group of S relative to
the Q-action on S, and the classical results, suitably rephrased in terms of a
generalized Teichmueller cocycle map defined on the abelian group XB(S,Q) and
crucially involving crossed 2-fold extensions, extend to the more general
situation. The Teichmueller cocycle map is even defined on the abelian group
kRep(Q,B((S,Q))) of classes of representations of Q in the Q-graded Brauer
category B((S,Q)) of S relative to the Q-action on S, and the obvious
homomorphism from XB(S,Q) to kRep(Q,B((S))) is injective, an isomorphism when
the image of Q in the group of automorphisms of S is a finite group.
Furthermore, in that case, the equivariant and crossed Brauer groups fit into
various exact sequences generalizing among others the corresponding low degree
group cohomology five term exact sequence in the classical case over a field.
Crossed pair algebras defined relative to a suitable notion of Q-equivariant
Galois extension of commutative rings lead to a comparison of the theory with
the appropriate group cohomology groups and with the corresponding abelian
group of classes of crossed pairs defined relative to the data. A number of
examples illustrating the theory are included.
