Normality of algebras over commutative rings, crossed pairs, and the Teichmueller class

Abstract

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.