Normality of algebras over commutative rings and the Teichmüller class. II.

Abstract

Using a suitable notion of normal Galois extension of commutative rings, we develop the relative theory of the generalized Teichmüller cocycle map. We interpret the theory in terms of the Deuring embedding problem, construct an eight term exact sequence involving the relative Teichmüller cocycle map and suitable relative versions of generalized Brauer groups and compare the theory with the group cohomology eight term exact sequence involving crossed pairs. We also develop somewhat more sophisticated versions of the ordinary, equivariant and crossed relative Brauer groups and show that the resulting exact sequences behave better with regard to comparison of the theory with group cohomology than do the naive notions of the generalized relative Brauer groups.