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

Abstract

Let S be a commutative ring and Q a group that acts on S by ring automorphisms. Given an S-algebra endowed with an outer action of Q, we study the associated Teichmüller class in the appropriate third group cohomology group. We extend the classical results to this general setting. Somewhat more specifically, let R denote the subring of S that is fixed under Q. A Q-normal S-algebra consists of a central S-algebra A and a homomorphism \(\sigma :Q\rightarrow \mathop {\mathrm{Out}}\nolimits (A)\) into the group \(\mathop {\mathrm{Out}}\nolimits (A)\) of outer automorphisms of A that lifts the action of Q on S. With respect to the abelian group \(\mathrm {U}(S)\) of invertible elements of S, endowed with the Q-module structure coming from the Q-action on S, we associate to a Q-normal S-algebra \((A, \sigma )\) a crossed?2-fold extension \(\mathrm {e}_{(A, \sigma )}\) starting at \(\mathrm {U}(S)\) and ending at Q, the Teichmüller complex of \((A, \sigma )\), and this complex, in turn, represents a class, the Teichmüller class of \((A, \sigma )\), in the third group cohomology group \(\mathrm {H}^3(Q,\mathrm {U}(S))\) of Q with coefficients in \(\mathrm {U}(S)\). We extend some of the classical results to this general setting. Among others, we relate the Teichmüller cocycle map with the generalized Deuring embedding problem and develop a seven term exact sequence involving suitable generalized Brauer groups and the generalized Teichmüller cocycle map. We also relate the generalized Teichmüller cocycle map with a suitably defined abelian group of classes of representations of Q in the Q-graded Brauer category of S with respect to the given action of Q on S.