The Rosenberg-Zelinsky Sequence of Dyslectic Yetter-Drinfel’d \((S,H)\) -Modules

The purpose of this paper is to establish the Rosenberg-Zelinsky sequence in the category \(Dys\mbox{-}_S\mathcal {Q}^H\) of dyslectic Hopf Yetter-Drinfel’d \((S,H)\) -module Azumaya algebras. This sequence describes a homomorphism from the group of H-inner (resp. H-INNER) S-algebra automorphisms of an Azumaya algebra in \(Dys\mbox{-}_S\mathcal {Q}^H\) to the group of isomorphism classes of invertible S-modules (resp. invertible dyslectic Hopf Yetter-Drinfeld \((S,H)\) -modules). When H is a commutative, cocommutative Hopf algebra, the dyslectic \((S,H)\) -dimodule version of the Rosenberg-Zelinsky exact sequence is also given.