Anti-isomorphism between Brauer groups BQ(S, H) AND BQ(Sop , H∗)

Abstract

For a commutative ring R and a Hopf algebra H which is finitely generated projective as an R-module, it is established that there is an (anti)-isomorphism of groups between the Brauer group BQ(R, H) of Hopf Yetter-Drinfel’d H-module algebras and the Brauer group BQ ( R , H * ) of Hopf Yetter-Drinfel’d H * -module algebras, where H * is the linear dual of H. In this paper, we generalize this result by establishing an anti-isomorphism of groups between BQ(S, H), the Brauer group of dyslectic Hopf Yetter-Drinfel’d (S, H)-module algebras and BQ ( S o p , H * ) , the Brauer group of dyslectic Hopf Yetter-Drinfel’d ( S o p , H * ) -module algebras, where S is an H-commutative Hopf Yetter-Drinfel’d H-module algebra and Sop is the opposite algebra of S. Communicated by Alberto Elduque