The Schur group of a commutative ring
Let R denote a commutative ring and B(R) the Brauer group of classes of Azumaya R-algebras as developed by \textit{M. Auslander} and \textit{O. Goldman} [Trans. Am. Math. Soc. 97(1960), 367-409 (1961; Zbl 0100.263)]. This paper introduces and gives basic facts about the Schur subgroup of B(R). An Azumaya R-algebra A represents a class in the Schur subgroup S(R) of B(R) when there is a finite group G and an R-algebra epimorphism \(f: RG\to A.\) If R is a commutative ring of positive characteristic, \(S(R)=0\). On the other hand, any finite abelian group is the Schur group of a commutative ring which is finitely generated as an algebra over the rational integers. Some standard facts about the Schur group of a field are generalized to commutative rings.
scientific article; zbMATH DE number 3884279
scientific article