The Schur subgroup of the Brauer group of cyclotomic rings of integers

Abstract

Let K K be a finite abelian extension of the rational numbers Q \mathbb {Q} . Let S {\mathbf {S}} be a finite set of primes of K K including the infinite ones, and let o \mathfrak {o} be the ring of S {\mathbf {S}} -integers in K K . Then the Schur subgroup S ( o ) S(\mathfrak {o}) of the Brauer group B ( o ) B(\mathfrak {o}) is defined, in analogy with S ( K ) S(K) , via representations of finite groups on finitely generated projective o \mathfrak {o} -modules. It is easy to see that S ( o ) ⊆ S ( K ) ∩ B ( o ) S(\mathfrak {o}) \subseteq S(K) \cap B(\mathfrak {o}) . We shall show that there is equality in the case of K K a purely cyclotomic extension Q ( ε m ) \mathbb {Q}({\varepsilon _m}) of Q \mathbb {Q} (where ε m {\varepsilon _m} is an m m th root of 1).