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

Abstract:

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