The linear and quadratic Schur subgroups over the $S$-integers of a number field

Abstract:

Let $K$ be an algebraic number field and let $\mathfrak {O}$ be a ring of $S$-integers in $K$ (where $S$ is a set of primes of $K$ containing all the archimedean primes); that is to say, $\mathfrak {O}$ is a Dedekind domain whose field of quotients is $K$. In analogy with a theorem of T. Yamada in the case of a field of characteristic 0, it is shown that if $S\left ( \mathfrak {O} \right )$ is the Schur subgroup of the Brauer group $B\left ( \mathfrak {O} \right )$ and if $\mathfrak {o} = \mathfrak {O} \cap k$, where $k$ is any field containing the maximal abelian extension of $\mathbb {Q}$ in $K$, then $S\left ( \mathfrak {O} \right ) = \mathfrak {O} \otimes S\left ( \mathfrak {o} \right )$, i.e. the Brauer classes in $S\left ( \mathfrak {O} \right )$ are those obtained from $S\left ( \mathfrak {o} \right )$ by extension of the scalars to $\mathfrak {O}$. A similar theorem is proved as well in the case of the Schur subgroup $S\left ( {\mathfrak {O},\omega } \right )$ of the quadratic Brauer group $B\left ( {\mathfrak {O},\omega } \right )$, where $\omega$ is an involution of $\mathfrak {O}$.