Unramified Kummer classes with bounded prime support ($K(S,m)=\{[a]:v_{\mathfrak{p}}(a)\equiv 0(\mathrm{mod}m)\text{ outside }S\}$)

For a number field $K$, a finite set $S$ and $m\ge 2$, let $K(S,m)$ consist of classes $[a]\in K^{\ast }/K^{\ast m}$ whose valuations outside $S$ are divisible by $m$. It fits into $0\to {\mathcal{O}}_{K,S}^{\times }/({\mathcal{O}}_{K,S}^{\times }{)}^m\to K(S,m)\to \mathrm{Cl}({\mathcal{O}}_{K,S})[m]\to 0$. The S-unit group is finitely generated and the ideal class group is finite, so $K(S,m)$ is finite. When $S$ contains primes dividing $m$, this controls unramified multiplicative Kummer classes outside $S$.