Let $F$ be a number field. Let $A=\mathcal O_F$, or let $S$ be a finite set of nonzero prime ideals of $\mathcal O_F$ and let $A=\mathcal O_{F,S}$ be the ring of $S$-integers, equivalently
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\begin{align*}
\mathcal O_{F,S}=\{x\in F: v_{\mathfrak p}(x)\geq 0\text{ for every nonzero prime ideal }\mathfrak p\notin S\}.
\end{align*}
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Then the stable determinant induces an isomorphism of abelian groups