= Class-field unit sequence
{title2=$0\to\overline E_\infty\to U_\infty\to X\to Y\to0$}
Passing to norm limits in <Artin reciprocity> gives an <exact sequence> connecting closures of global units, local pro-$p$ units at primes over $p$, the <p-ramified Iwasawa module> and the <unramified Iwasawa module>. It explains arithmetic <Iwasawa-module ranks> by comparing local and global unit ranks. Roots of unity and the precise splitting conventions must be retained when seeking exact integral statements.
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