= Coleman unit-measure exact sequence
{c}
{title2=$0\to\mathbb Z_p(1)\to V_\infty\to\mathbb Z_p[[\mathbb Z_p^\times]]\to\mathbb Z_p(1)\to0$}
Let $V_\infty$ be the norm limit of local <principal units> in the $p$-power <cyclotomic field> tower, for odd $p$. The <Coleman power series> $f_u$ gives the <Frobenius-corrected logarithm> $p^{-1}\log(f_u^p/\phi f_u)$, whose inverse <Amice transform> is supported on $\mathbb Z_p^\times$. This is the middle map in the displayed <exact sequence>; the last map is the first moment $\mu\mapsto\int x\,d\mu$. Its kernel consists of norm-compatible $p$-power <roots of unity>. The logarithmic difference equation has one resonant degree, namely degree one in $\log(1+T)$, accounting for the last term. <Complex conjugation> acts by $-1$ on both copies of $\mathbb Z_p(1)$. Taking its even part therefore identifies the local principal-unit limit in the real tower with $\mathbb Z_p[[\mathbb Z_p^\times/\{\pm1\}]]$.
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