11.4. Generators for the local cyclotomic units
We pass to the local cyclotomic units \CC_n (the p-adic closure of \DD_n in
\sU_n) and their principal/real refinements \CC_{n,1}^+,
\CC_{\infty,1}^+=\varprojlim_n\CC_{n,1}^+. Since \CC_n^+ is not a \Zp-module,
one must work with the principal units \CC_{n,1}^+ to get a \Zp-structure. The
following lemma identifies p-adic closures with \Zp-spans.
Let g_1,\dots,g_r\in\sU_{n,1} and let X=\ang{g_1,\dots,g_r} be the (multiplicative)
\Z-module they generate. Then the p-adic closure \overline X of X in
\sU_{n,1} is the \Zp-submodule generated by g_1,\dots,g_r.
If a\in\Zp and a_j\to a with a_j\in\Z, then since g_i-1\equiv 0\pmod{\pri_n}
the binomial series gives
g_i^{a_j}=\sum_k\binom{a_j}{k}(g_i-1)^k\to\sum_k\binom{a}{k}(g_i-1)^k=g_i^a, so the
\Zp-span lies in \overline X. Conversely, for g\in\overline X choose integer
exponent vectors (a_{1,j},\dots,a_{r,j}) with \prod_i g_i^{a_{i,j}}\to g; by
compactness of \Zp^r a subsequence of exponents converges to some
(b_1,\dots,b_r)\in\Zp^r, and by the same continuity
\prod_i g_i^{b_i}=g. Hence g is in the \Zp-span.
Let a\in\Z be a topological generator of \Zpx and w\in\mu_{p-1}\subset\sU_n
with aw\equiv 1\pmod{\pri_n}. Then: (i) w\gamma_{n,a}\in\sU_{n,1}; and (ii)
(w\gamma_{n,a})^{p-1}=\gamma_{n,a}^{p-1}\in\sU_{n,1}^+ generates the cyclic
\Z[\GGam_n^+]-module (p-1)\DD_n^+=\set{\gamma^{p-1}:\gamma\in\DD_n^+}.
(i) We claim \gamma_{n,a}\equiv a\pmod{\pri_n}. Since
\gamma_{n,a}=\xi_{p^n}^{a/2}c_n(a) and \xi_{p^n}^{a/2}\equiv 1\pmod{\pri_n}, it
suffices to show c_n(a)\equiv a. For any unit u, u_n=f_u(\pi_n)\equiv
f_u(0)\pmod{\pri_n}; applied to the Coleman power series
f_{c(a)}=((1+T)^a-1)/T this gives c_n(a)\equiv f_{c(a)}(0)=a. Hence w is the
unique root of unity with w\gamma_{n,a}\equiv 1\pmod{\pri_n}, so
w\gamma_{n,a}\in\sU_{n,1}. (ii) By Corollary 11.3.2 \gamma_{n,a}
generates \DD_n^+, so \gamma_{n,a}^{p-1} generates (p-1)\DD_n^+; and
w^{p-1}=1 gives \gamma_{n,a}^{p-1}=(w\gamma_{n,a})^{p-1}.
Let a\in\Z be a topological generator of \Zpx and w\in\mu_{p-1} with
aw\equiv 1\pmod{\pri_n}. Then: (i) \CC_{n,1}^+ is a cyclic \Zp[\GGam_n^+]-module
generated by w\gamma_{n,a}; and (ii) \CC_{\infty,1}^+ is a cyclic
\Lam(\GGam^+)-module generated by (w\gamma_{n,a})_{n\ge 1}.
(i) By Lemma 11.4.2, (p-1)\DD_{n,1}^+\subset\sU_{n,1}^+ is
generated over \Z[\GGam_n^+] by (w\gamma_{n,a})^{p-1}. By
Lemma 11.4.1, its p-adic closure (p-1)\CC_{n,1}^+ is generated over
\Zp[\GGam_n^+] by the same element. As p-1 is invertible in \Zp,
(p-1)\CC_{n,1}^+=\CC_{n,1}^+; and since w\gamma_{n,a}\equiv 1\pmod{\pri_n} is the
unique (p-1)-th root of (w\gamma_{n,a})^{p-1} lying in \CC_{n,1}^+, the module
is generated by w\gamma_{n,a}. (ii) Taking the inverse limit,
\CC_{\infty,1}^+\cong\varprojlim_n\Zp[\GGam_n^+]\cdot w\gamma_{n,a}
\cong\Lam(\GGam^+)\cdot(w\gamma_{n,a})_n. This depends on Definition 9.3.1.