Bases for some modules of cyclotomic units

Abstract

Let $\mathbf{Was}(\mathbb{K})$ denote the group of Washington's cyclotomic units of any abelian number field $\mathbb{K}$. If $\mathbb{K}$ coincides with its genus field in the narrow sense, we give a $\hat{I}›$-basis of $\lim \_\leftarrow \mathbf{Was}(\mathbb{K}\_k^{+})$ where $(\mathbb{K}\_k)\_k⩾0$ denotes the cyclotomic $\mathbb{Z}\_p$-tower of $\mathbb{K}$ and $\hat{I}›$ denotes the Iwasawa's algebra. This results from a $\mathbb{Z}[1/2]$-basis of $\mathbf{Was}(\mathbb{K})\otimes \_\mathbb{Z}\mathbb{Z}[1/2]$ that we give under the same hypothesis.