Bases for some modules of cyclotomic units

Let $\mathbf{W}\mathbf{a}\mathbf{s}(\mathbb{K})$ 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 $\mathrm{\Lambda }$-basis of $\lim \mathrm{\_}\stackrel{}{\leftarrow }\mathbf{W}\mathbf{a}\mathbf{s}(\mathbb{K}\mathrm{\_}{k}^{+})$ where $(\mathbb{K}\mathrm{\_}k)\mathrm{\_}k⩾0$ denotes the cyclotomic $\mathbb{Z}\mathrm{\_}p$-tower of $\mathbb{K}$ and $\mathrm{\Lambda }$ denotes the Iwasawa’s algebra. This results from a $\mathbb{Z}[1\mathrm{/}2]$-basis of $\mathbf{W}\mathbf{a}\mathbf{s}(\mathbb{K})\otimes \mathrm{\_}\mathbb{Z}\mathbb{Z}[1\mathrm{/}2]$ that we give under the same hypothesis.