  Let $F$ be a number field, abelian over the rational field, and fix a odd
prime number $p$. Consider the cyclotomic $Z_p$-extension $F_\infty/F$ and
denote $F_n$ the ${n}^{\rm th}$ finite subfield and $C_n$ its group of circular
units. Then the Galois groups $G_{m,n}=\Gal(F_m/F_n)$ act naturally on the
$C_m$'s (for any $m\geq n>> 0$). We compute the Tate cohomology groups
$\Hha^i(G_{m,n}, C_m)$ for $i=-1,0$ without assuming anything else neither on
$F$ nor on $p$.
