Minus class groups of the fields of the $l$-th roots of unity

We show that for any prime number $l>2$ the minus class group of the field of the $l$-th roots of unity $\overline{\mathbf{Q}_p} (\zeta _l)$ admits a finite free resolution of length 1 as a module over the ring $\widehat{\mathbf{Z}} [G]/(1+\iota)$. Here $\iota$ denotes complex conjugation in $G={{Gal}}( \overline{\mathbf{Q}_p} (\zeta _l)/\overline{\mathbf{Q}_p} )\cong(\mathbf{Z} /l\mathbf{Z} )^*$. Moreover, for the primes $l\le 509$ we show that the minus class group is cyclic as a module over this ring. For these primes we also determine the structure of the minus class group.