Cohomology groups of cyclotomic units
Let \(d\) be a positive integer, \(p\) an odd prime not dividing \(d\) or \(\varphi(d)\). Set \(\zeta_ n=e^{2\pi i/n}\), \(K_n=\mathbb{Q}(\zeta_{p^{n+1}d})\). Set \(\Delta=\text{Gal}(K_{-1}/\mathbb{Q})\), let \(D\) be the decomposition group of \(\Delta\) for \(p\), and \(k\) the fixed field of \(D\). Let \(G_{m,n}=\text{Gal}(K_m/K_n)\) for \(m>n\geq 0\). The author computes the Tate cohomology groups \(H^i(G_{m,n},C_m)\) for \(i=0,1\), where \(C_m\) is the group of cyclotomic units of \(K_m\), namely \[ C_m=\Bigl\langle\pm\zeta^a_{p^{m+1}d},\;1-\zeta^b_{p^{m+1}d}\mid a,\;b\text{ integers}\Bigr\rangle\cap\{\text{units of }K_ m\}. \] The author shows that \[ H^0(G_{m,n}, C_m)\cong (\mathbb{Z}/p^{m-n}\mathbb{Z})^{\ell-1},\quad\text{and}\quad H^1(G_{m,n}, C_m)\cong (\mathbb{Z}/p^{m-n}\mathbb{Z})^\ell, \] where \(\ell=[k:\mathbb{Q}]\) if \(k\) is totally real, and \(\ell=[k:\mathbb{Q}]/2\) otherwise. These results extend work of the author and \textit{R. Gold} [Compos. Math. 71, 13--27 (1989; Zbl 0687.12003)].
scientific article; zbMATH DE number 130482