If \(Y\to X\) is a \(G\)-covering of smooth projective curves over an algebraically closed field \(k\), then the Tate module \(T_\ell(Y)=\text{projlim}_n\text{Pic}^0(Y)[\ell^n]\) is naturally a module over \(\mathbb Z_\ell[G]\). The subject of the present paper is to determine this module for the case where \(G\) is a cyclic \(\ell\)-group, and \(\ell\) is prime to \(\mathrm{char}(k)\). The result depends on the ramification of the covering, and we will not give details here. The general argument is complicated and uses an intricate theorem of Yakovlev in integral representation theory. However, and this makes the paper quite readable, the authors do the cases \(|G|=\ell\) and \(|G|=\ell^2\) before the general case. In these special cases, one can completely classify lattices over \(\mathbb Z_\ell[G]\) (for \(|G|=\ell\), this is called Reiner's theorem); calculating the cohomology of the Tate module, and comparing, suffices to obtain the result (prop. 5 and 6). The starting point of the cohomology calculation is Tsen's theorem and Hilbert 90. One also uses the Riemann-Hurwitz formula for the genus of a covering curve. The general result was presumably inspired by the special cases; it is to be found in theorem 14. It is interesting to note that in a number-theoretic setting (where \(T_\ell\) is replaced by an Iwasawa module, and the genus roughly corresponds to the \(\lambda\)-invariant) one does not have a Riemann-Hurwitz formula to begin with, but one can prove versions of it by methods related to the ones in the paper under review (Kida, Kuz'min, Iwasawa, Wingberg, Nguyen Quang Do, and others). Some small remarks: I suppose that the term ``component'' (p.81) just means ``direct summand''. There is a slightly irritating misprint in the seventh line of the introduction: \(J_L(\ell)\) is an inductive limit, so one should read \(\lim_\to\) instead of \(\lim_\leftarrow\). Of course the projective limit in the twelfth line is correct.