On the equivariant structure of ideals in abelian extensions of local fields (with an appendix by W. Bley)
The subject of study is the module structure of ideals in \(O_L\) over an order \(A\) in \(E[G]\), where \(L/K\) is a \(G\)-Galois extension of \(p\)-adic fields, and \(E\) is a subfield of \(K\). The first case that comes to mind is \(E=K\) and \(A=O_K[G]\). In this situation, Vostokov found a rather simple criterion for the existence of ideals \(I\) that are decomposable as \(A\)-modules. In the tame case, \(I\) is always free over \(O_K[G]\), but this is far from true in the wild case. This motivates looking at another larger order \(A\) that acts on \(I\), and it is natural to consider the largest order that acts on \(I\), the so-called associated order \(A({\mathbb Q}_p, I)\) (we have chosen \(E\) to be \({\mathbb Q}_p\)). It is more likely for an ideal \(I\) to be free over its associated order than over \({\mathbb Z}_p[G]\), and one should see the main result of the paper against this backdrop: it makes a strong negative statement, to the effect that freeness is rare even over the associated order. The main result (Theorem 1.3) distinguishes five cases, and we somewhat arbitrarily pick case (i), perhaps the most generic one. Here the exact statement is: Suppose \(L/K\) is totally ramified and the first ramification group is not cyclic. Then there exists an ideal \(I\) in \(O_L\) that is free over its associated order, if and only if the extension \(L/K\) is weakly ramified. The latter means, by definition, that the second ramification group (in the lower numbering!) is already trivial. This is a very strong restriction. For instance, \({\mathbb Q}_p(\zeta_{p^2})/{\mathbb Q}_p\) is not weakly ramified, and the degree \(p^n\) subfield of \({\mathbb Q}_p(\zeta_{p^{n+1}})/{\mathbb Q}_p\) is weakly ramified if and only if \(n\leq 1\). The methods employed in the paper include standard ramification theory but go beyond this, e.g. by the use of factorisability techniques. The author presents an enormous amount of results, case distinctions, and comments. The writing is terse, and the reviewer does not pretend to have worked through the details. Apart from its strong theoretical results, the paper also features an appendix by Werner Bley, which gives an algorithm for determining the Galois module structure of ideals in the analogous global situation. Two examples are presented where a computer calculation yields global freeness for those ideals which are known to be locally free by theory.
scientific article; zbMATH DE number 1455387