Galois structure of ideals in wildly ramified abelian \(p\)-extensions of a \(p\)-adic field, and some applications
It is shown that fairly frequently ideals in the ring of integers of on extension \(L\) of \(K\) fail to be free over their associated order. Here \(K\) is a \(p\)-adic field and \(L/K\) is a finite abelian \(p\)-extension with ramification index \(p^n\). An efficient criterion for an ideal \(I\) not to be free over its associated order \(\mathcal A\) is given in terms of ramification numbers: if \(I\) is free, then, under a very mild extra condition which we don't spell out here, all ramification numbers of \(L/K\) have to be congruent to \(-1\) modulo \(p^n\). The first step towards this result is a very simple, yet clever statement (Theorem 2.1), which is worth being quoted in full: if a module \(M\) over the local ring \(\mathcal A\) has a system of generators \(m_1,\ldots,m_n\) such that none of the \(m_i\) is a generator individually, then \(M\) is not free of rank one. One other main source is \textit{S. Vostokov}'s criterion for an ideal \(I\) to be indecomposable as a module over \({\mathcal O}_K[\)Gal\((L/K)]\) [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklov 57, 64-84 (1976; Zbl 0355.12012) and ibid. 46, 14-35 (1974; Zbl 0345.12009)]. This makes the associated order of \(I\) local. These ingredients are combined with some further ideas in order to prove the main result. Several applications are given. Firstly, it is a direct consequence that all ramification numbers are congruent to \(-1\) modulo \(p^n\) if \(S={\mathcal O}_L\) is Hopf Galois over \(R={\mathcal O}_K\). This is a pleasant generalization of previous results in the case \([L:K]=p^2\) due to \textit{N. P. Byott} [Math. Z. 220, No. 4, 495-522 (1995; Zbl 0841.16021)], and to \textit{C. Greither} [Math. Z. 210, No. 1, 37-68 (1992; Zbl 0737.11038)]. Secondly, the author considers division fields \(K=k_r\) and \(L=k_{m+r}\) associated to a Lubin-Tate group. Then \(L/K\) carries a priori two Hopf Galois structures: the first is the structure of a classical Galois extension, coming simply from the fact that \(L\) is already Galois over \(k=k_0\); the second comes from the theory of formal groups, in fact Spec\((L)\) is principal homogeneous under the group scheme of \(p^m\)-torsion points of the given Lubin-Tate group. If \(m\leq r\), these two structures are the same (this is analogous to Kummer theory), but for \(m>r\) they differ. Indeed this paper shows that they lead to different behaviour at integral level: the second structure always leads to a Hopf Galois extension \(S/R\), whereas for the first structure the main result allows to show that \(S\) is not free over its associated order if \(m>r\) and \(k\) is bigger than \({\mathbb{Q}}_p\). This quite well-written paper is an important contribution to local Galois module theory.
scientific article; zbMATH DE number 1132894
scientific article