On Galois structure of the integers in cyclic extensions of local number fields
This article presents considerable progress on the following problem: For a Galois extension \(L/K\) of \(p\)-adic fields whose Galois group \(G\) is cyclic of \(p\)-power order \(p^n\), find the decomposition of the \({\mathbb Z}_p[G]\)-module \(O_L\) into indecomposable summands. By E. Noether's result, this module is free iff \(L/K\) is tamely ramified, so the interesting case is when \(L/K\) is totally (hence wildly) ramified. Previous work by Rzedowski-Caldéron, Villa-Salvador, Madan, and by Elder, Madan respectively, considered the cases \(n=2\), respectively, \(n\) general, both times with essential restrictions on the first ramification number \(b_1=b_1(L/K)\). (A little context: If \(b_1\) takes on its maximal value \(pe_0/(p-1)\) or one less, then all idempotents of the algebra \(K[G]\) act on \(O_L\), so this module is free rank one over the maximal order of \(O_K[G]\) in \(K[G]\), which completely solves the problem.) In the present article, \(b_1\) is, loosely speaking, restricted to the upper half of its possible range, and the main theorem 1.1 gives the indecomposable decomposition of \(O_L\) in a formulation which is (quite reasonably, to reduce complexity) recursive in \(n\). It should be pointed out that the removal of summands (``direct difference'') does make sense, given uniqueness of decomposition due to the Krull-Schmidt theorem. The result, and the whole theory, are intrisincally complex and technical. The author does a good job of explaining the involved modules, doing the necessary bookkeeping, and comparing his notation to other work. All the same, the technical parts of the paper are quite challenging, and the reviewer has not gone through the details. There are at least two features of the main result which inspire confidence in spite of the technical complications: firstly, one retrieves, in a nice way, the earlier results for \(n\leq 2\), and secondly the formula of the main result is consistent with passing from \(n\) to \(n-1\): if \(H\) is the order \(p\) subgroup of \(G\) and you take \(H\)-invariants on both sides, you end up with a formula which turns out to be obviously true. The writing is careful, often quite elegant, sometimes abruptly terse. In Lemma 3.4, the reviewer thinks that instead of an \(O_T[G]\)-basis, an \(O_T[\text{Gal}(K_j/K)]\)-basis is meant. The proof could be a little clearer. Actually the reasoning used to pass from a set of generators to a basis seems to be: some generators have a nontrivial annihilator, so they cannot be in any basis, so we can delete them. The first ``so'' is clearly correct, but the second demands justification (for instance locality of the ring \(O_T[\text{Gal}(K_j/K)]\) plus Nakayama's lemma). This justifies the claim (labelled as ``it is easily shown'' but not given further explanation) that the remaining elements continue to span, and then the author's counting argument indeed shows that we end up with a basis. Reviewer's comment: It is a nice bit of luck that this fairly general way of reasoning is sufficient at this important point. In two instructive final sections 6 and 7 the author makes his results more explicit for \(n=3\), and for arbitrary \(n\) under supplementary hypotheses. It is interesting to note that under the conditions of the main result, at most \(2^{n+1}-1\) isomorphism classes of indecomposables occur, and all of them have \(\mathbb{Z}_p\)-rank at most twice the order of \(G\). Actually it seems that the main theorem even gives the decomposition into indecomposables over \(O_T[G]\) where \(T\) is the maximal absolutely unramified subfield of \(K\).
scientific article; zbMATH DE number 1943331