Ramification ideals for products of pure subgroups

Abstract

Let $\mathcal E=(L/K,v)$ be a finite Galois extension of henselian valued fields. We study the ramification ideals $I_H$, for subgroups $H\leq {\rm Gal}(L/K)$, when the Galois group is a product of subgroups $H_i$ such that $L/K_{H_i}$ is pure (depth one). We first recall an explicit formula for ramification ideals of pure extensions and use it to obtain a lower bound for the ideals attached to arbitrary subgroups of a product of pure subgroups. A natural question is whether every $I_H$ coincides with one of the ideals coming from the pure factors. We show that this is not true, already for a defectless extension with Galois group $C_p\times C_p$. The counterexample is a compositum of two Artin--Schreier extensions with different ramification breaks; the failure comes from choosing a decomposition which is not compatible with the ramification filtration. Motivated by this example, we introduce ramification-adapted decompositions and prove a filtration-theoretic substitute for the conjectural statement in elementary abelian $p$-extensions. Since every flag of $\mathbb F_p$-vector spaces admits an adapted basis, every elementary abelian $p$-extension admits such a decomposition, and every subgroup ideal is represented by one adapted cyclic factor. If the adapted factors are pure, this representation can be written in the distance-set form occurring in the original conjecture. We also prove that, for an adapted decomposition $\mathcal G=H_1\times\cdots\times H_r$, all ramification ideals are principal if and only if every degree-$p$ extension $L/K_{H_i}$ is defectless. Finally, we discuss the degree-$p^2$ example constructed by Kuhlmann in \cite[Section 3.5]{Topics}. Consequently every basis is ramification-adapted, while principality of all ramification ideals still does not characterize defectlessness.