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Ramification ideals for products of pure subgroups

  • Josnei Novacoski
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Abstract

We study ramification ideals of finite Galois extensions of henselian valued fields whose Galois groups are products of pure subgroups. We give a criterion showing that the ideals supplied by the factors need not exhaust all subgroup ideals, and construct Cp2-counterexamples of rank one with each defect 1,p,p2. For ramification-adapted families, whose intersections generate each ramification group, every subgroup ideal is represented by a cyclic subgroup of one member; when the members are pure, this gives a description in terms of distance sets. Such adapted decompositions always exist for elementary abelian groups. We also characterize principality of ramification ideals through defectlessness of the pure adapted factors. Kuhlmann's example shows that these conditions do not imply defectlessness of the full extension.

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Cite

BibTeXRIS
BibTeX
@misc{novacoski2026ramification,
  title = {{Ramification ideals for products of pure subgroups}},
  author = {Josnei Novacoski},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.21210},
  eprint = {2608.21210},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.21210},
}
RIS
TY  - UNPB
TI  - Ramification ideals for products of pure subgroups
AU  - Josnei Novacoski
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2608.21210
UR  - https://arxiv.org/abs/2608.21210
AB  - We study ramification ideals of finite Galois extensions of henselian valued fields whose Galois groups are products of pure subgroups. We give a criterion showing that the ideals supplied by the factors need not exhaust all subgroup ideals, and construct C_p^2-counterexamples of rank one with each defect 1,p,p^2. For ramification-adapted families, whose intersections generate each ramification group, every subgroup ideal is represented by a cyclic subgroup of one member; when the members are pure, this gives a description in terms of distance sets. Such adapted decompositions always exist for elementary abelian groups. We also characterize principality of ramification ideals through defectlessness of the pure adapted factors. Kuhlmann's example shows that these conditions do not imply defectlessness of the full extension.
ER  -