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PreprintCC BY 4.0arXiv

On the equivariant KUG-local sphere for finite abelian groups

  • Yingxin Li
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Abstract

We study KUG-localization for finite groups. For a finite nilpotent group G, we identify the KUG/p-local sphere as the fiber of an Adams operation, and reduce the computation of its homotopy Mackey functors to the corresponding computation for the Sylow p-subgroup of G. At the prime $2$, we compute underlineπ* LKUG/2SG for abelian $2$-groups, resolve the resulting extension problems, and determine the degree-zero Hurewicz image. As a consequence, we completely determine the ℤ-graded homotopy Mackey functors of LKUGSG for all finite abelian groups. Finally, for an arbitrary finite group G, we prove that the KUG/p-localization of G-spectra splits as a wedge of height-one equivariant Morava K-localizations, indexed by conjugacy classes of cyclic subgroups of order prime to p.

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Cite

BibTeXRIS
BibTeX
@misc{li2026equivariant,
  title = {{On the equivariant $KU_G$-local sphere for finite abelian groups}},
  author = {Yingxin Li},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2605.30285},
  eprint = {2605.30285},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.30285},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On the equivariant KU_G-local sphere for finite abelian groups
AU  - Yingxin Li
PY  - 2026
DA  - 2026/08/31
PB  - arXiv
DO  - 10.48550/arxiv.2605.30285
UR  - https://arxiv.org/abs/2605.30285
AB  - We study KU_G-localization for finite groups. For a finite nilpotent group G, we identify the KU_G/p-local sphere as the fiber of an Adams operation, and reduce the computation of its homotopy Mackey functors to the corresponding computation for the Sylow p-subgroup of G. At the prime $2$, we compute underlineπ_* L_KU_G/2S_G for abelian $2$-groups, resolve the resulting extension problems, and determine the degree-zero Hurewicz image. As a consequence, we completely determine the ℤ-graded homotopy Mackey functors of L_KU_GS_G for all finite abelian groups. Finally, for an arbitrary finite group G, we prove that the KU_G/p-localization of G-spectra splits as a wedge of height-one equivariant Morava K-localizations, indexed by conjugacy classes of cyclic subgroups of order prime to p.
ER  -