On the equivariant $KU_G$-local sphere for finite abelian groups
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866910275573121024 |
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| author | Li, Yingxin |
| author_facet | Li, Yingxin |
| contents | Given a finite abelian group $G$ and a Sylow $p$-subgroup $N_p$, we prove that the $KU_G/p$-local sphere spectrum is equivalent to the homotopy fixed points of a $p$-complete $KO_{N_p}$-module spectrum. Then we compute the $\mathbb{Z}$-graded homotopy Mackey functors of the $KU_G$-local sphere spectrum. This result generalizes the computation of arXiv:2303.12271 for finite $p$-groups, where $p$ is an odd prime. Finally, by comparing the Bousfield classes of $KU_G/p$ and $G$-equivariant Morava $K$-theory, we prove that the $KU_G/p$-local sphere spectrum is equivalent to a wedge sum of equivariant Morava $K$-theory localized sphere spectra, and describe the $RO(G)$-graded homotopy Mackey functors of the $KU_G/p$-local sphere spectrum. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_30285 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the equivariant $KU_G$-local sphere for finite abelian groups Li, Yingxin Algebraic Topology 55P91, 55Q91 Given a finite abelian group $G$ and a Sylow $p$-subgroup $N_p$, we prove that the $KU_G/p$-local sphere spectrum is equivalent to the homotopy fixed points of a $p$-complete $KO_{N_p}$-module spectrum. Then we compute the $\mathbb{Z}$-graded homotopy Mackey functors of the $KU_G$-local sphere spectrum. This result generalizes the computation of arXiv:2303.12271 for finite $p$-groups, where $p$ is an odd prime. Finally, by comparing the Bousfield classes of $KU_G/p$ and $G$-equivariant Morava $K$-theory, we prove that the $KU_G/p$-local sphere spectrum is equivalent to a wedge sum of equivariant Morava $K$-theory localized sphere spectra, and describe the $RO(G)$-graded homotopy Mackey functors of the $KU_G/p$-local sphere spectrum. |
| title | On the equivariant $KU_G$-local sphere for finite abelian groups |
| topic | Algebraic Topology 55P91, 55Q91 |
| url | https://arxiv.org/abs/2605.30285 |