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PreprintarXiv licencearXiv

On the Anderson Dual of Hunderlineℤ and Hunderline𝔽2

  • Hana Jia Kong
  • Achim Krause
  • Yutao Liu
  • XiaoLin Danny Shi
  • Zhouli Xu
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Abstract

In this paper, we give complete classifications of the finite groups G for which the constant Eilenberg--Mac Lane spectra Hunderlineℤ and Hunderline𝔽2 are Anderson self-dual up to a virtual representation shift, and we determine all such shifts. For both of these spectra, we determine the representation periodicity lattice for every finite group. As applications, we obtain RO(G)-graded universal coefficient theorems relating equivariant homology and cohomology, together with formulas for the Anderson duals of Hunderlineℤ-modules. In the Hunderline𝔽2 self-dual cases, we identify the graded 𝔽2-linear dual of the coefficient ring π⋆G Hunderline𝔽2 with a representation shift of the ring as a module over itself. This gives perfect multiplication pairings and graded self-injectivity.

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Cite

BibTeXRIS
BibTeX
@misc{kong2026anderson,
  title = {{On the Anderson Dual of $H\underline\mathbb Z$ and $H\underline\mathbb F_2$}},
  author = {Hana Jia Kong and Achim Krause and Yutao Liu and XiaoLin Danny Shi and Zhouli Xu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.35796},
  eprint = {2610.35796},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.35796},
}
RIS
TY  - UNPB
TI  - On the Anderson Dual of Hunderlineℤ and Hunderline𝔽_2
AU  - Hana Jia Kong
AU  - Achim Krause
AU  - Yutao Liu
AU  - XiaoLin Danny Shi
AU  - Zhouli Xu
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.35796
UR  - https://arxiv.org/abs/2610.35796
AB  - In this paper, we give complete classifications of the finite groups G for which the constant Eilenberg--Mac Lane spectra Hunderlineℤ and Hunderline𝔽_2 are Anderson self-dual up to a virtual representation shift, and we determine all such shifts. For both of these spectra, we determine the representation periodicity lattice for every finite group. As applications, we obtain RO(G)-graded universal coefficient theorems relating equivariant homology and cohomology, together with formulas for the Anderson duals of Hunderlineℤ-modules. In the Hunderline𝔽_2 self-dual cases, we identify the graded 𝔽_2-linear dual of the coefficient ring π_⋆^G Hunderline𝔽_2 with a representation shift of the ring as a module over itself. This gives perfect multiplication pairings and graded self-injectivity.
ER  -