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Cyclic length and chromatic fixed-point loss
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Abstract
For a finite p-group G and a subgroup H, we prove that the optimal chromatic fixed-point loss equals the shortest length of a subnormal chain from H to G with cyclic quotients. The equality holds at every prime and every nonnegative height, resolving positively the equality proposed by Kuhn and Lloyd.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Related prior literature mentioned by others
- Nicholas J. Kuhn and Christopher J. R. Lloyd, Chromatic fixed point theory and the Balmer spectrum for extraspecial 2-groups (arXiv:2008.00330, Amer. J. Math. 146 (2024)). The VibeMathed entry names this as the source of the question; it is the paper's reference [KL], where the equality $r_n(G,H)=\ell(G,H)$ is proposed and proved for central extensions of elementary abelian 2-groups by $C_2$.
Possibly relevant literature (found by the agent, not checked)
- Nicholas J. Kuhn, A short proof of the chromatic Smith Fixed Point Theorem (arXiv:2112.05001, 2021-12-09). Gives a simplified proof of the rank-based Morava K-theory acyclicity implication for fixed points of finite abelian p-group actions, i.e. the abelian case of the upper bound $r_n\le\ell$ that the paper attributes to [BHNNNS] and [BK].
- Nicholas J. Kuhn and Christopher J. R. Lloyd, Computing the Morava K-theory of real Grassmanians using chromatic fixed point theory (arXiv:2111.08812, Algebr. Geom. Topol. 24 (2024)). This applies the chromatic Floyd-type dimension inequalities for $C_4$-actions; the paper uses the equivalence of acyclicity implications with such inequalities from [KL, Theorem 2.2].
- Gregory Arone, Tobias Barthel, Drew Heard and Beren Sanders, The spectrum of excisive functors (arXiv:2402.04244, Invent. Math. 241 (2025)). This proves a non-abelian blueshift theorem for generalized Tate constructions and a functor-calculus analogue of transchromatic Smith–Floyd theory; it may be compared with the paper's nonabelian sharpness argument, which goes through the Balmer–Sanders tt-geometric framework.
Digests and expositions
- Latent Seminar, Chromatic Smith theory: cyclic length and fixed-point loss (2026-10-07). Explains the claimed cyclic-length formula and its character, profinite-group, and support arguments; the source states that it reviews statements and selected proof mechanisms rather than independently verifying the complete proof. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 314
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Cyclic-Length-and-Chromatic-Fixed-Point-Loss-September-24-2026,
author = {{OpenAI}},
title = {{Cyclic length and chromatic fixed-point loss}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Cyclic-Length-and-Chromatic-Fixed-Point-Loss-September-24-2026/Cyclic-Length-and-Chromatic-Fixed-Point-Loss-September-24-2026.pdf}{OAI:Cyclic-Length-and-Chromatic-Fixed-Point-Loss-September-24-2026}},
year = {2026}
}Family 314: Cyclic length and chromatic fixed-point loss
Determines the optimal chromatic loss from geometric H-fixed points to geometric G-fixed points for every subgroup H of a finite p-group G. At every nonnegative height, the loss equals the shortest subnormal-chain length from H to G with cyclic quotients. Each quotient counts once regardless of order, and finite spectra witness sharpness.
This is the only paper in this family.