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Filtered chromatic splitting at generic primes

OpenAI · September 25, 2026

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Abstract

For every n ≥ 1 and prime $p\gt n+1$, we construct a $2^n$-stage ordered filtration of $L_{n-1}L_{K(n)}S_p^\wedge$ with the classical chromatic-splitting cofibers. The map from the first stage to the target is the canonical localization unit.

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Possibly relevant literature (found by the agent, not checked)

  • Rin Ray, Modeling Group Actions on Stacks (Especially the Lubin-Tate Action) (arXiv:2507.00309, v2 20 October 2026). Develops the two-tower method for the Lubin–Tate action and describes itself as a companion to the proof of the coheight-one weak chromatic splitting conjecture; the paper cites the relative two-tower description only through Barthel–Mann–Ray–Schlank–Senger–Weinstein–Zhou.

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Cite

@misc{OAI:Filtered-chromatic-splitting-at-generic-primes-September-25-2026,
  author = {{OpenAI}},
  title = {{Filtered chromatic splitting at generic primes}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Filtered-chromatic-splitting-at-generic-primes-September-25-2026/paper.pdf}{OAI:Filtered-chromatic-splitting-at-generic-primes-September-25-2026}},
  year = {2026}
}

Family 318: Chromatic splitting: filtrations and counterexamples

Disproves strong chromatic splitting at height three for primes p ≥ 5, and weak splitting for the derived p-completed sphere at heights p (p ≥ 5) and $p+1$ (p ≥ 7). Nevertheless, for n ≥ 1 and $p\gt n+1$, the overlap $L_{n-1}L_{K(n)}S_p^\wedge$ admits a $2^n$-stage filtration by the predicted localized-sphere pieces. At height three and prime three, even finite assembly from such pieces fails in the category of $E(2)$-local modules over the derived completed sphere.

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