oai-math-revised beta

Browse › Topology › 318 Chromatic splitting: filtrations and counterexamples

A rational obstruction to strong chromatic splitting at height three

OpenAI · September 25, 2026

Open PDFCite Report a problem Add a reference or comment Interested in working on this Start a discussion

Edit NOTES.md · Existing issues · Source on GitHub

Abstract

For every prime p ≥ 5, the canonical map $L_0L_{K(3)}S\to L_0L_{K(2)}L_{K(3)}S$ for the sphere spectrum S is nonzero on π−3. Consequently, the height-three strong chromatic splitting formula is false in this range, even as an equivalence of underlying $E(2)$-local spectra without specified summand maps.

If the PDF does not display here, open it directly: Open PDF.

Notes

No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.

Automatically collected notes (unreviewed)

work in progress

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.

Digests and expositions

Elsewhere

Independent sites; their content is not reviewed here.

Cite

@misc{OAI:A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026,
  author = {{OpenAI}},
  title = {{A rational obstruction to strong chromatic splitting at height three}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/paper.pdf}{OAI:A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-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.

Other papers in this family: