Browse › Topology › 308 Finite Smith–Toda complexes at every height
A finite Smith–Toda complex V(4) at the prime 1009
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
We construct a Smith–Toda complex $V(4)$ at the prime 1009. It is an ordinary finite 1009-local spectrum whose Brown–Peterson homology is $BP_*/(1009,v_1,v_2,v_3,v_4)$, with its canonical comodule structure and generator in degree zero.
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.
Follow-up works
- Vigleik Angeltveit, The Smith-Toda complex V(4) exists for p ≥ 11 (arXiv:2610.23215, 2026-10-16). Extends the existence of V(4) from p = 1009 to all primes p ≥ 11, reusing this paper's framework with shorter arguments (a reference to Toda, a shorter May spectral sequence calculation, a Toda bracket argument); the author reports finding no mathematical errors but calls 1009 "suspiciously high", and states that ChatGPT was used as a research assistant. [family-level]
- acprk (GitHub account; author name a placeholder), Smith–Toda V(4) exists if and only if p ≥ 11 (GitHub remark R4, October 2026). Reruns this paper's finite inequalities at a general prime p ≥ 11 by exact enumeration, with existence conditional on this manuscript, and collects nonexistence arguments for p < 11; it states that it does not verify the manuscript's homotopy-theoretic steps. [AI-generated per source]
Comments and discussion
- Vigleik Angeltveit, A closer look at one of the OpenAI papers (ANU Algebra and Topology Seminar, 2026-10-20). The seminar abstract describes using ChatGPT as a research assistant to understand this manuscript, simplify its arguments, and extend the result to all primes p ≥ 11.
Possibly relevant literature (found by the agent, not checked)
- Douglas C. Ravenel, open letter on Zhou Xueguang's preprint "Smith-Toda spectrum V(∞) exists for all p ≥ 5" (algtop-l archive, 15 July 2002). Discusses a disputed earlier existence claim for all V(n) and defends Toda's relation α₁β₁ᵖ = 0, the relation of [Toda1967] that this paper cites as a topological antecedent of its β-power annihilation bound.
- Andrew Salch, The cohomology of the height four Morava stabilizer group at large primes (arXiv:1607.01108, July 2016). Announces a large-prime computation of height-four Morava stabilizer group cohomology and of K(4)-local homotopy of V(3), which bears on the height-four Adams–Novikov obstruction analysis for v₄ on V(3) in this paper.
Digests and expositions
- Latent Seminar, Smith–Toda complexes: varying primes and an explicit height four (2026-10-07). Explains both constructions, including finite descent and the timing of Adams–Novikov obstructions, and explicitly disclaims independent verification of the complete proofs. [family-level]
News and blogs
- Post-Cutoff, Smith–Toda V(4) for p ≥ 11, after OpenAI’s p = 1009 (filed 2026-10-19). Reports Angeltveit's height-four improvement and his comments on both OpenAI manuscripts, with links to the primary sources. [AI-generated per source] [family-level]
Elsewhere
- Emergent Mind: family 308
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026,
author = {{OpenAI}},
title = {{A finite Smith--Toda complex $V(4)$ at the prime $1009$}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/paper.pdf}{OAI:A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026}},
year = {2026}
}Family 308: Finite Smith–Toda complexes at every height
For every n ≥ 0, constructs a finite Smith–Toda spectrum at a prime p depending on n, with Brown–Peterson homology $BP_*/(p,v_1,\ldots,v_n)$ and the canonical comodule structure. Thus Smith–Toda complexes exist at every height when the prime may vary; an explicit example realizes $V(4)$ at p = 1009.
Other papers in this family:
- Finite Smith–Toda Complexes at Varying Primes · September 23, 2026