Result 308, Topology

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,v1,…,vn)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)V(4) at p = 1009.

Proof

The bigger picture

Why it matters

Topology often asks whether a prescribed algebraic pattern comes from an actual geometric object. These manuscripts report finite realizations for an entire sequence of such patterns, provided the prime used to study them can change.

What changes?

The manuscript reports that for every nonnegative integer n, there is a prime p, depending on n, and a finite p-local spectrum V(n): a stable topological object built from finitely many cells and studied at p. Its Brown-Peterson homology, an algebraic invariant, is the coefficient ring with p and the generators v1 through vn set to zero, each to its first power. It has the canonical comodule structure, encoding homology operations, and its homology generator is in degree zero.

What does that help mathematicians do?

The claim establishes that these specific algebraic quotients can be realized by finite spectra at every index n, without replacing the generators by higher powers. It separates existence somewhere among the primes from existence at a prescribed prime: the latter does not follow. The companion manuscript makes one case explicit, reporting V(4) at prime 1009, with 1009 and v1 through v4 killed to their first powers.

Are there practical applications?

The immediate value is foundational: these complexes connect explicit algebraic relations to finite objects in stable topology. Their specified homology and comodule structure give researchers models in which those relations and homology operations coexist. The supplied abstracts establish no practical application or computational speedup; their contribution is this precise realization claim.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

Finite Smith–Toda Complexes at Varying Primes

September 23, 2026 54 pages

For every nonnegative integer n, we construct a Smith–Toda complex V(n)V(n) at some prime p depending on n. It is a finite p-local spectrum whose Brown–Peterson homology is BP∗/(p,v1,…,vn)\mathrm{BP}_*/(p,v_1,\ldots,v_n) with its canonical comodule structure and generator in degree zero. Each listed generator is killed to its first power.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Finite-Smith-Toda-Complexes-at-Varying-Primes-September-23-2026,
  author = {{OpenAI}},
  title = {{Finite Smith--Toda Complexes at Varying Primes}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Finite-Smith-Toda-Complexes-at-Varying-Primes-September-23-2026/paper.pdf}{OAI:Finite-Smith-Toda-Complexes-at-Varying-Primes-September-23-2026}},
  year = {2026}
}

A finite Smith–Toda complex V(4) at the prime 1009

September 23, 2026 44 pages

We construct a Smith–Toda complex V(4)V(4) at the prime 1009. It is an ordinary finite 1009-local spectrum whose Brown–Peterson homology is BP∗/(1009,v1,v2,v3,v4)BP_*/(1009,v_1,v_2,v_3,v_4), with its canonical comodule structure and generator in degree zero.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.