Topology · Result 308

Finite Smith–Toda complexes at every height

2 papers

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.

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▶Introduction: the problem and why it mattersin preparation

▶The presented solutionin preparation

Papers 2

Finite Smith–Toda Complexes at Varying Primes

OpenAI · September 23, 2026

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

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@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

OpenAI · September 23, 2026

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.

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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}
}

Formal verification

No Lean formalization is listed for this result yet. The repository notes that unformalized results could have issues.