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Finite Smith–Toda Complexes at Varying Primes
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Abstract
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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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 family's explicit height-four construction to every prime p ≥ 11, building on the companion p = 1009 paper rather than proving the all-height assertion; its introduction reports finding no mathematical errors in the two manuscripts, but suggests stating this result for all sufficiently large p and questions the use of ultrafilters. [family-level]
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). Records an earlier, disputed claim that V(n) exists for all n at primes p ≥ 5, which is relevant to this paper's description of the history of the all-height existence problem.
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]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 308
Independent sites; their content is not reviewed here.
Cite
@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}
}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:
- A finite Smith–Toda complex V(4) at the prime 1009 · September 23, 2026