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.

Manuscripts (2)

Finite Smith–Toda Complexes at Varying Primes

September 23, 2026 · source · cite as \cite{OAI:Finite-Smith-Toda-Complexes-at-Varying-Primes-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.

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

September 23, 2026 · source · cite as \cite{OAI:A-finite-Smith-Toda-complex-V4-at-the-prime-1009-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.

Result 308 · Topology. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.