Finite Smith–Toda complexes at every height

1 · The essentials

The manuscripts describe a family of finite mathematical objects with a precisely prescribed algebraic description. The family is indexed by a nonnegative whole number, called its height. For every height, the main manuscript claims to construct an object of the required kind. Each construction uses a prime number, and that prime is allowed to depend on the height. A companion manuscript gives the specific example of height four at the prime 1009. The claim is therefore about reaching every height by allowing the prime to vary, not about reaching every height at one fixed prime. The main result has not been formalized in Lean or formally verified.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine making a finite model for each level of a design challenge, with permission to choose a different material at each level. Here the level stands for the height and the material stands for the prime: the claim is that every level can be reached with some choice, not that one choice works for all levels.

Why it matters

The claimed result gives a uniform existence statement across all nonnegative heights while allowing the prime to change. It also specifies the algebraic structure of the objects, rather than merely asserting that some finite objects exist. The companion manuscript makes this concrete with a height-four example at the specified prime 1009.

2 · Going further

The precise construction problem is to find, for every integer $n \geq 0$ , a finite $p$ -local spectrum $V (n)$ whose Brown–Peterson homology is $BP_{*} / (p, v_{1}, \dots, v_{n})$ . The required homology carries its canonical comodule structure, with its generator in degree zero. The main manuscript claims such a construction for some prime $p$ depending on $n$ .

The quotient is specified exactly: each of $p, v_{1}, \dots, v_{n}$ is killed to its first power. Thus the claim is not merely about a related quotient in which higher powers of these elements are set to zero. The essential scope distinction is between existence at every height with a varying prime, which is claimed, and existence at every height for a fixed prime, which is not asserted here.

The companion manuscript claims an ordinary finite 1009-local spectrum $V (4)$ with homology $BP_{*} / (1009, v_{1}, v_{2}, v_{3}, v_{4})$ , again with the canonical comodule structure and generator in degree zero. It supplies a specific instance of the general family. The supplied abstracts state the construction targets and outcomes but do not describe the construction method or the argument, so no further account of the mechanism can be given from this source.

Useful notions

Prime numbers
Quotient rings and setting elements to zero
Spectra and finiteness
Localization at a prime
Brown–Peterson homology and comodule structure

What was known before / what is new

Before

The supplied source identifies the target objects as Smith–Toda complexes and specifies their required Brown–Peterson homology. It gives no account of earlier existence results, historical developments, or previously known height ranges.

New

The main manuscript claims a finite Smith–Toda complex $V (n)$ for every nonnegative integer $n$ , at a prime that may depend on $n$ . Its homology has the stated quotient, canonical comodule structure, and degree-zero generator, with each listed element killed to its first power. The companion manuscript claims the explicit case $V (4)$ at the prime 1009.

Limits and assumptions

The general claim assumes $n \geq 0$ and permits the prime to depend on $n$ ; it does not assert existence at every height for a fixed prime. The objects must be finite and local at the chosen prime, with exactly the specified homology and comodule structure. The abstracts give neither a construction method nor a general rule for choosing the prime, and they do not identify further open problems. The main result is not formalized in Lean and has not been formally verified.

Mini glossary

Height: In this family, the nonnegative integer $n$ indexing the desired complex $V (n)$ .
Smith–Toda complex: Here, a finite $p$ -local spectrum $V (n)$ with Brown–Peterson homology $BP_{*} / (p, v_{1}, \dots, v_{n})$ , its canonical comodule structure, and a generator in degree zero.
Brown–Peterson homology: The algebraic description denoted by $BP_{*}$ whose prescribed quotient is a defining requirement of the constructions in these manuscripts.
Killed to its first power: An element itself is set to zero in the quotient, rather than only a higher power of that element.

Keywords : Smith–Toda complexes · Brown–Peterson homology · Varying primes

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

For every n ≥ 0, constructs a finite Smith–Toda spectrum at a prime p depending on n, with Brown–Peterson homology $B P_{*} / (p, v_{1}, \dots, v_{n})$ 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.

To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.

Manuscripts in this family

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Finite Smith–Toda Complexes at Varying Primes

23/09/2026

To be verified

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 $BP_{*} / (p, v_{1}, \dots, v_{n})$ with its canonical comodule structure and generator in degree zero. Each listed generator is killed to its first power.

PDF GitHub folder

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

23/09/2026

To be verified

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 $B P_{*} / (1009, v_{1}, v_{2}, v_{3}, v_{4})$ , with its canonical comodule structure and generator in degree zero.

PDF GitHub folder