For each height n, the category of spectra of fp-type n is the thick subcategory generated by the truncated Brown–Peterson spectrum BPlangle n rangle; equivalently, FP_n=operatornameThick(BPlangle n rangle), where FP_n denotes the category of spectra of fp-type n.

Hahn–Wilson conjecture

For each height $n$ , the category of spectra of fp-type $n$ is the thick subcategory generated by the truncated Brown–Peterson spectrum $\mathrm{BP}\langle n\rangle$ ; equivalently, $\mathrm{FP}_n=\operatorname{Thick}(\mathrm{BP}\langle n\rangle)$ , where $\mathrm{FP}_n$ denotes the category of spectra of fp-type $n$ .

References

Primary source

Inventiones mathematicae

Additional references

The monochromatic Hahn-Wilson conjecture — Inventiones mathematicae — David Jongwon Lee, Piotr Pstrągowski

chromatic-homotopy-theory stable-homotopy-theory

Posted 1 month ago

Progress summary

Refreshed 4 weeks ago

Claimed progress

A 2024 paper proves a localized version and the first nontrivial height, but the conjecture at higher heights remains open.

The Hahn–Wilson conjecture asserts that spectra of fp-type $n$ are exactly the thick subcategory generated by $\mathrm{BP}\langle n\rangle$ . The authors learned it from Dylan Wilson in 2021 and give its first written formulation.

October 2024 advance

Burklund, Hahn, Levy, and Schlank prove the $K(n)$ -local analogue: locally fp spectra are generated by $L_{K(n)}\mathrm{BP}\langle n\rangle$ . Using the height- $1$ telescope conjecture, they deduce $\mathrm{FP}_1=\operatorname{Thick}(\mathrm{BP}\langle 1\rangle)$ . The paper does not claim the original conjecture for all heights; the result is therefore reported here as unverified progress.

Current status (as of September 2026): The $K(n)$ -local analogue and the original height- $1$ case are claimed proved; the original conjecture for heights $n>1$ remains open.

Sources

No solutions have been posted yet.