Uniqueness of $p$-local truncated Brown-Peterson spectra

TL;DR

This paper proves that for odd primes, the p-local truncated Brown-Peterson spectrum is uniquely determined by its mod p cohomology as a Steenrod algebra module, establishing its dependence solely on its p-completion.

Contribution

It demonstrates the uniqueness of p-local truncated Brown-Peterson spectra based on their cohomology and p-completion, extending understanding of their homotopy types.

Findings

Cohomology determines the p-local spectrum uniquely.

p-local spectrum depends only on its p-completion.

Vanishing line for odd degree classes in Adams spectral sequence.

Abstract

When $p$ is an odd prime, we prove that the $\mathbb F_p$-cohomology of $\mathrm{BP}\langle n\rangle$ as a module over the Steenrod algebra determines the $p$-local spectrum $\mathrm{BP}\langle n\rangle$. In particular, we prove that the $p$-local spectrum $\mathrm{BP}\langle n\rangle$ only depends on its $p$-completion $\mathrm{BP}\langle n\rangle_p^\wedge$. As a corollary, this proves that the $p$-local homotopy type of $\mathrm{BP}\langle n\rangle$ does not depend on the ideal by which we take the quotient of $\mathrm{BP}$. In the course of the argument, we show that there is a vanishing line for odd degree classes in the Adams spectral sequence for endomorphisms of $\mathrm{BP}\langle n\rangle$. We also prove that there are enough endomorphisms of $\mathrm{BP}\langle n\rangle$ in a suitable sense. When $p=2$, we obtain the results for $n\leq 3$.