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

Abstract

When $p$ is an odd prime, we prove that the ${\mathbb{F}}_p$-cohomology of $\mathrm{BP}⟨n⟩$ as a module over the Steenrod algebra determines the $p$-local spectrum $\mathrm{BP}⟨n⟩$. In particular, we prove that the $p$-local spectrum $\mathrm{BP}⟨n⟩$ only depends on its $p$-completion $\mathrm{BP}⟨n{⟩}_p^{\wedge }$. As a corollary, this proves that the $p$-local homotopy type of $\mathrm{BP}⟨n⟩$ 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 $\mathrm{BP}⟨n⟩$. We also prove that there are enough endomorphisms of $\mathrm{BP}⟨n⟩$ in a suitable sense. When $p=2$, we obtain the results for $n\le 3$.

27 pages, comments welcome