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

Abstract: When $p$ is an odd prime, we prove that the $\mathbb F_p$-cohomology of $\mathrm{BP}\langle n\rangle$ 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 $n\leq 3$.