Uniqueness of p-local truncated Brown-Peterson spectra

When p is an odd prime, we prove that the Fₚ-cohomology of BP n as a module over the Steenrod algebra determines the p-local spectrum BP n. In particular, we prove that the p-local spectrum BP n only depends on its p-completion BP nₚ^. As a corollary, this proves that the p-local homotopy type of BP n does not depend on the ideal by which we take the quotient of 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 BP n. We also prove that there are enough endomorphisms of BP n in a suitable sense. When $p=2$, we obtain the results for n≤ 3.