Uniqueness of p-local truncated Brown-Peterson spectra

Abstract

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⟨n⟩p̂. 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.