Uniqueness of p-local truncated Brown–Peterson spectra

Abstract When p is an odd prime, we prove that the $$\mathbb {F}_p$$ F p -cohomology of $$\textrm{BP}\langle n\rangle $$ BP ⟨ n ⟩ as a module over the Steenrod algebra determines the p -local spectrum $$\textrm{BP}\langle n\rangle $$ BP ⟨ n ⟩ . In particular, we prove that the p -local spectrum $$\textrm{BP}\langle n\rangle $$ BP ⟨ n ⟩ only depends on its p -completion $$\textrm{BP}\langle n\rangle _p^\wedge $$ BP ⟨ n ⟩ p ∧ . As a corollary, this proves that the p -local homotopy type of $$\textrm{BP}\langle n\rangle $$ BP ⟨ n ⟩ does not depend on the ideal by which we take the quotient of $$\textrm{BP}$$ 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 $$\textrm{BP}\langle n\rangle $$ BP ⟨ n ⟩ . We also prove that there are enough endomorphisms of $$\textrm{BP}\langle n\rangle $$ BP ⟨ n ⟩ in a suitable sense. When $$p=2$$ p = 2 , we obtain the results for $$n\le 3$$ n ≤ 3 .