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