2024/05/01 by Lee, David Jongwon
#Algebraic Topology (math.AT) #FOS: Mathematics
paper · doi:10.48550/arxiv.2405.00889
When p is an odd prime, we prove that the \mathbb 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⟩p^\wedge. 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.