A counterexample to a 𝐵𝑃-analogue of the chromatic splitting conjecture

Abstract

We prove that, if n ≥ 2 n\ge 2 , the E ( n − 1 ) ∗ E(n-1)_\ast -localization of the K ( n ) ∗ K(n)_\ast -localization map B P p → L K ( n ) B P BP_p\to L_{K(n)}BP is not a split monomorphism in the stable category by exhibiting spectra Z Z for which the map π ∗ ( L n − 1 ( B P p ) ∧ Z ) → π ∗ ( L n − 1 ( L K ( n ) B P ) ∧ Z ) \pi _\ast (L_{n-1}(BP_p)\wedge Z)\to \pi _\ast (L_{n-1}(L_{K(n)}BP)\wedge Z) is not injective. If p ≥ max { 1 2 ( n 2 − 2 n + 2 ) , n + 1 } p\ge \max \{\frac {1}{2}(n^2-2n +2), n+1\} and n ≥ 3 n\geq 3 , we show that Z Z may be taken to be a two-cell complex in the sense of E ( n − 1 ) ∗ E(n-1)_\ast -local homotopy theory. The question of whether the map L n − 1 ( B P p ) → L n − 1 L K ( n ) B P L_{n-1}(BP_p)\to L_{n-1}L_{K(n)}BP splits was asked by Hovey and is in some sense a B P BP -analogue of Hopkins’ chromatic splitting conjecture.