For each prime p and integer n, the author builds spectra X_n with X_n^{⊗n} ≠ 0 but X_n^{⊗(n+1)} ≃ 0, refuting the Hovey-Palmieri retract conjecture and showing the Bousfield lattice has 2^{2^{ℵ_0}} elements.
Hodel,Cardinal functions I, in: Handbook of Set-Theoretic Topology, K
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Tensor Nilpotence and the Size of the Bousfield Lattice
For each prime p and integer n, the author builds spectra X_n with X_n^{⊗n} ≠ 0 but X_n^{⊗(n+1)} ≃ 0, refuting the Hovey-Palmieri retract conjecture and showing the Bousfield lattice has 2^{2^{ℵ_0}} elements.