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.
Inhabitants of interesting subsets of the Bousfield lattice
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abstract
The set of Bousfield classes has some important subsets such as the distributive lattice $\mathbf{DL}$ of all classes $\langle E\rangle$ which are smash idempotent and the complete Boolean algebra $\mathbf{cBA}$ of closed classes. We provide examples of spectra that are in $\mathbf{DL}$, but not in $\mathbf{cBA}$; in particular, for every prime $p$, the Bousfield class of the Eilenberg-MacLane spectrum $\langle H\mathbb{F}_p\rangle\in\mathbf{DL}{\setminus}\mathbf{cBA}$.
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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.