Tensor Nilpotence and the Size of the Bousfield Lattice

Abstract

For each prime $p$ and integer $n \geq 1$, we construct a $p$-local spectrum $X_n$ with $X_n^{\otimes n} \not\simeq 0$ but $X_n^{\otimes (n+1)} \simeq 0$. This refutes the retract conjecture of Hovey-Palmieri. Our construction also allows us to determine the cardinality of the Bousfield lattice to be $2^{2^{\aleph_0}}$, answering a question of Dwyer-Palmieri.