For an arbitrary prime \(p\), the author studies the set \(\mathcal L\) of Bousfield classes \(\langle E\rangle\) of \(p\)-local spectra \(E\), which has the structure of an ordered semiring by declaring \(\langle E\rangle\leq\langle F\rangle\) if each \(F_*\)-acyclic spectrum is \(E_*\)-acyclic. The goal is to shed light on Ravenel’s Telescope Conjecture. The method of attack is a two-stage construction. First, the author builds an ordered semiring quotient \(\pi:\mathcal L \to\overline{\mathcal L}\) which would be an isomorphism in case the Telescope Conjecture is true, meaning one can regard \(\overline{\mathcal L}\) as the largest quotient of \(\mathcal L\) in which the Telescope Conjecture holds. Second, the author gives an explicit, combinatorial construction of another ordered semiring \(\mathcal A\) that injects into the quotient \(\overline{\mathcal L}\) and such that \(\pi(\langle E\rangle)\) is in the image of the injection for most known and studied classes \(\langle E\rangle\in\mathcal L\).
The author notes it is reasonable to conjecture that a number of new Bousfield classes introduced in [M. Mahowald et al., Contemp. Math. 271, 217–284 (2001; Zbl 0984.55009)] lie in the image of the injection \(\mathcal A\hookrightarrow\overline{\mathcal L}\), thereby opening up a potential avenue for further investigation.