On the dual of a $P$-algebra and its comodules, with applications to comparison of some Bousfield classes

TL;DR

This paper explores the relationships between Bousfield classes of spectra, especially focusing on the spectra MSp and MU, using $P$-algebra theory to derive new cohomological vanishing results.

Contribution

It introduces a novel approach using $P$-algebras and comodules to analyze Bousfield classes, extending Ravenel's work with new algebraic techniques.

Findings

Established connections between Bousfield classes of MSp and MU spectra.

Derived vanishing results for cohomology using $P$-algebra duality.

Applied methods to the mod 2 Steenrod algebra and subHopf algebras.

Abstract

In his seminal work on localisation of spectra, Ravenel initiated the study of Bousfield classes of spectra related to the chromatic perspective. In particular he showed that there were infinitely many distinct Bousfield classes between $\langle MU\rangle$ and $\langle S^0\rangle$. The main topological goal of this paper is investigate how these Bousfield classes are related to that of another classical Thom spectrum $MSp$, and in particular how $\langle MSp\rangle$ is related to $\langle MU\rangle$. We follow the approach of Ravenel, but adapt it using the theory of $P$-algebras to give vanishing results for cohomology. Our work involves dualising and considering comodules over duals of $P$-algebras; these ideas are then applied to the mod~$2$ Steenrod algebra and certain subHopf algebras.