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In his seminal work on localisation of spectra, Ravenel initiated the studyof Bousfield classes of spectra related to the chromatic perspective. Inparticular he showed that there were infinitely many distinct Bousfield classesbetween $\langle MU\rangle$ and $\langle S^0\rangle$. The main topological goalof this paper is investigate how these Bousfield classes are related to that ofanother classical Thom spectrum $MSp$, and in particular how $\langleMSp\rangle$ 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 involvesdualising and considering comodules over duals of $P$-algebras; these ideas arethen applied to the mod~$2$ Steenrod algebra and certain subHopf algebras.

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