$I_n$-local Johnson-Wilson spectra and their Hopf algebroids

Abstract

We consider a generalization $\mathcal{E}(n)$ of the Johnson-Wilson spectrum $E(n)$ for which $\mathcal{E}(n{)}_{\ast }$ is a local ring with maximal ideal $I_n$. We prove that the spectra $E(n),\mathcal{E}(n)$ and $\widehat{E(n)}$ are Bousfield equivalent. We also show that the Hopf algebroid $\mathcal{E}(n{)}_{\ast }\mathcal{E}(n)$ is a free $\mathcal{E}(n{)}_{\ast }$-module, generalizing a result of Adams and Clarke for $K{U}_{\ast }KU$.