Towards topological Hochschild homology of Johnson-Wilson spectra
In order to apply trace methods to the study of algebraic \(K\)-theory, it is useful to be able to compute the topological Hochschild homology of structured ring spectra. The paper under review is concerned with such computations for the Johnson-Wilson spectra \(E(n)\) for \(n \geq 1\) at a fixed odd prime \(p\). When \(n = 1\), the \(p\)-completion of \(E(1)\) is the \(p\)-completed periodic Adams summand. In this case, \textit{J. E. McClure} and \textit{R. E. Staffeldt} [Am. J. Math. 115, No. 1, 1--45 (1993; Zbl 0770.55010)] have shown that \(\mathrm{THH}(E(1)_p)\) decomposes as a sum of \(E(1)_p\) and the rational spectrum \(\Sigma^{2p-1}E(1)_{\mathbb Q}\). One of the main results of the present paper generalizes this decomposition: the authors show that if \(E(2)\) admits an \(E_3\)-structure and the unit map \(E(2) \to \mathrm{THH}(E(2))\) splits, then \(\mathrm{THH}(E(2))\) decomposes as an \(E(2)\)-module into a sum of \(4\) terms given by \(E(2)\) itself and shifts of lower chromatic localizations of \(E(2)\). The authors formulate a conjecture about a generalization of this result and the McClure-Staffeldt computation to \(\mathrm{THH}(E(n))\) for higher \(n\). Another main result of the paper under review is a computation of the localization of \(\mathrm{THH}(E(n))\) at the \(i\)-th Morava \(K\)-theory spectrum \(K(i)\). More precisely, the authors show that when \(E(n)\) is an \(E_3\)-ring spectrum and \(1\leq i \leq n\), then there is an isomorphism of \(K(i)_*E(n)\)-algebras identifying \(K(i)_*\mathrm{THH}(E(n))\) with the tensor product over \(\mathbb F_p\) of \(K(i)_*E(n)\) and an exterior power algebra on \(n-i\) generators.
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