#### Volume 20, issue 1 (2020)

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Towards topological Hochschild homology of Johnson–Wilson spectra

### Christian Ausoni and Birgit Richter

Algebraic & Geometric Topology 20 (2020) 375–393
##### Abstract

We present computations in Hochschild homology that lead to results on the $K\left(i\right)$–local behaviour of $THH\left(E\left(n\right)\right)$ for all $n\ge 2$ and $0\le i\le n$, where $E\left(n\right)$ is the Johnson–Wilson spectrum at an odd prime. This permits a computation of $K{\left(i\right)}_{\ast }THH\left(E\left(n\right)\right)$ under the assumption that $E\left(n\right)$ is an ${E}_{3}$–ring spectrum. We offer a complete description of $THH\left(E\left(2\right)\right)$ as an $E\left(2\right)$–module in the form of a splitting into chromatic localizations of $E\left(2\right)$, under the assumption that $E\left(2\right)$ carries an ${E}_{\infty }$–structure. If $E\left(2\right)$ is admits an ${E}_{3}$–structure, we obtain a similar splitting of the cofiber of the unit map $E\left(2\right)\to THH\left(E\left(2\right)\right)$.

##### Keywords
topological Hochschild homology, Johnson–Wilson spectra, $E_\infty$–structures on ring spectra, chromatic squares
##### Mathematical Subject Classification 2010
Primary: 55N35, 55P43