Whitehead Filtrations for Computations in Topological Hochschild Homology

Abstract

We discuss spectral sequences coming from Whitehead filtrations in the computation of topological Hochschild homology of ring spectra. Using cyclic invariance, this makes for simple computations of $THH$ of connective rings $R$ with coefficients in discrete ring spectra. In particular, we show how to use this to compute $THH(tmf,\mathbb{F}_2)$, and $THH(tmf,\mathbb{Z}_{(2)})$, where $tmf$ denotes the $\mathbb{E}_\infty$ ring spectrum of topological modular forms. Then, we obtain a description of $THH(\ell/v_1^n)$ in terms of $THH(\ell,\ell/v_1^n)$, where the latter can be computed by results of arXiv:0710.4368. We next explain how the methods of this computation generalize to give us information about $THH(cofib(x^k:\Sigma^{k|x|}R\to R))$ for $R$ and $cofib(x^k)$ suitably structured connective ring spectra, $k>1$, and $x\in \pi_{*}(R)$ an arbitrary element in positive degree. Finally, we examine the general framework to describe the topological Hochschild homology of 2-local connective self-conjugate K-theory, $ksc_2$.