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, $ksc_2$.
Categories: math.AT math.KT