\right.$$ in nonnegative degrees after $p$-completion when $q$ is a prime power that topologically generates $\mathbb{Z}_p^{\times}$. Therefore, there are equivalences $K(\mathbb{F}_q)_p \simeq \tau_{\ge 0}L_{K(1)}S\simeq (\tau_{\ge 0}L_{E(1)}S)_p$, where $\tau_{\ge 0}L_{K(1)}S$ denotes the connective cover of the $K(1)$-local sphere.

We compute mod $(p,v_1)$ homotopy of $THH(\tau_{\ge 0}L_{K(1)}S)$ because this is the natural first step in approaching the chromatic red-shift conjecture of C. Ausoni and J. Rognes [@1403.4838]. Recall that there are homology theories $K(n)$ for each prime $p$ and nonnegative integer $n$ with coefficients $$K(n)_*\cong  \left \{ \begin{array}{ll} \mathbb{F}_p[v_n^{\pm 1}] & \text{ when } n>0  \\ \mathbb{Q} & \text{ when  } n=0. \end{array} \right.$$ We say that a $p$-local finite cell $S$-module $V$ has type $n$ if $K(i)_*V\cong 0$ for all $i<n$, but $K(n)_*V$ is nontrivial. Due to the thick subcategory theorem of E. Devinatz, M. Hopkins, J. Smith, the category of $p$-local finite cell $S$-modules $\mathcal{F}_{(p)}$ can be filtered into proper thick subcategories $$0 \subset \dots \cC_2 \subset \cC_1 \subset \cC_0 \subset \mathcal{F}_{(p)}$$ where $\cC_n$ contains exactly spectra of type $>n$; i.e., it consists of the $K(n)$-acyclic spectra [@MR960945]. The first examples of type $n$ spectra for small $n$ can be constructed by an iterative procedure by taking cofibers of $v_n$-self 