Abstract
For a prime number $p$ and a $p$-quasisyntomic commutative ring $R$, Bhatt–Morrow–Scholze defined motivic filtrations on the $p$-completions of $\mathrm{THH}(R)$, $\mathrm{TC}^-(R)$,$\mathrm{TP}(R)$, and $\mathrm{TC}(R)$, with the associated graded objects or $\mathrm{TP}(R)$ and $\mathrm{TC}(R)$ recovering the prismatic and syntomic cohomology of $R$, respectively. We give an alternate construction of thesee filtrations that applies also when $R$ is a well-behaved commutative ring spectrum; for example, we can take $R$ to be $\mathbb{S}$, $\mathrm{MU}$, $\mathrm{ku}$, $\mathrm{ko}$, or $\mathrm{tmf}$. We compute the $\mathrm{mod}(p,v_1)$ syntomic cohomology of the Adams summand $\ell$ and observe that, when $p\ge 3$, the motivic spectral sequenc for $V(1)_\ast \mathrm{TC}(\ell)$ collapses at the $\mathrm{E}_2$-page.
