A motivic filtration on the topological cyclic homology of commutative ring spectra

For a prime number $p$ and a $p$-quasisyntomic commutative ring $R$, Bhatt–Morrow–Scholze defined motivic filtrations on the $p$-completions of $\mathrm{T}\mathrm{H}\mathrm{H}(R),{\mathrm{T}\mathrm{C}}^{-}(R),\mathrm{T}\mathrm{P}(R),$ and $\mathrm{T}\mathrm{C}(R)$, with the associated graded objects for $\mathrm{T}\mathrm{P}(R)$ and $\mathrm{T}\mathrm{C}(R)$ recovering the prismatic and syntomic cohomology of $R$, respectively. We give an alternate construction of these filtrations that applies also when $R$ is a well-behaved commutative ring spectrum; for example, we can take $R$ to be $\mathbb{S}$, $\mathrm{M}\mathrm{U}$, $\mathrm{k}\mathrm{u}$, $\mathrm{k}\mathrm{o}$, or $\mathrm{t}\mathrm{m}\mathrm{f}$. We compute the mod $(p,v_1)$ syntomic cohomology of the Adams summand $\mathrm{\ell }$ and observe that, when $p\ge 3$, the motivic spectral sequence for $V(1{)}_{\ast }\mathrm{T}\mathrm{C}(\mathrm{\ell })$ collapses at the ${\mathrm{E}}_2$-page.