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

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 for $\mathrm{TP}(R)$ and $\mathrm{TC}(R)$ 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 $\mathbb{S}$, $\mathrm{MU}$, $\mathrm{ku}$, $\mathrm{ko}$, or $\mathrm{tmf}$. We compute the mod $(p,v_1)$ syntomic cohomology of the Adams summand $\ell$ and observe that, when $p\ge 3$, the motivic spectral sequence for $V(1{)}_{\ast }\mathrm{TC}(\ell )$ collapses at the ${\mathrm{E}}_2$-page.

to appear in Annals of Mathematics

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• Real topological Hochschild homology of schemes