Cyclotomic synthetic spectra

TL;DR

This paper introduces a new $$-category of cyclotomic synthetic spectra and shows that the motivic filtration on topological Hochschild homology naturally fits into this framework, leading to new bounds on syntomic cohomology.

Contribution

It defines the $$-category $ ext{CycSyn}$ of cyclotomic synthetic spectra and demonstrates its application to the motivic filtration on $ ext{THH}$, connecting it to syntomic cohomology bounds.

Findings

Established the $$-category $ ext{CycSyn}$ of cyclotomic synthetic spectra.

Proved the motivic filtration on $ ext{THH}$ admits a cyclotomic synthetic spectrum structure.

Derived new bounds on the syntomic cohomology of certain $$-ring spectra.

Abstract

We define an $\infty$-category $\mathrm{CycSyn}$ of $p$-typical cyclotomic synthetic spectra and prove that the motivic filtration on $\mathrm{THH}(R;\mathbf{Z}_p)$, defined by Bhatt, Morrow, and Scholze when $R$ is quasisyntomic and by Hahn, Raksit, and Wilson in the chromatically quasisyntomic case, naturally admits the structure of a $p$-typical cyclotomic synthetic spectrum. As a consequence, we obtain new bounds on the syntomic cohomology of connective chromatically quasisyntomic $\mathbf{E}_\infty$-ring spectra.