Syntomic cohomology and real topological cyclic homology

Abstract

We define the motivic filtrations on real topological Hochschild homology and its companions. In particular, we prove that real topological cyclic homology admits a natural complete filtration whose graded pieces are equivariant suspensions of syntomic cohomology. As an application, we compute the equivariant slices of $p$-completed real $K$-theories of $\mathbb{F}_p[x]/x^e$ and $\mathbb{Z}/p^n$ after certain suspensions assuming a real refinement of the Dundas-McCarthy-Goodwillie theorem and an announced result of Antieau-Krause-Nikolaus.