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, assuming a real refinement of the Dundas--Goodwillie--McCarthy theorem, we compute the the $RO(\mathbb{Z}/2)$-graded homotopy groups of $\mathrm{KR}({\mathbb{F}}_p[x]/x^e;{\mathbb{Z}}_p)$, and we compute the equivariant slices of ${\Sigma }^2{\tau }_{\ge 1}\mathrm{KR}(\mathbb{Z}/p^n;{\mathbb{Z}}_p)$.

arXiv commentAccepted version, 41 pages, to appear in Proc. Lond. Math. Soc