Topological cyclic homology of Cartier smooth rings

We study algebraic K-theory, syntomic cohomology, and prismatic cohomology of Cartier smooth rings. As an application, we provide an alternative proof of Kelly-Morrow’s generalization of the Geisser-Levine theorem computing $p$-adic algebraic K-theory of Cartier smooth local rings; our approach relies on the description of topological cyclic homology through the motivic filtration.