TR and the $r$-Nygaard filtered prismatic cohomology

Given an animated ring $S$, we define a filtration on its absolute prismatic cohomology ${\mathcal{N}}_r^{\ge i}{\mathbb{\Delta }}_S$, which we call the $r$-Nygaard filtration and study some of its main properties using a mixture of algebraic and homotopy theoretic techniques. This filtration is obtained by suitably gluing $r$-copies of the usual Nygaard filtration and corresponds to the ${\xi }_r$-adic filtration on ${\mathbb{A}}_{\mathrm{i}\mathrm{n}\mathrm{f}}$, in the case that $S$ is a perfectoid ring. Using this, we study the motivic filtration of topological restriction homology ${\mathrm{T}\mathrm{R}}^r(S;{\mathbb{Z}}_p)$ and of its $S^1$-homotopy fixed points. We also pursue connections with the theory of topological cyclic homology. Finally we discuss connections with the de Rham–Witt complex, towards a prismatic - de Rham–Witt comparison theorem.