On the Motivic Filtration of TR

Given a p-complete, animated ring S, we define a filtration on the Nygaard-complete, absolute prismatic cohomology, which we call the r-Nygaard filtration. This is obtained by suitably gluing r-copies of the usual Nygaard filtration and, in the case that S is a perfectoid ring, it corresponds to the ξr-adic filtration on Ainf.Using this, we study the motivic filtration of topological restriction homology and of its S^1-homotopy fixed points. We also explore connections with topological cyclic homology. Finally, we apply our results to the case of AΩ-cohomology, drawing comparisons to the first results of Bhatt–Morrow–Scholze, which also marked the beginning of the prismatic story.