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arXiv · 2412.00914

TR and the $r$-Nygaard filtered prismatic cohomology

Faidon Andriopoulos
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Abstract

Given an animated ring $S$, we define a filtration on its absolute prismatic cohomology $\mathcal{N}_r^{\geq i} \mathbbΔ_S$, 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 $ξ_r$-adic filtration on $\mathbb{A}_{\mathrm{inf}}$, in the case that $S$ is a perfectoid ring. Using this, we study the motivic filtration of topological restriction homology $\mathrm{TR}^r (S;\mathbb{Z}_p)$ 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.

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Faidon Andriopoulos. 2024-12-03. TR and the $r$-Nygaard filtered prismatic cohomology. https://arxiv.org/abs/2412.00914

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