TR and the $r$-Nygaard filtered prismatic cohomology
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909414869434368 |
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| author | Andriopoulos, Faidon |
| author_facet | Andriopoulos, Faidon |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_00914 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | TR and the $r$-Nygaard filtered prismatic cohomology Andriopoulos, Faidon Algebraic Geometry K-Theory and Homology Number Theory 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. |
| title | TR and the $r$-Nygaard filtered prismatic cohomology |
| topic | Algebraic Geometry K-Theory and Homology Number Theory |
| url | https://arxiv.org/abs/2412.00914 |