[Paper Review] $p$-adic vanishing cycles as Frobenius-fixed points

Matthew Morrow|arXiv (Cornell University)|Feb 9, 2018

Algebraic Geometry and Number Theory15 references5 citations

TL;DR

This paper establishes that $p$-adic vanishing cycles for smooth formal schemes over the ring of integers of a mixed-characteristic perfectoid field are isomorphic to the Frobenius-fixed points of a certain pro-system of de Rham–Witt complexes. By leveraging the $p$-adic Cartier isomorphism and the homotopy fiber of the Frobenius minus restriction map, the authors identify $p$-adic vanishing cycles as the kernel of $F - R$, providing a new interpretation of these cohomological invariants via Frobenius-fixed structures in integral $p$-adic Hodge theory.

ABSTRACT

Given a smooth formal scheme over the ring of integers of a mixed-characteristic perfectoid field, we study its $p$-adic vanishing cycles via de Rham--Witt and $q$-de Rham complexes.

Motivation & Objective

• To interpret $p$-adic vanishing cycles as Frobenius-fixed points of integral cohomology theories in $p$-adic Hodge theory.
• To extend the framework of de Rham–Witt and $q$-de Rham complexes to formal schemes over perfectoid rings.
• To provide a new perspective on $p$-adic vanishing cycles by relating them to the homotopy fiber of $F - R$ on pro-systems of $\widetilde{W_r\Omega}_{\mathfrak{X}}\{j\}$.
• To generalize Geisser–Hesselholt's result from discretely valued fields to algebraic closures via perfectoid geometry.
• To establish a structural link between $p$-adic vanishing cycles and the $q$-de Rham complex $\mathbb{A}\Omega_{\mathfrak{X}}\{j\}$ via Frobenius-fixed points.

Proposed method

• The authors use the pro-étale topology and construct pro-complexes $\tau^{\leq j}\widetilde{W_r\Omega}_{\mathfrak{X}}\{j\}$ of $W_r(\mathcal{O})$-modules on the étale site of the special fiber.
• They apply the $p$-adic Cartier isomorphism to relate cohomology of $\widetilde{W_r\Omega}_{\mathfrak{X}}\{j\}$ to relative de Rham–Witt complexes $W_r\Omega^{\bullet}_{\mathfrak{X}/\mathcal{O}}$.
• The key construction is the homotopy fiber of the map $F - R: \tau^{\leq j}\widetilde{W_r\Omega}_{\mathfrak{X}}\{j\} \to \tau^{\leq j}\widetilde{W_{r-1}\Omega}_{\mathfrak{X}}\{j\}$, which captures $p$-adic vanishing cycles.
• They prove that the fiber sequence involving $\tau^{\leq j}Rb_*\mathbb{Z}/p^N\mathbb{Z}(j)$ and the pro-system of $F - R$ maps is exact, identifying the vanishing cycles as the kernel.
• The proof relies on showing that the long exact sequence of $p$-adic cohomology splits into short exact sequences, using surjectivity of $F - R$ on de Rham–Witt sheaves.
• They use the $q$-de Rham complex $\mathbb{A}\Omega_{\mathfrak{X}}\{j\} = \varprojlim_{r, F} \widetilde{W_r\Omega}_{\mathfrak{X}}\{j\}$ to recover the known result from Bhatt–Morrow–Scholze on $p$-adic vanishing cycles as $\ker(1 - \varphi^{-1})$.

Experimental results

Research questions

• RQ1 How can $p$-adic vanishing cycles be interpreted as Frobenius-fixed points in the context of de Rham–Witt complexes over perfectoid rings?
• RQ2 What is the precise relationship between the pro-system of $\widetilde{W_r\Omega}_{\mathfrak{X}}\{j\}$ and the $p$-adic étale cohomology $Rb_*\mathbb{Z}/p^N\mathbb{Z}(j)$?
• RQ3 Does the map $F - R$ on the truncated de Rham–Witt complexes induce a fiber sequence that realizes $p$-adic vanishing cycles?
• RQ4 Can the surjectivity of $F - R$ on de Rham–Witt sheaves over $\mathbb{F}_p$-schemes be used to prove the splitting of the $p$-adic cohomology long exact sequence?
• RQ5 How does the $q$-de Rham complex $\mathbb{A}\Omega_{\mathfrak{X}}\{j\}$ encode $p$-adic vanishing cycles via Frobenius-fixed points?

Key findings

• The fiber sequence $\tau^{\leq j}Rb_*\mathbb{Z}/p^N\mathbb{Z}(j) \to \varprojlim'_{r, R} \tau^{\leq j}\widetilde{W_r\Omega}_{\mathfrak{X}}\{j\} \xrightarrow{F-R} \varprojlim'_{r, R} \tau^{\leq j}\widetilde{W_{r-1}\Omega}_{\mathfrak{X}}\{j\}$ identifies $p$-adic vanishing cycles as the kernel of $F - R$ on the pro-system.
• The $p$-adic Cartier isomorphism ensures that $H^j(\widetilde{W_r\Omega}_{\mathfrak{X}}\{j\}) \cong W_r\Omega^j_{\mathfrak{X}/\mathcal{O}}$, linking the cohomology of the complexes to de Rham–Witt forms.
• The long exact sequence of $p$-adic cohomology splits into short exact sequences due to the surjectivity of $F - R$, which is proven via reduction to the $\mathbb{F}_p$-case and the smooth case.
• The map $F - R: W_r\Omega^j_{Y/\mathbb{F}_p} \to W_{r-1}\Omega^j_{Y/\mathbb{F}_p}$ is surjective for affine, finite-type $\mathbb{F}_p$-schemes $Y$, ensuring no $\varprojlim^1$ obstruction.
• The pro-system $\mathbb{A}\Omega_{\mathfrak{X}}\{j\} = \varprojlim_{r, F} \widetilde{W_r\Omega}_{\mathfrak{X}}\{j\}$ satisfies $\tau^{\leq j}(\mathbb{A}\Omega_{\mathfrak{X}}\{j\}/p^N) \xrightarrow{1 - \varphi^{-1}} \tau^{\leq j}(\mathbb{A}\Omega_{\mathfrak{X}}\{j\}/p^N)$ with kernel $\tau^{\leq j}R\nu_*\mathbb{Z}/p^N\mathbb{Z}(j)$, recovering the known result from Bhatt–Morrow–Scholze.
• The Frobenius-fixed points of $\mathbb{A}\Omega_{\mathfrak{X}}\{j\}/p^N$ are isomorphic to the $p$-adic vanishing cycles $\tau^{\leq j}Rb_*\mathbb{Z}/p^N\mathbb{Z}(j)$, providing a new cohomological interpretation.