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$p$-Adic Weight Spectral Sequences of Strictly Semi-stable Schemes over Formal Power Series Rings via Arithmetic $\mathcal{D}$-modules

Yuanmin Liu

TL;DR

This work constructs a $p$-adic weight spectral sequence for strictly semi-stable schemes over $k[[t]]$ using the framework of arithmetic $D$-modules. The $E_1$-terms are described by rigid cohomology of the closed-fiber strata and their intersections, while the $E_ abla$-terms are conjecturally controlled by the unipotent nearby cycle of Lazda–Pál’s rigid cohomology over the bounded Robba ring $\mathcal{E}^\dag_K$. The paper establishes six-functor formalism, purity, and the nearby cycle $\Psi$, and proves functoriality by pushforward, with conjectural pullback and dual statements. Under a comparison conjecture, the $E_1$-terms align with Lazda–Pál rigid cohomology, yielding a pathway to relate arithmetic $D$-module cohomology to rigid cohomology in the $p$-adic setting. The results provide a robust framework for understanding $p$-adic weight filtrations in the strictly semi-stable context and set the stage for further development of functoriality and comparison theorems.

Abstract

Let $k$ be a perfect field of characteristic $p > 0$. For a strictly semi-stable scheme over $k[[t]]$, we construct the weight spectral sequence in $p$-adic cohomology using the theory of arithmetic $\mathcal{D}$-modules, whose $E_1$ terms are described by rigid cohomologies of irreducible components of the closed fiber and whose $E_\infty$ terms are conjecturally described by the (unipotent) nearby cycle of Lazda-Pál's rigid cohomology over the bounded Robba ring. We also show its functoriality by pushforward and state the conjecture of its functoriality by pullback and dual.

$p$-Adic Weight Spectral Sequences of Strictly Semi-stable Schemes over Formal Power Series Rings via Arithmetic $\mathcal{D}$-modules

TL;DR

This work constructs a -adic weight spectral sequence for strictly semi-stable schemes over using the framework of arithmetic -modules. The -terms are described by rigid cohomology of the closed-fiber strata and their intersections, while the -terms are conjecturally controlled by the unipotent nearby cycle of Lazda–Pál’s rigid cohomology over the bounded Robba ring . The paper establishes six-functor formalism, purity, and the nearby cycle , and proves functoriality by pushforward, with conjectural pullback and dual statements. Under a comparison conjecture, the -terms align with Lazda–Pál rigid cohomology, yielding a pathway to relate arithmetic -module cohomology to rigid cohomology in the -adic setting. The results provide a robust framework for understanding -adic weight filtrations in the strictly semi-stable context and set the stage for further development of functoriality and comparison theorems.

Abstract

Let be a perfect field of characteristic . For a strictly semi-stable scheme over , we construct the weight spectral sequence in -adic cohomology using the theory of arithmetic -modules, whose terms are described by rigid cohomologies of irreducible components of the closed fiber and whose terms are conjecturally described by the (unipotent) nearby cycle of Lazda-Pál's rigid cohomology over the bounded Robba ring. We also show its functoriality by pushforward and state the conjecture of its functoriality by pullback and dual.

Paper Structure

This paper contains 21 sections, 48 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $p$-basis
  4. Strictly semi-stable scheme over $k[[t]]$
  5. Six functor formalism and t-structures
  6. Six functor formalism of frames and couples
  7. Constructible t-structure by devissage
  8. Another definition of constructible t-structure
  9. Dual constructible t-structure
  10. Purity
  11. Calculation of $\mathbb{R}\underline{\Gamma}^\dag_Y(\mathcal{O}_{\mathfrak{P}, \mathbb{Q}})$
  12. Purity theorem
  13. Nearby cycle
  14. Construction of $\Psi$
  15. Filtrations on $\Psi$
  16. ...and 6 more sections

Key Result

Proposition 2.1.2

Let $f : X \rightarrow Y$ be a morphism of $k$-schemes. Consider following properties: (i) $f$ is etale; (ii) $f$ is relatively perfect; (iii) $f$ is formally etale. Then we have $(i) \Rightarrow (ii) \Rightarrow (iii).$

Theorems & Definitions (102)

  • Definition 2.1.1
  • Proposition 2.1.2: rel-perf
  • Definition 2.1.3
  • Definition 2.1.4: intro
  • Definition 3.1.1
  • Definition 3.1.2: cf. intro
  • Theorem 3.1.3: intro
  • Remark 3.1.4
  • Definition 3.1.5
  • Definition 3.1.6
  • ...and 92 more