[Paper Review] Slope filtrations in families

Ruochuan Liu|arXiv (Cornell University)|Sep 1, 2008

Rings, Modules, and Algebras5 references3 citations

TL;DR

This paper establishes the lower semicontinuity of slope polygons for families of $φ$-modules over reduced affinoid spaces in mixed characteristic $(0,p)$, and proves that when slope polygons are locally constant at a rigid point, a global slope filtration exists after base change to an extended Robba ring. The key contribution is a local-to-global filtration result under slope constancy, extending classical slope theory to arithmetic families of $p$-adic representations.

ABSTRACT

This paper concerns arithmetic families of $φ$-modules over reduced affinoid spaces. For such a family, we first prove that the slope polygons is lower semicontinuous around any rigid point. If the slope polygons are locally constant around a rigid point, we further prove that around this point, the family has a global slope filtration after base change to some extended Robba ring.

Motivation & Objective

• To generalize the classical slope filtration theorem for $φ$-modules to arithmetic families over reduced affinoid spaces.
• To understand the variation of Harder-Narasimhan (HN) polygons in families of $φ$-modules.
• To establish conditions under which a global slope filtration exists in such families, particularly when slope polygons are locally constant.
• To extend the theory to families of $p$-adic representations via $(φ,\Gamma)$-modules, with applications to relative $p$-adic Hodge theory.

Proposed method

• Uses the Robba ring and its extended version to define families of $φ$-modules over affinoid algebras over $\mathbb{Q}_p$.
• Applies the theory of Dieudonné-Manin decomposition and slope filtrations to $φ$-modules over the Robba ring and its extensions.
• Introduces Weierstrass subdomains to localize the study around rigid points and analyze slope polygon behavior.
• Employs the notion of $(c,d)$-pure models in the extended Robba ring to construct global filtrations when slope polygons are constant.
• Utilizes cohomological techniques, including the complex $C^\bullet_{\varphi,\gamma}(M)$, to study extensions and étale properties.
• Applies the theory of HN-polygons and their comparison across fibers to analyze semicontinuity and constancy.

Experimental results

Research questions

• RQ1 Under what conditions does the HN-polygon of a $φ$-module vary semicontinuously in a family over a reduced affinoid space?
• RQ2 When is the slope polygon locally constant around a rigid point in a family of $φ$-modules?
• RQ3 Can a global slope filtration be constructed for a family of $φ$-modules after base change when the slope polygon is locally constant?
• RQ4 What is the structure of the locus where the HN-polygon of a fiber matches that of a given fiber in a family?

Key findings

• The HN-polygon of a family of $φ$-modules is lower semicontinuous: for any rigid point $x$, there exists a Weierstrass subdomain $M(B)$ around $x$ such that the HN-polygon of $M_y$ lies above that of $M_x$ for all $y \in M(B)$.
• If the fiber $M_x$ at a rigid point $x$ is pure and $k(x) \subset A$, then $M_A$ is globally pure of the same slope around $x$ after base change to a suitable Weierstrass subdomain.
• When the slope polygon is locally constant at $x$, the set of points $y$ where $M_y$ has the same HN-polygon as $M_x$ forms a Zariski closed subset of a Weierstrass subdomain around $x$.
• On this Zariski closed subset, after base change to an extended Robba ring, the family admits a global slope filtration.
• The construction is compatible with the $(\varphi,\Gamma)$-module formalism, providing a framework for relative $p$-adic Hodge theory in arithmetic families.
• An explicit example shows that slope polygons are not always locally constant, but the semicontinuity result still holds, and the locus of constancy is Zariski closed.