[Paper Review] Rigid analytic reconstruction of Hyodo--Kato theory

Veronika Ertl, Kazuki Yamada|arXiv (Cornell University)|Jul 25, 2019

Algebraic Geometry and Number Theory14 references4 citations

TL;DR

This paper presents a new rigid analytic construction of Hyodo–Kato cohomology and the Hyodo–Kato map using logarithmic rigid cohomology, achieving independence from the choice of uniformiser and dependence only on the branch of the $p$-adic logarithm. The construction is explicit, computable via Čech cocycles, and compatible with both classical crystalline Hyodo–Kato theory and Gro{̈}sse-Kl{"o}nne's rigid approach.

ABSTRACT

We give a new and very intuitive construction of Hyodo--Kato cohomology and the Hyodo--Kato map, based on logarithmic rigid cohomology. We show that it is independent of the choice of a uniformiser and study its dependence on the choice of a branch of the $p$-adic logarithm. Moreover, we show the compatibility with the classical construction of Hyodo--Kato cohomology and the Hyodo--Kato map.)

Motivation & Objective

• To develop a computationally explicit and geometrically intuitive rigid analytic version of Hyodo–Kato cohomology.
• To eliminate dependence on the choice of uniformiser in the Hyodo–Kato map, replacing it with dependence on the branch of the $p$-adic logarithm.
• To establish compatibility with both classical crystalline Hyodo–Kato theory and Gro{̈}sse-Kl{"o}nne's rigid analytic construction.
• To provide a framework amenable to generalization to cohomology with coefficients in log overconvergent $F$-isocrystals.

Proposed method

• The authors use a complex $\omega^{\bullet}_{\mathcal{Z}/W^{\varnothing},\mathbb{Q}}[u]$ on a dagger space over the open unit disk, inspired by Kim and Hain, to define rigid Hyodo–Kato cohomology.
• They define the rigid Hyodo–Kato map $\Psi_{\pi,q}$ using a natural morphism from the rigid cohomology of a dagger space to de Rham cohomology, enabling explicit computation via Čech cocycles.
• The construction relies on logarithmic rigid cohomology and weak formal schemes with pseudo-wcfg algebras, allowing a relative notion of weak completeness.
• The map $\Psi_{\pi,q}$ is shown to be independent of the uniformiser $\pi$, depending only on the branch of the $p$-adic logarithm encoded by $q \in \mathfrak{m}_V \setminus \{0\}$.
• A commutative diagram involving rigid cohomology of simplicial log schemes and their special fibers is used to compare the new map with classical and Gro{̈}sse-Kl{"o}nne constructions.
• Frobenius compatibility and functoriality in the axiomatization of rigid complexes are used to prove that the new map coincides with the classical section $s_\pi$, ensuring compatibility.

Experimental results

Research questions

• RQ1 How can Hyodo–Kato cohomology be reconstructed in a rigid analytic setting that is both computationally explicit and independent of the choice of uniformiser?
• RQ2 What is the precise dependence of the Hyodo–Kato map on the choice of a branch of the $p$-adic logarithm?
• RQ3 Is the new rigid analytic construction compatible with the classical crystalline Hyodo–Kato theory and with Gro{̈}sse-Kl{"o}nne's rigid analytic approach?
• RQ4 Can the new construction be extended to cohomology with coefficients in log overconvergent $F$-isocrystals?

Key findings

• The rigid Hyodo–Kato cohomology $R\Gamma_{\mathrm{HK}}^{\mathrm{rig}}(\mathcal{X})$ is constructed via a complex on a dagger space, enabling explicit computation using Čech cocycles.
• The rigid Hyodo–Kato map $\Psi_{\pi,q}$ is independent of the uniformiser $\pi$ and depends only on the branch of the $p$-adic logarithm defined by $q$, a non-zero element in the maximal ideal of $V$.
• The construction of $\Psi_{\pi,q}$ is compatible with the classical crystalline Hyodo–Kato map $\Psi_{\pi}^{\mathrm{cris}}$, as shown by a commutative diagram involving quasi-isomorphisms and Frobenius compatibility.
• The map $\Psi_{\pi,q}$ for $q = \pi$ coincides with Gro{̈}sse-Kl{"o}nne's rigid Hyodo–Kato map, establishing consistency with prior rigid analytic approaches.
• The new construction is compatible with the Frobenius action, and the composition of maps in the diagram commutes with Frobenius, ensuring consistency with the $\varphi$-structure in $(\varphi,N)$-modules.
• The authors prove that the composition of maps in the diagram is a section of the specialization map, and by Frobenius compatibility, it coincides with the classical section $s_\pi$, thus proving full compatibility.