Semistable Reduction Theorem for Overconvergent $F$-isocrystals over Laurent Series Fields

TL;DR

This paper proves a semistable reduction theorem for overconvergent $F$-isocrystals over Laurent series fields and demonstrates the finite dimensionality of their rigid cohomology with compact support.

Contribution

It establishes the semistable reduction theorem for overconvergent $F$-isocrystals over Laurent series fields, extending previous work and applications.

Findings

Proved the semistable reduction theorem for overconvergent $F$-isocrystals.

Established finite dimensionality of $ E^{ ext{dag}}_K$-valued rigid cohomology with compact support.

Extended the theory to $k((t))$-varieties introduced by Lazda and Pál.

Abstract

We prove the semistable reduction theorem for $\mathcal{E}^{\dag}_K$-valued and $K$-valued overconvergent $F$-isocrystals over $k((t))$-varieties which were introduced by Lazda and P\'{a}l. As an application, we prove the finite dimensionality of $\mathcal{E}^{\dag}_K$-valued rigid cohomology with compact support.