Semistable reduction for overconvergent F-isocrystals, II: A valuation-theoretic approach

Abstract

We introduce a valuation-theoretic approach to the problem of semistable reduction (i.e. existence of logarithmic extensions on suitable covers) of overconvergent isocrystals with Frobenius structure. The key tool is the quasicompactness of the Riemann-Zariski space associated to the function field of a variety. We also make some initial reductions, which allow attention to be focused on valuations of height 1 and transcendence degree 0. © 2008 Foundation Compositio Mathematica.