Log-decay F-isocrystals on higher dimensional varieties

2019/02/13 by Kramer-Miller, Joe
#14F30 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1902.04730

Abstract

Let k be a perfect field of positive characteristic and let X be a smooth irreducible quasi-compact scheme over k. The Drinfeld-Kedlaya theorem states that for an irreducible F-isocrystal on X, the gap between consecutive generic slopes is bounded by one. In this note we provide a new proof of this theorem. Our proof utilizes the theory of F-isocrystals with r-log decay. We first show that a rank one F-isocrystal with r-log decay is overconvergent if r<1. Next, we establish a connection between slope gaps and the rate of log-decay of the slope filtration. The Drinfeld-Kedlaya theorem then follows from a simple patching argument.

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