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Semistable reduction for overconvergent $F$-isocrystals I: Unipotence and logarithmic extensions

Published online by Cambridge University Press:  20 September 2007

Kiran S. Kedlaya
Kiran S. Kedlaya
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA kedlaya@mit.edu
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Abstract

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Let $X$ be a smooth variety over a field $k$ of characteristic $p>0$, and let $\mathcal{E}$ be an overconvergent isocrystal on $X$. We establish a criterion for the existence of a ‘canonical logarithmic extension’ of $\mathcal{E}$ to a smooth compactification $\overline{X}$ of $X$ whose complement is a strict normal crossings divisor. We also obtain some related results, including a form of Zariski–Nagata purity for isocrystals.

Keywords

rigid $p$-adic cohomologyoverconvergent isocrystalslogarithmic extensionsZariski–Nagata purity14F30 (primary)14F40 (secondary)
Type
Research Article
Information
Compositio Mathematica , Volume 143 , Issue 5 , September 2007 , pp. 1164 - 1212
Copyright
Foundation Compositio Mathematica 2007