Compositio Mathematica

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

Kiran S. Kedlaya a1
a1 Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA

Article author query
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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.

(Received January 20 2005)
(Accepted January 11 2007)

Key Words: rigid $p$-adic cohomology; overconvergent isocrystals; logarithmic extensions; Zariski–Nagata purity.

Maths Classification

14F30 (primary); 14F40 (secondary).

Dedicated to Pierre Berthelot