Semistable reduction for overconvergent \(F\)-isocrystals. II: A valuation-theoretic approach. (English) Zbl 1153.14015

This article is the second in a series of four (“Semistable reduction for overconvergent \(F\)-isocrystals I [Compos. Math. 143, No. 5, 1164–1212 (2007; Zbl 1144.14012)], II, III [Preprint 2006, arXiv:math/0609645 and IV [Preprint 2007, arXiv:0712.3400]”) in which the author gives a proof of the “semistable reduction theorem” for overconvergent \(F\)-isocrystals (Shiho’s conjecture). The reader should consult the introduction of the first paper for a detailed discussion of the general strategy. In the first paper, the author reduced the semistable reduction theorem to some sort of local monodromy condition. In the present paper, he gives a valuation theoretic interpretation of this concept and uses some properties of Riemann-Zariski spaces to reduce the problem of semistable reduction to a question that is local around a particular valuation. The actual application of this work to the semistable reduction theorem is in the next two papers.


14F30 \(p\)-adic cohomology, crystalline cohomology
14F40 de Rham cohomology and algebraic geometry


Zbl 1144.14012
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