Kedlaya, Kiran S. Semistable reduction for overconvergent \(F\)-isocrystals. II: A valuation-theoretic approach. (English) Zbl 1153.14015 Compos. Math. 144, No. 3, 657-672 (2008). 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. Reviewer: Laurent Berger (Lyon) Cited in 3 ReviewsCited in 21 Documents MSC: 14F30 \(p\)-adic cohomology, crystalline cohomology 14F40 de Rham cohomology and algebraic geometry Keywords:rigid cohomology; overconvergent isocrystals; semistable reduction; valuations; Riemann-Zariski spaces Citations:Zbl 1144.14012 PDFBibTeX XMLCite \textit{K. S. Kedlaya}, Compos. Math. 144, No. 3, 657--672 (2008; Zbl 1153.14015) Full Text: DOI arXiv