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Crystalline fundamental groups. II: Log convergent cohomology and rigid cohomology. (English) Zbl 1057.14025
From the text: This is a continuation of a previous paper of the author [J. Math. Sci. Univ. Tokyo 7, 509–656 (2000; Zbl 0984.14009)]. There, we gave a definition of crystalline fundamental groups for certain fine log schemes over a perfect field of positive characteristic and proved some fundamental properties of them. Here, we investigate the log convergent cohomology in detail. In particular, we prove the log convergent Poincaré lemma and the comparison theorem between log convergent cohomology and rigid cohomology in the case that the coefficient is an $$F^a$$-isocrystal. We also give applications to finiteness of rigid cohomology with coefficient, Berthelot-Ogus theorem for crystalline fundamental groups [A. Shiho, loc. cit.] and independence of compactification for crystalline fundamental groups.

##### MSC:
 14F30 $$p$$-adic cohomology, crystalline cohomology 14F35 Homotopy theory and fundamental groups in algebraic geometry 14G22 Rigid analytic geometry 14G20 Local ground fields in algebraic geometry 14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
##### Keywords:
rigid analytic geometry; isocrystals
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