https://hal-ens-lyon.archives-ouvertes.fr/ensl-01420358Colmez, PierrePierreColmezIMJ - Institut de Mathématiques de Jussieu - UPMC - Université Pierre et Marie Curie - Paris 6 - UPD7 - Université Paris Diderot - Paris 7 - CNRS - Centre National de la Recherche ScientifiqueNiziol, WieslawaWieslawaNiziolENS Lyon - École normale supérieure - LyonSYNTOMIC COMPLEXES AND p-ADIC NEARBY CYCLESHAL CCSD2017[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT][MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG]niziol, wieslawa2016-12-20 14:48:582023-03-24 14:53:032016-12-21 10:02:32enJournal articlesapplication/pdf1We compute syntomic cohomology of semistable affinoids in terms of cohomology of (ϕ, Γ)-modules which, thanks to work of Fontaine-Herr, Andreatta-Iovita, and Kedlaya-Liu, is known to compute Galois cohomology of these affinoids. For a semistable scheme over a mixed characteristic local ring this implies a comparison isomorphism, up to some universal constants, between truncated sheaves of p-adic nearby cycles and syntomic cohomology sheaves. This generalizes the comparison results of Kato, Kurihara, and Tsuji for small Tate twists (where no constants are necessary) as well as the comparison result of Tsuji that holds over the algebraic closure of the field. As an application, we combine this local comparison isomorphism with the theory of finite dimensional Banach Spaces and finiteness ofétale ofétale cohomology of rigid analytic spaces proved by Scholze to prove a Semistable conjecture for formal schemes with semistable reduction.