Syntomic complexes and p-adic nearby cycles

We 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...

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Bibliographic Details
Published in:Inventiones mathematicae Vol. 208; no. 1; pp. 1 - 108
Main Authors: Colmez, Pierre, Nizioł, Wiesława
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.04.2017
Springer Verlag
Subjects:
Algebraic Geometry
Constants
Homology
Isomorphism
Mathematical analysis
Mathematics
Mathematics and Statistics
Number Theory
Rings (mathematics)
Sheaves
Texts
Topology
ISSN:0020-9910, 1432-1297
Online Access:Get full text
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Summary:We 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 of étale cohomology of rigid analytic spaces proved by Scholze to prove a Semistable conjecture for formal schemes with semistable reduction.
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ISSN:0020-9910
1432-1297
DOI:10.1007/s00222-016-0683-3