Syntomic complexes and p-adic nearby cycles

• 成果类型: Article
• 署名作者: Colmez, Pierre; Niziol, Wieslawa
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Universite Paris Cite; Sorbonne Universite; Ecole Normale Superieure de Lyon (ENS de LYON); Centre National de la Recherche Scientifique (CNRS)
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-016-0683-3
• 发表日期: 2017
• 页码: 1-108
• 关键词: semi-stable reduction; log smooth families; K-THEORY; etale cohomology; galois cohomology; residue field; REPRESENTATIONS; conjecture; Isomorphism; EXTENSIONS

摘要

We compute syntomic cohomology of semistable affinoids in terms of cohomology of (phi, Gamma)-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 ,tale cohomology of rigid analytic spaces proved by Scholze to prove a Semistable conjecture for formal schemes with semistable reduction.