Algebra & Number Theory

Syntomic cohomology and $p$-adic regulators for varieties over $p$-adic fields

Jan Nekovář and Wiesława Nizioł

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We show that the logarithmic version of the syntomic cohomology of Fontaine and Messing for semistable varieties over p-adic rings extends uniquely to a cohomology theory for varieties over p-adic fields that satisfies h-descent. This new cohomology — syntomic cohomology — is a Bloch–Ogus cohomology theory, admits a period map to étale cohomology, and has a syntomic descent spectral sequence (from an algebraic closure of the given field to the field itself) that is compatible with the Hochschild–Serre spectral sequence on the étale side and is related to the Bloch–Kato exponential map. In relative dimension zero we recover the potentially semistable Selmer groups and, as an application, we prove that Soulé’s étale regulators land in the potentially semistable Selmer groups.

Our construction of syntomic cohomology is based on new ideas and techniques developed by Beilinson and Bhatt in their recent work on p-adic comparison theorems.

Article information

Algebra Number Theory, Volume 10, Number 8 (2016), 1695-1790.

Received: 26 February 2016
Accepted: 5 July 2016
First available in Project Euclid: 16 November 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14F30: $p$-adic cohomology, crystalline cohomology
Secondary: 11G25: Varieties over finite and local fields [See also 14G15, 14G20]

syntomic cohomology regulators


Nekovář, Jan; Nizioł, Wiesława. Syntomic cohomology and $p$-adic regulators for varieties over $p$-adic fields. Algebra Number Theory 10 (2016), no. 8, 1695--1790. doi:10.2140/ant.2016.10.1695.

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