Cycle classes and the syntomic regulator

Chiarellotto, Bruno; Ciccioni, Alice; Mazzari, Nicola

doi:10.2140/ant.2013.7.533

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Cycle classes and the syntomic regulator

Bruno Chiarellotto, Alice Ciccioni and Nicola Mazzari

Vol. 7 (2013), No. 3, 533–566

DOI: 10.2140/ant.2013.7.533

Abstract

Let $V = Spec (R)$ and $R$ be a complete discrete valuation ring of mixed characteristic $(0, p)$ . For any flat $R$ -scheme $X$ , we prove the compatibility of the de Rham fundamental class of the generic fiber and the rigid fundamental class of the special fiber. We use this result to construct a syntomic regulator map ${reg}_{syn} : {C H}^{i} (X ∕ V, 2 i - n) \to H_{syn}^{n} (X, i)$ when $X$ is smooth over $R$ with values in the syntomic cohomology defined by A. Besser. Motivated by the previous result, we also prove some of the Bloch–Ogus axioms for the syntomic cohomology theory but viewed as an absolute cohomology theory.

Keywords

syntomic cohomology, cycles, regulator map, rigid cohomology, de Rham cohomology

Mathematical Subject Classification 2010

Primary: 14F43

Secondary: 14F30, 19F27

Milestones

Received: 12 October 2010

Revised: 22 December 2011

Accepted: 3 May 2012

Published: 23 August 2013

Authors

Bruno Chiarellotto
Università degli Studi di Padova Via Trieste 63 35100 Padova Italy

Alice Ciccioni
Università degli Studi di Padova Via Trieste 63 35100 Padova Italy

Nicola Mazzari
IMB Université Bordeaux 1 351, cours de la Libération 33405 Talence cedex France

Vol. 7, No. 3, 2013