Finite polynomial cohomology for general varieties
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Finite polynomial cohomology for general varieties Amnon Besser1 · David Loeffler2 · Sarah Livia Zerbes3 In honour of Glenn Stevens’ 60th birthday
Received: 16 December 2014 / Accepted: 20 January 2015 © The Author(s) 2016. This article is published with open access at Springerlink.com
Abstract Nekováˇr and Nizioł (Syntomic cohomology and p-adic regulators for varieties over p-adic fields, 2013) have introduced in a version of syntomic cohomology valid for arbitrary varieties over p-adic fields. This uses a mapping cone construction similar to the rigid syntomic cohomology of (Besser, Israel J Math 120(1):291–334, 2000) in the good-reduction case, but with Hyodo–Kato (log-crystalline) cohomology in place of rigid cohomology. In this short note, we describe a cohomology theory which is a modification of the theory of Nekováˇr–Nizioł, modified by replacing 1 − ϕ with other polynomials in ϕ. This is the analogue for bad-reduction varieties of the finite-polynomial cohomology of (Besser, Invent Math 142(2):397–434, 2000); and we use this cohomology theory to give formulae for p-adic regulator maps, extending the results of (Besser, Invent Math 142(2):397–434, 2000; Besser, Israel J Math 120(1):335–360, 2000; Besser, Israel J Math 190(1):29–66, 2012) to varieties over p-adic fields, without assuming any good reduction hypotheses. Keywords
Finite-polynomial cohomology · Syntomic cohomology · Regulators
The authors’ research was supported by the following Grants: Royal Society University Research Fellowship (Loeffler); EPSRC First Grant EP/J018716/1 (Zerbes).
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Sarah Livia Zerbes [email protected] Amnon Besser [email protected] David Loeffler [email protected]
1
Department of Mathematics, Ben-Gurion University of the Negev, P.O.B. 653, Be’er-Sheva 84105, Israel
2
Mathematics Institute, University of Warwick, Zeeman Building, Coventry CV4 7AL, UK
3
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK
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A. Besser et al.
Résumé Nekovar et Niziol (Syntomic cohomology and p-adic regulators for varieties over p-adic fields, 2013) ont introduit une version de la cohomologie syntomique pour les variétés definis sur un corps p-adique. Leur construction généralise la cohomologie rigide syntomique de (Besser, Israel JMath 120(1):291–334, 2000) dans le cas de bonne reduction, remplacant la cohomologie rigide par la cohomologie de Hyodo-Kato. Nouse décrivons une modification de la theorie de Nekovar-Niziol, remplacant l’operateur 1- par d’autres polynomes en . Ceci est l’analogue de la “finite-polynomial cohomology” de (Besser, Invent. Math 142(2):397 434, 2000) pour les variétés de mauvaise réduction. Nous utilisons cette cohomologie pour obtenir des formules pour les régulateurs p-adiques, généralisant les résultats de (Besser, Invent. Math 142(2):397–434, 2000; Besser, 11 Israel J Math 120(1):335–360, 2000; Besser, Israel J Math 190(1):29–66, 2012) dans le cas de bonne réduction. Mathematics Subject Classification 11G25 (Arithmetic geometry: Varieties over finite
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