×

zbMATH — the first resource for mathematics

Syntomic regulators and \(p\)-adic integration. I: Rigid syntomic regulators. (English) Zbl 1001.19003
The author constructs a new version of syntomic cohomology defined for smooth schemes \(X\) of finite type over the ring of integers \(\mathcal V\) of a \(p\)-adic field extending previous definitions of M. Gros [Invent. Math. 115, No. 1, 61-79 (1994; Zbl 0799.14010)] and W. Nizioł [Invent. Math. 127, No. 2, 375-400 (1997; Zbl 0928.14014)]. The definition of the syntomic cohomology groups \(H^i_{\text{syn}}(X,n)\) is more refined than the previous ones – it allows, e.g., log-singularities –, and the cohomology groups turn out to be finite. In addition the author constructs syntomic Chern classes \[ c_j^p:K_p(X) \rightarrow H^{2j-p}_{\text{syn}}(X,j) \] and Chern characters based on the general framework provided in A. Huber’s “Mixed motives and their realization in derived categories” [Lect. Notes Math. 1604 (1995; Zbl 0938.14008)]. See also the second part of thie paper [A. Besser, Isr. J. Math. 120, Pt. B, 335-359 (2000; Zbl 1001.19004)].

MSC:
19E20 Relations of \(K\)-theory with cohomology theories
14F30 \(p\)-adic cohomology, crystalline cohomology
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] A. Besser and R. de Jeu,The syntomic regulator for K-theory of fields, in preparation, 1999. · Zbl 1106.11024
[2] Beilinson, A. A., Higher regulators and values of L-functions, Journal of Soviet Mathematics, 30, 2036-2070, (1985) · Zbl 0588.14013
[3] Beilinson, A. A., Notes on absolute Hodge cohomology, 35-68, (1986), Providence, RI · Zbl 0621.14011
[4] P. Berthelot,Cohomologie rigide et cohomologie rigide a support propres, premier partie, Preprint 96-03 of the University of Rennes, available at http://www.maths.univ-rennes1.fr/ berthelo/, 1996.
[5] P. Berthelot,Finitude et pureté cohomologique en cohomologie rigide, Inventiones Mathematicae128 (1997), 329-377, with an appendix in English by A. J. de Jong. · Zbl 0908.14005
[6] A. Besser,A generalization of Coleman’s integration theory, Preprint available at http://www.math.bgu.ac.il/ bessera/padic/coleman.html, 1999. · Zbl 1053.14020
[7] A. Besser,Syntomic regulators and p-adic integration II: K_{2}of curves, this volume. · Zbl 1001.19004
[8] Coleman, R.; Shalit, E., P-adic regulators on curves and special values of p-adic L-functions, Inventiones Mathematicae, 93, 239-266, (1988) · Zbl 0655.14010
[9] Chiarellotto, B., Weights in rigid cohomology applications to unipotent F-isocrystals, Annales Scientifiques de l’École Normale Supérieure, 31, 683-715, (1998) · Zbl 0933.14008
[10] B. Chiarellotto and B. le Stum,Sur la pureté de la cohomologie cristalline, Comptes Rendus de l’Académie des Sciences, Paris, Série I326 (1998), 961-963. · Zbl 0936.14016
[11] Coleman, R., Torsion points on curves and p-adic abelian integrals, Annals of Mathematics, 121, 111-168, (1985) · Zbl 0578.14038
[12] P. Deligne,Théorie de Hodge. II, Publications Mathématiques de l’Institut des Hautes Études Scientifiques40 (1971), 5-57. · Zbl 0219.14007
[13] P. Deligne,Théorie de Hodge. III, Publications Mathématiques de l’Institut des Hautes Études Scientifiques44 (1974), 5-77. · Zbl 0237.14003
[14] Jeu, R., Zagier’s conjecture and wedge complexes in algebraic K-theory, Compositio Mathematica, 96, 197-247, (1995) · Zbl 0868.19002
[15] Fontaine, J. M.; Messing, W., P-adic periods and p-adic étale cohomology, No. 67, 179-207, (1987), Providence, RI · Zbl 0632.14016
[16] M. Gros,Régulateurs syntomiques et valeurs de fonctions L p-adiques. I, Inventiones Mathematicae99 (1990), 293-320, with an appendix by Masato Kurihara. · Zbl 0667.14006
[17] M. Gros,Régulateurs syntomiques et valeurs de fonctions L p-adiques. II, Inventiones Mathematicae115 (1994), 61-79. · Zbl 0799.14010
[18] A. Huber,Mixed motives and their realization in derived categories, Lecture Notes in Mathematics1604, Springer-Verlag, Berlin, 1995. · Zbl 0938.14008
[19] U. Jannsen,Continuous étale cohomology, Mathematische Annalen280 (1988), 207-245. · Zbl 0649.14011
[20] Kolster, M.; Nguyen Quang Do, T., Syntomic regulators and special values of p-adic l-functions, Inventiones Mathematicae, 133, 417-447, (1998) · Zbl 0910.11046
[21] Monsky, P.; Washnitzer, G., Formal cohomology. I, Annals of Mathematics, 88, 181-217, (1968) · Zbl 0162.52504
[22] J. Nekovář,Syntomic cohomology and p-adic regulators, 1998. Preprint available from http://can.dpmms.cam.ac.uk/ nekovar/papers/
[23] Nizioł, W., Cohomology of crystalline representations, Duke Mathematical Journal, 71, 747-791, (1993) · Zbl 0803.14008
[24] Nizioł, W., On the image of p-adic regulators, Inventiones Mathematicae, 127, 375-400, (1997) · Zbl 0928.14014
[25] W. Nizioł,Cohomology of crystalline smooth sheaves, Compositio Mathematica, to appear, 1999. · Zbl 1066.14022
[26] Ogus, A., The convergent topos in characteristic p, 133-162, (1990), Boston
[27] B. Perrin-Riou,Fonctions L p-adiques des représentations p-adiques, Astérisque229 (1995), 1-198. · Zbl 0656.10019
[28] Saint-Donat, B., Exp. v\^{}{bis}: techniques de descente cohomologique, No. 270, 83-162, (1972), Berlin
[29] Somekawa, M., Log-syntomic regulators and p-adic polylogarithms, K-Theory, 17, 265-294, (1999) · Zbl 0978.19004
[30] Put, M., The cohomology of Monsky and washnitzer, Mémoires de la Société Mathématique de France (N.S.), 23, 33-59, (1986) · Zbl 0606.14018
[31] Put, M.; Schneider, P., Points and topologies in rigid geometry, Mathematische Annalen, 302, 81-103, (1995) · Zbl 0867.11049
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.