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An explicit basis for the rational higher Chow groups of abelian number fields. (English) Zbl 1388.14033

The paper under review provides a tour de force cycle-theoretic description of an explicit basis for the motivic cohomology of a point over a cyclotomic field (and by extension, of an abelian number field), paralleling A. A. Beilinson’s construction of such a basis (section 7 of his seminal work on higher regulators [J. Sov. Math. 30, 2036–2070 (1985; Zbl 0588.14013); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 24, 181–238 (1984)], albeit in terms of \(K\)-theory). Beilinson’s (and that of the authors) relies on a theorem of Borel, together with the coincidence of the Borel and Beilinson regulators (up to a factor of \(1/2\)), proven in detail by J. I. Burgos Gil [The regulators of Beilinson and Borel. Providence, RI: American Mathematical Society (AMS) (2002; Zbl 0994.19003)], and which was modelled after Beilinson’s original sketch of his ‘proof’. The ‘calculus’ of expressing the Beilinson regulator in terms of polylogarithmic currents, was established by Kerr, Lewis and Mueller-Stach [M. Kerr et al., Compos. Math. 142, No. 2, 374–396 (2006; Zbl 1123.14006)], but there is a caveat in order here. The latter (viz., an integral version of the Beilinson regulator) requires higher Chow cycles that are admissible in the sense of meeting the real negative cube and its faces properly. This required a moving lemma which was only proven over \(\mathbb{Q}\) [M. Kerr and the reviewer, Invent. Math. 170, No. 2, 355–420 (2007; Zbl 1139.14010)]; but an integral moving lemma is now established.

MSC:

14C25 Algebraic cycles
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
19E15 Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects)
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[1] ; Beĭlinson, Current problems in mathematics, 181, (1984)
[2] ; Beĭlinson, Applications of algebraic K-theory to algebraic geometry and number theory, Part I. Contemporary Math., 55, 35, (1986)
[3] 10.1016/0001-8708(86)90081-2 · Zbl 0608.14004
[4] ; Bloch, J. Algebraic Geom., 3, 537, (1994)
[5] 10.24033/asens.1269 · Zbl 0316.57026
[6] ; Borel, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 4, 613, (1977)
[7] ; Burgos Gil, The regulators of Beilinson and Borel. CRM Monograph Series, 15, (2002) · Zbl 0994.19003
[8] 10.1007/BF00534143 · Zbl 0697.14006
[9] 10.1007/s00222-007-0066-x · Zbl 1139.14010
[10] 10.1112/S0010437X05001867 · Zbl 1123.14006
[11] ; Levine, K-theory. Astérisque, 226, 235, (1994)
[12] ; Neukirch, Beĭlinson’s conjectures on special values of L-functions. Perspectives in Math., 4, 193, (1988)
[13] 10.1016/j.jnt.2009.04.009 · Zbl 1257.11064
[14] ; Rapoport, Beĭlinson’s conjectures on special values of L-functions. Perspectives in Math., 4, 169, (1988)
[15] ; Zagier, Arithmetic algebraic geometry. Progress in Math., 89, 391, (1991)
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