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The syntomic regulator for the $$K$$-theory of fields. (English) Zbl 1106.11024
Summary: We define complexes analogous to Goncharov’s complexes for the $$K$$-theory of discrete valuation rings of characteristic zero. Under suitable assumptions in $$K$$-theory, there is a map from the cohomology of those complexes to the $$K$$-theory of the ring under consideration. In case the ring is a localization of the ring of integers in a number field, there are no assumptions necessary. We compute the composition of our map to the $$K$$-theory with the syntomic regulator. The result can be described in terms of a $$p$$-adic polylogarithm. Finally, we apply our theory in order to compute the regulator to syntomic cohomology on Beilinson’s cyclotomic elements. The result is again given by the $$p$$-adic polylogarithm. This last result is related to one by Somekawa and generalizes work by Gros.

##### MSC:
 11G55 Polylogarithms and relations with $$K$$-theory 14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) 19F27 Étale cohomology, higher regulators, zeta and $$L$$-functions ($$K$$-theoretic aspects)
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