Based on the unpublished preprint [BD] of Beilinson and Deligne, we give the
construction of the classical polylogarithm in the motivic cohomology of a
certain simplicial scheme and compute its regulators in absolute Hodge and
etale cohomology.
As a consequence, we obtain an alternative proof of Beilinson's theorem on
the regulator of the cyclotomic elements in the K-theory of an abelian number
field.
Another consequence is the validity of Conjecture 6.2 of [BK], and hence, of
the Tamagawa number conjecture for Tate twists up to powers of two, also for
twists of odd parity.
[BD] A.A. Beilinson, P. Deligne, ``Motivic Polylogarithm
and Zagier Conjecture'', preprint, 1992.
[BK] S. Bloch, K. Kato, ``L-functions and Tamagawa numbers
of Motives'', The Grothendieck Festschrift, Volume I.