#### DOCUMENTA MATHEMATICA,
Vol. Extra Volume: Andrei A. Suslin's Sixtieth Birthday (2010), 525-594

** S. Saito and K. Sato **
A $p$-adic Regulator Map and Finiteness Results
for Arithmetic Schemes

A main theme of the paper is a conjecture of Bloch-Kato on the image of
$p$-adic regulator maps for a proper smooth variety $X$ over an algebraic
number field $k$. The conjecture for a regulator map of particular degree
and weight is related to finiteness of two arithmetic objects: One is the
$p$-primary torsion part of the Chow group in codimension $2$ of $X$. Another
is an unramified cohomology group of $X$. As an application, for a regular
model ${\mathscr X}$ of $X$ over the integer ring of $k$, we prove an injectivity
result on the torsion cycle class map of codimension $2$ with values in
a new $p$-adic cohomology of ${\mathscr X}$ introduced by the second author,
which is a candidate of the conjectural étale motivic cohomology with
finite coefficients of Beilinson-Lichtenbaum.

2010 Mathematics Subject Classification: Primary 14C25, 14G40; Secondary 14F30, 19F27, 11G25.

Keywords and Phrases: $p$-adic regulator, unramified cohomology, Chow groups, $p$-adic étale
Tate twists

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