#### 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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