. A G ] 2 2 Ja n 20 09 A p - ADIC REGULATOR MAP AND FINITENESS RESULTS FOR ARITHMETIC SCHEMES 1

A main theme of the paper is a conjecture of Bloch-Kato on the image of padic 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 X of X over the integer ring of k, we show an injectivity result on torsion of a cycle class map from the Chow group in codimension 2 of X to a new p-adic cohomology of X introduced by the second author, which is a candidate of the conjectural étale motivic cohomology with finite coefficients of Beilinson-Lichtenbaum.