K-THEORY AND LOGARITHMIC HODGE-WITT SHEAVES OF FORMAL SCHEMES IN CHARACTERISTIC p

We describe the mod p(r) pro K-groups {K-n, (A/I-s)/p(r)}(s) of a regular local F-p-algebra A modulo powers of a suitable ideal I, in terms of logarithmic Hodge-Witt groups, by proving pro analogues of the theorems of Geisser-Levine and Bloch-Kato-Gabber. This is achieved by combining the pro Hochschild-Kostant-Rosenberg theorem in topological cyclic homology with the development of the theory of de Rham-Witt complexes and logarithmic Hodge-Witt sheaves on formal schemes in characteristic p. Applications include the following: the infinitesimal part of the weak Lefschetz conjecture for Chow groups; a p-adic version of Kato-Saito's conjecture that their Zariski and Nisnevich higher dimensional class groups are isomorphic; continuity results in K-theory; and criteria, in terms of integral or torsion etale-motivic cycle classes, for algebraic cycles on formal schemes to admit infinitesimal deformations. Moreover, in the case n = 1, we compare the etale cohomology of W-r Omega(1)(log)and the fppf cohomology of mu p(r) on a formal scheme, and thus present equivalent conditions for line bundles to deform in terms of their classes in either of these cohomologies.