K-theory and logarithmic Hodge-Witt sheaves of formal schemes in characteristic p

2015/12/15 by Morrow, Matthew
#Algebraic Geometry (math.AG) #FOS: Mathematics #K-Theory and Homology (math.KT)

paper · doi:10.48550/arxiv.1512.04703

Abstract

We describe the mod p^r pro K-groups \K_n(A/I^s)/p^r\_s of a regular local \mathbb 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 étale-motivic cycle classes, for algebraic cycles on formal schemes to admit infinitesimal deformations. Moreover, in the case n=1, we compare the étale cohomology of W_rΩ¹_log and the fppf cohomology of \mathbfμ_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.

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