p-adic deformation of algebraic cycle classes
  We study the p-adic deformation properties of algebraic cycle classes modulo rational equivalence. We show that the crystalline Chern character of a vector bundle lies in a certain part of the Hodge filtration if and only if, rationally, the class of the vector bundle lifts to a formal pro-class in K-theory on the p-adic scheme. 
Spencer Bloch, H\'el\`ene Esnault, Moritz Kerz
math.AG