Pro cdh-descent for cyclic homology and $K$-theory
  In this paper we prove that cyclic homology, topological cyclic homology, and algebraic $K$-theory satisfy a pro Mayer--Vietoris property with respect to abstract blow-up squares of varieties, in both zero and finite characteristic. This may be interpreted as the well-definedness of $K$-theory with compact support. 
Matthew Morrow
math.KT math.AG