Toric varieties, monoid schemes and $cdh$ descent
  We give conditions for the Mayer-Vietoris property to hold for the algebraic K-theory of blow-up squares of toric varieties in any characteristic, using the theory of monoid schemes. These conditions are used to relate algebraic K-theory to topological cyclic homology in characteristic p. To achieve our goals, we develop for monoid schemes many notions from classical algebraic geometry, such as separated and proper maps. 
Guillermo Corti\~nas, Christian Haesemeyer, Mark E. Walker and Charles   A. Weibel
math.KT math.AG