Etale and motivic cohomology and ultraproducts of schemes

This paper is a continuation of the authors article "Enlargements of schemes" (Log. Anal.1 (2007), no. 1, 1-60) We mainly study the behaviour of etale cohomology, algebraic cycles and motives under ultraproducts respectively enlargements. The main motivation for that is to find methods to transfer statements about etale cohomology and algebraic cycles from characteristic zero to positive characteristic and vice versa. We give one application to the independence of $l$ of Betti numbers in etale cohomology and applications to the complexity of algebraic cycles.