We associate weight complexes of (homological) motives, and hence
Euler characteristics in the Grothendieck group of motives, to
arithmetic varieties and Deligne-Mumford stacks; this extends the
results in the paper "Descent, Motives and K-theory" in volume 478 of
Crelle, where a similar result was proved for varieties over a field of
characteristic zero. We use K_0-motives with rational coefficients,
rather than Chow motives, because we cannot appeal to resolution of
singularities, but rather must use de Jong's results. In addition, for
varieties over a field we prove a general result on contravariance of
weight complexes, in particular showing that any morphism of finite
tor-dimension between varieties induces a morphism of weight complexes.
See http://arxiv.org/abs/0804.4853.