  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.
