Abstract
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 [H. Gillet, C. Soulé, Descent, motives and K-theory, J. Reine Angew. Math. 478 (1996) 127–176], 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 projective varieties induces a morphism of weight complexes.
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