Motives of Deligne–Mumford stacks

For every smooth and separated Deligne–Mumford stack FF, we associate a motive M(F)M(F) in Voevodsky’s category of mixed motives with rational coefficients DMeff(k,Q). When FF is proper over a field of characteristic 0, we compare M(F)M(F) with the Chow motive associated to FF by Toen (2000) ([31]). Without the properness condition we show that M(F)M(F) is a direct summand of the motive of a smooth quasi-projective variety.