ScalingStacks

Remark 2.9. The weight complex functor for $\operatorname{DM}_{gm}(k,\mathbb{Q})$ allows to decompose the motive $\operatorname{M}(X)$ of any (not necessarily smooth or projective) variety $X$ into a complex of motives of smooth projective varieties. Again, this reflects Deligne’s yoga of weights for mixed Hodge structures and $\ell$-adic cohomology: the cohomology of a variety admits a weight filtration whose graded pieces behave like the cohomology of smooth projective varieties. Similarly to Deligne’s approach, the existence of the Chow weight structure and the weight complex functor relies on resolutions of singularities or de Jong’s and Gabber’s theory of alterations, see [Bon11].