Smooth weight structures and birationality filtrations on motivic categories

Abstract

We study various triangulated motivic categories and introduce a vast family of aisles (these are certain classes of objects) in them. These aisles are defined in terms of the corresponding "motives" (or motivic spectra) of smooth varieties in them; we relate them to the corresponding homotopy t−structures. We describe our aisles in terms of stalks at function fields and prove that they widely generalize the ones corresponding to slice filtrations. Further, the filtrations on the "homotopy hearts" Ht ef f hom of the corresponding effective subcategories that are induced by these aisles can be described in terms of (Nisnevich) sheaf cohomology as well as in terms of the Voevodsky contractions − −1 . Respectively, we express the condition for an object of Ht ef f hom to be weakly birational (i.e., that its n + 1th contraction is trivial or, equivalently, the Nisnevich cohomology vanishes in degrees > n for some n ≥ 0) in terms of these aisles; this statement generalizes well-known results of Kahn and Sujatha. Next, these classes define weight structures w s Smooth (where s = (s j ) are non-decreasing sequences parameterizing our aisles) that vastly generalize the Chow weight structures w Chow defined earlier. Using general abstract nonsense we also construct the corresponding adjacent t−structures t s Smooth and prove that they give the birationality filtrations on Ht ef f hom . Moreover, some of these weight structures induce weight structures on the corresponding n−birational motivic categories (these are the localizations by the levels of the slice filtrations). Our results also yield some new unramified cohomology calculations.