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" $H{t}_{hom}^{eff}$ 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 $H{t}_{hom}^{eff}$ to be weakly birational (i.e., that its $n+1$th contraction is trivial or, equivalently, the Nisnevich cohomology vanishes in degrees $>n$ for some $n\ge 0$) in terms of these aisles; this statement generalizes well-known results of Kahn and Sujatha. Next, these classes define weight structures $w_{Smooth}^s$ (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 $t_{Smooth}^s$ and prove that they give the birationality filtrations on $H{t}_{hom}^{eff}$. 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.

To appear in Algebra i Analiz. Comments are very welcome!