Smooth weight structures and birationality filtrations on motivic categories

Abstract

In various triangulated motivic categories, a vast family of aisles (these are certain classes of objects) is introduced. These aisles are defined in terms of the corresponding “motives” (or motivic spectra) of smooth varieties; it is proved that they are expressed in terms of the corresponding homotopytt-structures. The aisles in question are described in terms of stalks at function fields, and it is shown that they widely generalize the ones corresponding to slice filtrations. Further, the filtrations on the “homotopy hearts”Ht_homeff{\underline {Ht}}_{\mathrm {hom}}^{\mathrm {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-_{-1}. Respectively, the condition for an object ofHt_homeff{\underline {Ht}}_{\mathrm {hom}}^{\mathrm {eff}}to be weakly birational (i.e., for its(n+1)(n+1)st contraction to be trivial, or equivalently, its Nisnevich cohomology to vanish in degrees strictly greater thannnfor somen≥0n\ge 0) are expressed in terms of these aisles; this statement generalizes well-known results of Kahn and Sujatha. Next, these classes give rise to weight structureswSmoothsw_{\mathrm {Smooth}}^{s}(where thes=(sj)s=(s_{j})are nondecreasing sequences parametrizing our aisles) that vastly generalize the Chow weight structureswChoww_{\operatorname {Chow}}defined earlier. By using general abstract nonsense, the correspondingadjacenttt-structurestSmoothst_{\mathrm {Smooth}}^{s}are constructed and it is proved that they give the birationality filtrations onHt_homeff{\underline {Ht}}^{\mathrm {eff}}_{\mathrm {hom}}.Moreover, some of these weight structures induce weight structures on the correspondingnn-birational motivic categories (these are the localizations by the levels of the slice filtrations). The results also yield some new unramified cohomology calculations.