Smooth weight structures and birationality filtrations on motivic categories

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