In this paper certain Chow weight structures on the "big" triangulated motivic categories DM ef f R ⊂ DM R are defined in terms of motives of all smooth varieties over the base field. This definition allows studying basic properties of these weight structures without applying resolution of singularities; thus we don't have to assume that the coefficient ring R contains 1/p in the case where the characteristic p of the base field is positive. Moreover, in the case where R satisfies the latter assumption our weight structures are "compatible" with the weight structures that were defined in previous papers in terms of Chow motives; it follows that a motivic complex has non-negative weights if and only if its positive Nisnevich hypercohomology vanishes. The results of this article yield certain Chow-weight filtration (also) on p-adic cohomology of motives and smooth varieties.

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In this paper we study in detail the so-called Chow-weight homology of Voevodsky motivic complexes and relate it to motivic homology. We generalize earlier results and prove that the vanishing of higher motivic homology groups of a motif M implies similar vanishing for its Chow-weight homology along with effectivity properties of the higher terms of its weight complex t(M) and of higher Deligne weight quotients of its cohomology. Applying this statement to motives with compact support we obtain a similar relation between the vanishing of Chow groups and the cohomology with compact support of varieties. Moreover, we prove that if higher motivic homology groups of a geometric motif or a variety over a universal domain are torsion (in a certain “range”) then the exponents of these groups are uniformly bounded. To prove our main results we study Voevodsky slices of motives. Since the slice functors do not respect the compactness of motives, the results of the previous Chow-weight homology paper are not sufficient for our purposes; this is our main reason to extend them to (wChow-bounded below) motivic complexes

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.

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 homotopy t t -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” H t _ h o m e f f {\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 of H t _ h o m e f f {\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 than n n for some n ≥ 0 n\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 structures w S m o o t h s w_{\mathrm {Smooth}}^{s} (where the s = ( s j ) s=(s_{j}) are nondecreasing sequences parametrizing our aisles) that vastly generalize the Chow weight structures w Chow w_{\operatorname {Chow}} defined earlier. By using general abstract nonsense, the corresponding adjacent t t -structures t S m o o t h s t_{\mathrm {Smooth}}^{s} are constructed and it is proved that they give the birationality filtrations on H t _ h o m e f f {\underline {Ht}}^{\mathrm {eff}}_{\mathrm {hom}} . Moreover, some of these weight structures induce weight structures on the corresponding n n -birational motivic categories (these are the localizations by the levels of the slice filtrations). The results also yield some new unramified cohomology calculations.