On Chow-weight homology of motivic complexes and its relation to motivic homology

Bondarko, Mikhail V.; Z., Kumallagov, David

doi:10.21638/spbu01.2020.401

Cited by 3 publications

(1 citation statement)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…3. In the papers [BoS14] and [BoK20] certain Chow-weight homology functors from the categories DM ef f gm (k) and DM ef f (k) were studied in detail. Using the results of this article (especially §4.3), one can generalize this theory to any motivic category.…”

Section: Supplementary Remarks and Questionsmentioning

confidence: 99%

Smooth weight structures and birationality filtrations on motivic categories

Bondarko¹,

Z.²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Supplementary Remarks and Questionsmentioning

confidence: 99%

Smooth weight structures and birationality filtrations on motivic categories

Bondarko¹,

Z.²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

On Morphisms Killing Weights and Stable Hurewicz-Type Theorems

Bondarko

2022

J. Inst. Math. Jussieu

Self Cite

View full text Add to dashboard Cite

For a weight structure w on a triangulated category $\underline {C}$ we prove that the corresponding weight complex functor and some other (weight-exact) functors are ‘conservative up to weight-degenerate objects’; this improves earlier conservativity formulations. In the case $w=w^{sph}$ (the spherical weight structure on $SH$ ), we deduce the following converse to the stable Hurewicz theorem: $H^{sing}_{i}(M)=\{0\}$ for all $i<0$ if and only if $M\in SH$ is an extension of a connective spectrum by an acyclic one. We also prove an equivariant version of this statement. The main idea is to study M that has no weights $m,\dots ,n$ (‘in the middle’). For $w=w^{sph}$ , this is the case if there exists a distinguished triangle $LM\to M\to RM$ , where $RM$ is an n-connected spectrum and $LM$ is an $m-1$ -skeleton (of M) in the sense of Margolis’s definition; this happens whenever $H^{sing}_i(M)=\{0\}$ for $m\le i\le n$ and $H^{sing}_{m-1}(M)$ is a free abelian group. We also consider morphisms that kill weights $m,\dots ,n$ ; those ‘send n-w-skeleta into $m-1$ -w-skeleta’.

show abstract

Smooth weight structures and birationality filtrations on motivic categories

Bondarko¹,

Z.²

2022

St. Petersburg Math. J.

View full text Add to dashboard Cite

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.

show abstract

On Chow-weight homology of motivic complexes and its relation to motivic homology

Cited by 3 publications

References 1 publication

Smooth weight structures and birationality filtrations on motivic categories

Smooth weight structures and birationality filtrations on motivic categories

On Morphisms Killing Weights and Stable Hurewicz-Type Theorems

Smooth weight structures and birationality filtrations on motivic categories

Contact Info

Product

Resources

About