TY - JOUR

T1 - On Chow-Weight Homology of Motivic Complexes and Its Relation to Motivic Homology

AU - Bondarko, M. V.

AU - Kumallagov, D. Z.

N1 - Funding Information: The work of the authors on Section 3 of the paper was supported by the Russian Science Foundation, project no. 16-11-10200. Sections 1 and 2 were written by D. Kumallagov; his work was supported by the Russian Foundation for Basic Research, project no. 19-31-90074. Publisher Copyright: © 2020, Pleiades Publishing, Ltd. Copyright: Copyright 2021 Elsevier B.V., All rights reserved.

PY - 2020/10

Y1 - 2020/10

N2 - Abstract: 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.

AB - Abstract: 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.

KW - Chow groups

KW - Chow-weight homology

KW - Deligne weight filtration

KW - effectivity

KW - motives

KW - triangulated categories

KW - weight structures

UR - http://www.scopus.com/inward/record.url?scp=85099780262&partnerID=8YFLogxK

U2 - 10.1134/S1063454120040032

DO - 10.1134/S1063454120040032

M3 - Article

AN - SCOPUS:85099780262

VL - 53

SP - 377

EP - 397

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 4

ER -