On Chow-weight homology of motivic complexes and its relation to motivic homology
DOI:
https://doi.org/10.21638/spbu01.2020.401Abstract
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.Keywords:
motives, triangulated categories, Chow groups, weight structures, Chow-weight homology, Deligne filtration
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Литература
1. Bondarko M. V., Sosnilo V. A. On Chow-weight homology of geometric motives. Preprint. 2020. URL: https://www.researchgate.net/publication/340849991_On_Chow-weight_homology_of_geometric_motives (accessed: September 7, 2020).
2. Bondarko M. V., Sosnilo V. A. Detecting the c-effectivity of motives, their weights, and dimension via Chow-weight (co)homology: a “mixed motivic decomposition of the diagonal”. Preprint. 2014. arxiv.org/abs/1411.6354
3. Bloch S. Lectures on algebraic cycles. In: Duke University Mathematics series IV, 1980.
4. Бондарко М. В., Кумаллагов Д. З. О весовых структурах Чжоу без проективности и разрешения особенностей// Алгебра и Анализ. 2018. Т. 30, №5. С. 57–83.
5. Kelly S. Voevodsky motives and ldh-descent // Asterisque. 2017. No. 391. P. 1–134.
6. Bondarko M. V., Sosnilo V. A. On constructing weight structures and extending them to idempotent extensions // Homology, Homotopy and Appl. 2018. Vol. 20, no. 1. P. 37–57.
7. Wildeshaus J. Chow motives without projectivity // Compositio Mathematica. 2009. Vol. 145, no. 5. P. 1196–1226.
8. Bondarko M. V. Weight structures vs. t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general) // J. of K-theory. 2010. Vol. 6, no. 3. P. 387–504. arxiv.org/abs/0704.4003
9. Pauksztello D. Compact cochain objects in triangulated categories and co-t-structures // Central European Journal of Mathematics. 2008. Vol. 6, no. 1. P. 25–42.
10. Bondarko M. V., Sosnilo V. A. On purely generated α-smashing weight structures and weightexact localizations // J. of Algebra. 2019. Vol. 535. P. 407–455.
11. Bondarko M. V. On weight complexes, pure functors, and detecting weights. Preprint. 2018. arxiv.org/abs/1812.11952
12. Bondarko M. V., Sosnilo V. A. Non-commutative localizations of additive categories and weight structures; applications to birational motives // J. of the Inst. of Math. of Jussieu. 2018. Vol. 17, no. 4. P. 785–821.
13. Sosnilo V. A. Theorem of the heart in negative K -theory for weight structures // Doc. Math. 2019. Vol. 24. P. 2137–2158.
14. Gillet H., Soul´e C. Descent, motives and K -theory // J. f. die reine und ang. Math. 1996. Vol. 478. P. 127–176.
15. Beilinson A., Vologodsky V. A DG guide to Voevodsky motives // Geom. Funct. Analysis. 2008. Vol. 17, no. 6. P. 1709–1787.
16. Cisinski D.-C., D´eglise F. Integral mixed motives in equal characteristic // Documenta Mathematica, Extra Volume: Alexander S. Merkurjev’s Sixtieth Birthday. 2015. P. 145–194.
17. Cisinski D.-C., D´eglise F. Triangulated categories of mixed motives. In: Springer Monographs in Mathematics. 2019.
18. Bondarko M. V., D´eglise F. Dimensional homotopy t-structures in motivic homotopy theory // Adv. in Math. 2017. Vol. 311. P. 91–189.
19. Bondarko M. V., Sosnilo V. A. On the weight lifting property for localizations of triangulated categories // Lobachevskii J. of Math. 2018. Vol. 39, no. 7. P. 970–984.
20. Bondarko M. V. Z[1/p]-motivic resolution of singularities // Compositio Math. 2011. Vol. 147, no. 5. P. 1434–1446.
21. Bondarko M. V. Differential graded motives: weight complex, weight filtrations and spectral sequences for realizations; Voevodsky vs. Hanamura // J. of the Inst. of Math. of Jussieu. 2009. Vol. 8, no. 1. P. 39–97. arxiv.org/abs/math.AG/0601713
22. Ayoub J. Motives and algebraic cycles: a selection of conjectures and open questions. In: Hodge theory and L2-analysis. P. 87–125. Adv. Lect. Math. (ALM). Vol. 39. Somerville, MA: Int. Press, 2017.
23. Bondarko M. V. Intersecting the dimension and slice filtrations for motives // Homology, Homotopy and Appl. 2018. Vol. 20, no. 1. P. 259–274.
24. Kahn B., Sujatha R. Birational motives, II: triangulated birational motives // Int. Math. Res. Notices. 2017. Vol. 2017, no. 22. P. 6778–6831.
25. Ayoub J. The slice filtration on DM(k) does not preserve geometric motives. Appendix to A. Huber’s “Slice filtration on motives and the Hodge conjecture” // Math. Nachr. 2008. Vol. 281, no. 12. P. 1764–1776.
26. Krause H. Smashing subcategories and the telescope conjecture — an algebraic approach // Invent. math. 2000. Vol. 139. P. 99–133.
27. D´eglise F. Modules homotopiques (Homotopy modules) // Doc. Math. 2011. Vol. 16. P. 411–455.
28. Bondarko M. V. Gersten weight structures for motivic homotopy categories; retracts of cohomology of function fields, motivic dimensions, and coniveau spectral sequences. Preprint. 2018. arxiv.org/abs/1803.01432
References
1. Bondarko M. V., Sosnilo V. A., “On Chow-weight homology of geometric motives”, preprint (2020). Available at: https://www.researchgate.net/publication/340849991_On_Chow-weight_homology_of_geometric_motives (accessed: September 7, 2020).
2. Bondarko M. V., Sosnilo V. A., “Detecting the c-effectivity of motives, their weights, and dimension via Chow-weight (co)homology: a “mixed motivic decomposition of the diagonal”, preprint (2014). arxiv.org/abs/1411.6354
3. Bloch S., Lectures on algebraic cycles, in: Duke University Mathematics series IV (1980).
4. Bondarko M. V., Kumallagov D. Z., “On Chow weight structures without projectivity and resolution of ingularities”, St. Petersburg Math. J. 30 (5), 803–819 (2019).
5. Kelly S., “Voevodsky motives and ldh-descent”, Asterisque (391), 1–134 (2017).
6. Bondarko M. V., Sosnilo V. A., “On constructing weight structures and extending them to idempotent extensions”, Homology, Homotopy and Appl. 20 (1), 37–57 (2018).
7. Wildeshaus J., Chow motives without projectivity”, Compositio Mathematica 145 (5), 1196–1226 (2009).
8. Bondarko M. V., “Weight structures vs. t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general)”, J. of K-theory 6 (3), 387–504 (2010). arxiv.org/abs/0704.4003
9. Pauksztello D., “Compact cochain objects in triangulated categories and co-t-structures”, Central European Journal of Mathematics 6 (1), 25–42 (2008).
10. Bondarko M. V., Sosnilo V. A., “On purely generated α-smashing weight structures and weightexact localizations”, J. of Algebra 535, 407–455 (2019).
11. Bondarko M. V., “On weight complexes, pure functors, and detecting weights”, preprint (2018). arxiv.org/abs/1812.11952
12. Bondarko M. V., Sosnilo V. A., “Non-commutative localizations of additive categories and weight structures; applications to birational motives”, J. of the Inst. of Math. of Jussieu 17 (4), 785–821 (2018).
13. Sosnilo V. A., “Theorem of the heart in negative K -theory for weight structures”, Doc. Math. 24, 2137–2158 (2019).
14. Gillet H., Soul´e C., “Descent, motives and K -theory”, J. f. die reine und ang. Math. 478, 127–176 (1996).
15. Beilinson A., Vologodsky V., “A DG guide to Voevodsky motives”, Geom. Funct. Analysis 17 (6), 1709–1787 (2008).
16. Cisinski D.-C., D´eglise F., “Integral mixed motives in equal characteristic”, Documenta Mathematica, Extra Volume: Alexander S. Merkurjev’s Sixtieth Birthday, 145–194 (2015).
17. Cisinski D.-C., D´eglise F., Triangulated categories of mixed motives, in: Springer Monographs in Mathematics (2019).
18. Bondarko M. V., D´eglise F., “Dimensional homotopy t-structures in motivic homotopy theory”, Adv. in Math. 311, 91–189 (2017).
19. Bondarko M. V., Sosnilo V. A., “On the weight lifting property for localizations of triangulated categories”, Lobachevskii J. of Math. 39 (7), 970–984 (2018).
20. Bondarko M. V., “Z[1/p]-motivic resolution of singularities”, Compositio Math. 147 (5), 1434–1446 (2011).
21. Bondarko M. V., “Differential graded motives: weight complex, weight filtrations and spectral sequences for realizations; Voevodsky vs. Hanamura”, J. of the Inst. of Math. of Jussieu 8 (1), 39–97 (2009). arxiv.org/abs/math.AG/0601713
22. Ayoub J., “Motives and algebraic cycles: a selection of conjectures and open questions”, in: Hodge theory and L2-analysis, 87–125 (Int. Press, Somerville, MA, 2017, vol. 39 of Adv. Lect. Math.).
23. Bondarko M. V., “Intersecting the dimension and slice filtrations for motives”, Homology, Homotopy and Appl. 20 (1), 259–274 (2018).
24. Kahn B., Sujatha R., “Birational motives, II: triangulated birational motives”, Int. Math. Res. Notices 2017 (22), 6778–6831 (2017).
25. Ayoub J., “The slice filtration on DM(k) does not preserve geometric motives. Appendix to A. Huber’s “Slice filtration on motives and the Hodge conjecture”, Math. Nachr. 281 (12), 1764–1776 (2008).
26. Krause H., “Smashing subcategories and the telescope conjecture — an algebraic approach”, Invent. math. 139, 99–133 (2000).
27. D´eglise F., “Modules homotopiques (Homotopy modules)”, Doc. Math. 16, 411–455 (2011).
28. Bondarko M. V., “Gersten weight structures for motivic homotopy categories; retracts of cohomology of function fields, motivic dimensions, and coniveau spectral sequences”, preprint (2018). arxiv.org/abs/1803.01432
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Published
2020-12-27
How to Cite
Bondarko, M. V., & Kumallagov, D. Z. (2020). On Chow-weight homology of motivic complexes and its relation to motivic homology. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 7(4), 560–587. https://doi.org/10.21638/spbu01.2020.401
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On the anniversary of S. V. Vostokov
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Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
