Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-19T07:20:24.462Z Has data issue: false hasContentIssue false
  • English
  • Français

ℤ[1/p]-motivic resolution of singularities

Part of: Abelian categories $K$-theory in geometry Arithmetic problems. Diophantine geometry Cycles and subschemes (Co)homology theory Birational geometry Commutative algebra: Homological methods

Published online by Cambridge University Press:  01 June 2011

M. V. Bondarko
M. V. Bondarko*
Affiliation:
Department of Mathematics and Mechanics, St. Petersburg State University, Bibliotechnaya Pl. 2, 198904, St. Petersburg, Russia (email: mbondarko@gmail.com)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The main goal of this paper is to deduce (from a recent resolution of singularities result of Gabber) the following fact: (effective) Chow motives with ℤ[1/p]-coefficients over a perfect field k of characteristic p generate the category DMeffgm[1/p] (of effective geometric Voevodsky’s motives with ℤ[1/p]-coefficients). It follows that DMeffgm[1/p] can be endowed with a Chow weight structure wChow whose heart is Choweff[1/p] (weight structures were introduced in a preceding paper, where the existence of wChow for DMeffgmℚ was also proved). As shown in previous papers, this statement immediately yields the existence of a conservative weight complex functor DMeffgm[1/p]→Kb (Choweff [1/p]) (which induces an isomorphism on K0-groups), as well as the existence of canonical and functorial (Chow)-weight spectral sequences and weight filtrations for any cohomology theory on DMeffgm[1/p] . We also mention a certain Chow t-structure for DMeff−[1/p] and relate it with unramified cohomology.

Keywords

motivestriangulated categoriescohomologyweight structuresresolution of singularitiesalterations

MSC classification

Secondary: 14E15: Global theory and resolution of singularities 14G17: Positive characteristic ground fields 14C15: (Equivariant) Chow groups and rings; motives 18E30: Derived categories, triangulated categories 18E40: Torsion theories, radicals 19E15: Algebraic cycles and motivic cohomology 14F42: Motivic cohomology; motivic homotopy theory 14F20: Étale and other Grothendieck topologies and (co)homologies 13D15: Grothendieck groups, $K$-theory 14C25: Algebraic cycles
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 5 , September 2011 , pp. 1434 - 1446
Copyright
Copyright © Foundation Compositio Mathematica 2011

References

[Bon09a]Bondarko, M. V., Differential graded motives: weight complex, weight filtrations and spectral sequences for realizations; Voevodsky vs. Hanamura, J. Inst. Math. Jussieu 8 (2009), 39–97, see also http://arxiv.org/abs/math.AG/0601713.CrossRefGoogle Scholar
[Bon09b]Bondarko, M. V., Weight structures and motives; comotives, coniveau and Chow-weight spectral sequences: a survey, Preprint, http://arxiv.org/abs/0903.0091.Google Scholar
[Bon10a]Bondarko, M. V., Weight structures vs. t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general), J. K-Theory 6 (2010), 387–504, see also http://arxiv.org/abs/0704.4003.CrossRefGoogle Scholar
[Bon10b]Bondarko, M. V., Motivically functorial coniveau spectral sequences; direct summands of cohomology of function fields, Doc. Math., extra volume: Andrei Suslin’s Sixtieth Birthday (2010), 33–117.CrossRefGoogle Scholar
[Bon10c]Bondarko, M. V., ℤ[1/p]-motivic resolution of singularities, and applications, Preprint, http://arxiv.org/abs/1002.2651.Google Scholar
[Bon10d]Bondarko, M. V., Weights for relative motives; relation with mixed complexes of sheaves, Preprint, http://arxiv.org/abs/1007.4543.Google Scholar
[Bon10e]Bondarko, M. V., Weights and t-structures: in general triangulated categories, for 1-motives, mixed motives, and for mixed Hodge complexes and modules, Preprint,http://arxiv.org/abs/1011.3507.Google Scholar
[CD09]Cisinski, D. and Deglise, F., Triangulated categories of mixed motives, Preprint, http://arxiv.org/abs/0912.2110.Google Scholar
[Deg08]Deglise, F., Around the Gysin triangle II, Doc. Math. 13 (2008), 613–675.CrossRefGoogle Scholar
[GS96]Gillet, H. and Soulé, C., Descent, motives and K-theory, J. Reine Angew. Math. 478 (1996), 127–176.Google Scholar
[GS09]Gillet, H. and Soulé, C., Motivic weight complexes for arithmetic varieties, J. Algebra 322 (2009), 3088–3141.CrossRefGoogle Scholar
[Heb10]Hébert, D., Structures de poids à la Bondarko sur les motifs de Beilinson, Preprint, http://arxiv4.library.cornell.edu/abs/1007.0219.Google Scholar
[HK06]Huber, A. and Kahn, B., The slice filtration and mixed Tate motives, Compositio Math. 142 (2006), 907–936.CrossRefGoogle Scholar
[Ill08]Illusie, L., On Gabber’s refined uniformization, Talks at the University of Tokyo, 17, 22, 31 January, 7 February, 2008, http://www.math.u-psud.fr/∼illusie/refined_uniformization3.pdf.Google Scholar
[KS]Kahn, B. and Sujatha, R., Birational motives, I, Preprint,http://www.math.uiuc.edu/K-theory/0596/.Google Scholar
[MVW06]Mazza, C., Voevodsky, V. and Weibel, Ch., Lecture notes on motivic cohomology, Clay Mathematics Monographs, vol. 2 (American Mathematical Society, Providence, RI, 2006).Google Scholar
[Pau08]Pauksztello, D., Compact cochain objects in triangulated categories and co-t-structures, Cent. Eur. J. Math. 6 (2008), 25–42.CrossRefGoogle Scholar
[SV00]Suslin, A. and Voevodsky, V., Relative cycles and Chow sheaves, in Cycles, transfers and motivic homology theories, Annals of Mathematics Studies, vol. 143, eds Voevodsky, V., Suslin, A. and Friedlander, E. (Princeton University Press, Princeton, NJ, 2000), 10–86.Google Scholar
[Voe00]Voevodsky, V., Triangulated category of motives, in Cycles, transfers and motivic homology theories, Annals of Mathematics Studies, vol. 143, eds Voevodsky, V., Suslin, A. and Friedlander, E. (Princeton University Press, Princeton, NJ, 2000), 188–238.Google Scholar
[Voe10]Voevodsky, V., Cancellation theorem, Doc. Math., extra volume: Andrei Suslin’s Sixtieth Birthday (2010), 671–685.CrossRefGoogle Scholar
[Wil09]Wildeshaus, J., Chow motives without projectivity, Compositio Math. 145 (2009), 1196–1226.CrossRefGoogle Scholar