Skip to main content Accessibility help

Mixed motivic sheaves (and weights for them) exist if ‘ordinary’ mixed motives do

  • Mikhail V. Bondarko (a1)


The goal of this paper is to prove that if certain ‘standard’ conjectures on motives over algebraically closed fields hold, then over any ‘reasonable’ scheme $S$ there exists a motivic $t$ -structure for the category $\text{DM}_{c}(S)$ of relative Voevodsky’s motives (to be more precise, for the Beilinson motives described by Cisinski and Deglise). If $S$ is of finite type over a field, then the heart of this $t$ -structure (the category of mixed motivic sheaves over $S$ ) is endowed with a weight filtration with semisimple factors. We also prove a certain ‘motivic decomposition theorem’ (assuming the conjectures mentioned) and characterize semisimple motivic sheaves over $S$ in terms of those over its residue fields. Our main tool is the theory of weight structures. We actually prove somewhat more than the existence of a weight filtration for mixed motivic sheaves: we prove that the motivic $t$ -structure is transversal to the Chow weight structure for $\text{DM}_{c}(S)$ (that was introduced previously by Hébert and the author). We also deduce several properties of mixed motivic sheaves from this fact. Our reasoning relies on the degeneration of Chow weight spectral sequences for ‘perverse étale homology’ (which we prove unconditionally); this statement also yields the existence of the Chow weight filtration for such (co)homology that is strictly restricted by (‘motivic’) morphisms.



Hide All
[Ayo07]Ayoub, J., The motivic vanishing cycles and the conservation conjecture, in Algebraic cycles and motives, Vol. 1, London Mathematical Society Lecture Note Series, vol. 343, eds Nagel, J., Peters, C. and Murre, J. P. (Cambridge University Press, Cambridge, 2007), 354.
[BK14]Barbieri-Viale, L. and Kahn, B., On the derived category of 1-motives, Preprint (2014),∼kahn/preprints/revder1mot1.pdf.
[Bei87]Beilinson, A., Height pairing between algebraic cycles, in K-theory, arithmetic and geometry, Lecture Notes in Mathematics, vol. 1289 (Springer, 1987), 126.
[Bei02]Beilinson, A., Remarks on n-motives and correspondences at the generic point, in Motives, polylogarithms and Hodge theory, Part I, Irvine, CA, 1998, International Press Lecture Series, 3, vol. I (International Press, Somerville, MA, 2002), 3546.
[Bei12]Beilinson, A., Remarks on Grothendieck’s standard conjectures, in Regulators III, Proceedings of the conference held at the University of Barcelona, July 11–23, 2010, Contemporary Mathematics, vol. 571, eds Burgos, J. I. and Lewis, J. (American Mathematical Society, Providence, RI, 2012), 2532.
[BBD82]Beilinson, A., Bernstein, J. and Deligne, P., Faisceaux pervers, Astérisque 100 (1982), 5171.
[Bon09]Bondarko, M. V., Differential graded motives: weight complex, weight filtrations and spectral sequences for realizations; Voevodsky vs. Hanamura, J. Inst. Math. Jussieu 8 (2009), 3997; see also arXiv:math.AG/0601713.
[Bon10a]Bondarko, M., Weight structures vs. t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general), J. K-Theory 6 (2010), 387504; see alsoarXiv:0704.4003.
[Bon10b]Bondarko, M. V., Motivically functorial coniveau spectral sequences; direct summands of cohomology of function fields, Doc. Math. (2010), 33117; extra volume: Andrei Suslin’s Sixtieth Birthday.
[Bon11]Bondarko, M. V., ℤ[1∕p]-motivic resolution of singularities, Compositio Math. 147 (2011), 14341446.
[Bon12]Bondarko, M. V., Weight structures and ‘weights’ on the hearts of t-structures, Homology, Homotopy Appl. 14 (2012), 239261.
[Bon14]Bondarko, M. V., Weights for relative motives: relation with mixed complexes of sheaves, Int. Math. Res. Not. IMRN 2014 (2014), 47154767; doi:10.1093/imrn/rnt088.
[CD09]Cisinski, D. and Déglise, F., Triangulated categories of mixed motives, Preprint (2009),arXiv:0912.2110.
[CD13]Cisinski, D. and Déglise, F., Étale motives, Compositio Math., to appear, arXiv:1305.5361v3.
[CH00]Corti, A. and Hanamura, M., Motivic decomposition and intersection Chow groups, I, Duke Math. J. 103 (2000), 459522.
[EGA4(3)]Dieudonné, J. and Grothendieck, A., Eléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas, Troisiéme Partie, Publ. Math. Inst. Hautes Études Sci. 28 (1966).
[Eke90]Ekedahl, T., On the adic formalism, in The Grothendieck Festschrift, Vol. II, Progress in Mathematics, vol. 87 (Birkhäuser, Boston, 1990), 197218.
[FK88]Freitag, E. and Kiehl, R., Etale cohomology and the Weil conjecture (Springer, 1988), 343.
[Han99]Hanamura, M., Mixed motives and algebraic cycles, III, Math. Res. Lett. 6 (1999), 6182.
[Heb11]Hébert, D., Structures de poids la Bondarko sur les motifs de Beilinson, Compositio Math. 147 (2011), 14471462.
[Hub97]Huber, A., Mixed perverse sheaves for schemes over number fields, Compositio Math. 108 (1997), 107121.
[HK06]Huber, A. and Kahn, B., The slice filtration and mixed Tate motives, Compositio Math. 142 (2006), 907936.
[ILO14]Illusie, L., Laszlo, Y. and Orgogozo, F., Travaux de Gabber sur l’uniformisation locale et la cohomologie étale des schémas quasi-excellents (Séminaire à l’École polytechnique 2006–2008), Astérisque, vol. 363–364 (Société Mathématique de France, 2014).
[Ito05]Ito, T., Weight-monodromy conjecture over equal characteristic local fields, Amer. J. Math. 127 (2005), 647658.
[Jan92]Jannsen, U., Motives, numerical equivalence and semisimplicity, Invent. Math. 107 (1992), 447452.
[Kle94]Kleinman, S., The standard conjectures, in Motives, Proceedings of Symposia in Pure Mathematics, vol. 55, Part 1 (American Mathematical Society, Providence, RI, 1994), 320.
[Nee01]Neeman, A., Triangulated categories, Annals of Mathematics Studies, vol. 148 (Princeton University Press, Princeton, NJ, 2001).
[Pop86]Popescu, D., General Néron desingularization and approximation, Nagoya Math. J. 104 (1986), 85115.
[Sch12]Scholbach, J., f-cohomology and motives over number rings, Kodai Math. J. 35 (2012), 132.
[SGA4(2)]Grothendieck, A., Expose VIII, in Séminaire de Géométrie Algébrique du Bois Marie 1963–1964, Théorie des topos et cohomologie étale des schémas (SGA 4), Tome 2, Lecture Notes in Mathematics, vol. 270 (Springer, 1972).
[SGA4(3)]Deligne, P., Expose XVIII, in Séminaire de Géométrie Algébrique du Bois Marie 1963–1964, Théorie des topos et cohomologie étale des schémas (SGA 4), Tome 3, Lecture Notes in Mathematics, vol. 305 (Springer, 1973).
[Sil94]Silverman, J., Advanced topics in the arithmetic of elliptic curves (Springer, 1994).
[Smi97]Smirnov, O., Graded associative algebras and Grothendieck standard conjectures, Invent. Math. 128 (1997), 201206.
[Voe00]Voevodsky, V., Triangulated categories of motives over a field, in Cycles, transfers and motivic homology theories, Annals of Mathematical Studies, vol. 143, eds Voevodsky, V., Suslin, A. and Friedlander, E. (Princeton University Press, Princeton, NJ, 2000), 188238.
[Wil08]Wildeshaus, J., Notes on Artin–Tate motives, Preprint (2008),
[Wil12]Wildeshaus, J., Intermediate extension of Chow motives of Abelian type, Preprint (2012),arXiv:1211.5327.
MathJax is a JavaScript display engine for mathematics. For more information see


MSC classification

Mixed motivic sheaves (and weights for them) exist if ‘ordinary’ mixed motives do

  • Mikhail V. Bondarko (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed