Pepin Lehalleur, Simon

Constructible \(1\)-Motives and Exactness of Realisation Functors

Doc. Math. 24, 1721-1737 (2019)
DOI: 10.25537/dm.2019v24.1721-1737
Communicated by Thomas Geisser

Summary

The triangulated category of cohomological \(1\)-motives with rational coefficients over a base scheme admits a motivic t-structure. We prove that this t-structure restricts to the subcategory of compact objects, and that pullbacks along arbitrary morphisms, as well as Betti and étale realisation functors, are t-exact relative to this t-structure. These exactness properties follow from a structural result: compact objects in the heart behave like a constructible sheaf of Deligne \(1\)-motives.

Mathematics Subject Classification

14C15, 19E15

Keywords/Phrases

Voevodsky motives, Deligne \(1\)-motives, motivic t-structure

References

  • 1. Joseph Ayoub, Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique. I \& II, Astérisque (2007), no. 314 \& 315. zbl 1146.14001 (no. 314); MR2423375 (no. 314); zbl 1153.14001 (no. 315).
  • 2. Joseph Ayoub, La réalisation étale et les opérations de Grothendieck, Ann. Sci. Éc. Norm. Supér. (4) 47 (2014), no. 1, 1-145. DOI 10.24033/asens.2210; zbl 1354.18016; MR3205601.
  • 3. Luca Barbieri-Viale and Bruno Kahn, On the derived category of 1-motives, Astérisque (2016), no. 381, xi+254. zbl 1356.19001; MR3545132; arxiv 1009.1900.
  • 4. A. A. Beilinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981), Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5-171. zbl 0536.14011; MR0751966.
  • 5. Michel Brion, Commutative algebraic groups up to isogeny, Doc. Math. 22 (2017), 679-725. https://www.elibm.org/article/10000435; zbl 1378.14040; MR3650225.
  • 6. A. Johan de Jong, Families of curves and alterations, Ann. Inst. Fourier (Grenoble) 47 (1997), no. 2, 599-621. DOI 10.5802/aif.1575; zbl 0868.14012; MR1450427.
  • 7. Philippe Gille and Patrick Polo (eds.), Schémas en groupes (SGA 3). Tome I. Propriétés générales des schémas en groupes, Documents Mathématiques (Paris) [Mathematical Documents (Paris)], 7, Société Mathématique de France, Paris, 2011, Séminaire de Géométrie Algébrique du Bois Marie 1962-64. [Algebraic Geometry Seminar of Bois Marie 1962-64], A seminar directed by M. Demazure and A. Grothendieck with the collaboration of M. Artin, J.-E. Bertin, P. Gabriel, M. Raynaud and J-P. Serre, Revised and annotated edition of the 1970 French original. zbl 1241.14002; MR2867621.
  • 8. Nicholas M. Katz, Space filling curves over finite fields, Math. Res. Lett. 6 (1999), no. 5-6, 613-624. DOI 10.4310/MRL.1999.v6.n6.a2; zbl 1016.11022; MR1739219.
  • 9. Sophie Morel, Cohomologie d'intersection des variétés modulaires de Siegel, Compos. Math. 147 (2011), no. 6, 1671-1740. DOI 10.1112/S0010437X11005409; zbl 1248.11042; MR2862060.
  • 10. Amnon Neeman, Triangulated categories, Annals of Mathematics Studies, vol. 148, Princeton University Press, Princeton, NJ, 2001. DOI 10.1515/9781400837212; zbl 0974.18008; MR1812507.
  • 11. Fabrice Orgogozo, Isomotifs de dimension inférieure ou égale à un, Manuscripta Math. 115 (2004), no. 3, 339-360. DOI 10.1007/s00229-004-0495-4; zbl 1092.14027; MR2102056.
  • 12. Simon Pepin Lehalleur, Triangulated categories of relative 1-motives, Adv. Math. 347 (2019), 473-596. DOI 10.1016/j.aim.2019.02.021; zbl 07044297; MR3920833; arxiv 1512.00266.
  • 13. Michael Temkin, Tame distillation and desingularization by p-alterations, Ann. of Math. (2) 186 (2017), no. 1, 97-126. DOI 10.4007/annals.2017.186.1.3; zbl 1370.14015; MR3665001; arxiv 1508.06255.
  • 14. V. Vaish, Punctual gluing of t-structures and weight structures, (2017), Preprint. arxiv 1705.00790.

Affiliation

Pepin Lehalleur, Simon
Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany

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