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Algebraic tori as Nisnevich sheaves with transfers
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 23 (2014) no. 3, pp. 699-715.

On relie la R-équivalence sur les tores aux faisceaux Nisnevich avec transferts invariants par homotopie et aux complexes motiviques effectifs, étudiés par Voevodsky.

We relate R-equivalence on tori with Voevodsky’s theory of homotopy invariant Nisnevich sheaves with transfers and effective motivic complexes.

@article{AFST_2014_6_23_3_699_0,
     author = {Bruno Kahn},
     title = {Algebraic tori as Nisnevich sheaves with transfers},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {699--715},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 23},
     number = {3},
     year = {2014},
     doi = {10.5802/afst.1421},
     zbl = {06374885},
     mrnumber = {3266710},
     language = {en},
     url = {afst.centre-mersenne.org/item/AFST_2014_6_23_3_699_0/}
}
Bruno Kahn. Algebraic tori as Nisnevich sheaves with transfers. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 23 (2014) no. 3, pp. 699-715. doi : 10.5802/afst.1421. https://afst.centre-mersenne.org/item/AFST_2014_6_23_3_699_0/

[1] Barbieri-Viale (L.), Kahn (B.).— On the derived category of 1-motives, arXiv:1009.1900.

[2] Borovoi (M.), Kunyavskǐi (B. E‘), Lemire (N.), Reichstein (Z.).— Stably Cayley groups in characteristic zero, IMRN (2013).

[3] Colliot-Thélène (J.-L.), Harari (D.), Skorobogatov (A.).— Compactification équivariante d’un tore (d’après Brylinski et Künnemann), Expo. Math. 23, p. 161-170 (2005). | MR 2155008 | Zbl 1078.14076

[4] Colliot-Thélène (J.-L.), Sansuc (J.-J.).— La R-équivalence sur les tores, Ann. Sci. Éc. Norm. Sup. 10, p. 175-230 (1977). | Numdam | MR 450280 | Zbl 0356.14007

[5] Colliot-Thélène (J.-L.), Sansuc (J.-J.).— Principal homogeneous spaces under flasque tori; applications, J. Alg. 106, p. 148-205 (1987). | MR 878473 | Zbl 0597.14014

[6] Gille (P.).— La R-équivalence pour les groupes algébriques réductifs définis sur un corps global, Publ. Math. IHÉS 86, p. 199-235 (1997). | Numdam | MR 1608570 | Zbl 0943.20044

[7] Gille (P.).— Spécialisation de la R-équivalence pour les groupes réductifs, Trans. Amer. Math. Soc. 356, p. 4465-4474 (2004). | MR 2067129 | Zbl 1067.20063

[8] Huber (A.) and Kahn (B.).— The slice filtration and mixed Tate motives, Compositio Math. 142, p. 907-936 (2006). | MR 2249535 | Zbl 1105.14022

[9] Kahn (B.).— Sur le groupe des classes d’un schéma arithmétique (avec un appendice de Marc Hindry), Bull. Soc. Math. France 134 (2006), p. 395-415. | Numdam | MR 2245999 | Zbl 1222.14048

[10] Kahn (B.), Yamazaki (T.).— Voevodsky’s motives and Weil reciprocity, Duke Math. J. 162, p. 2751-2796 (2013). | MR 3127813 | Zbl pre06305788

[11] Kahn (B.), Sujatha (R.).— Birational motives, I (preliminary version), preprint, 2002, http://www.math.uiuc.edu/K-theory/0596/.

[12] Merkurjev (A. S.).— R-equivalence on three-dimensional tori and zero-cycles, Algebra Number Theory 2, p. 69-89 (2008). | MR 2377363 | Zbl 1210.19005

[13] Merkurjev (A. S.).— Zero-cycles on algebraic tori, in The geometry of algebraic cycles, 119-122, Clay Math. Proc., 9, Amer. Math. Soc., Providence, RI (2010). | MR 2648668 | Zbl 1221.14008

[14] Morel (F.), Voevodsky (V.).— A 1 -homotopy theory of schemes, Publ. Math. IHÉS 90, p. 45-143 (1999). | Numdam | MR 1813224 | Zbl 0983.14007

[15] Mumford (D.).— Abelian varieties (corrected reprint), TIFR - Hindustan Book Agency (2008). | MR 2514037 | Zbl 1177.14001

[16] Serre J.-P..— Morphismes universels et variétés d’Albanese, in Exposés de séminaires, 1950-1989, Doc. mathématiques 1, SMF, p. 141-160 (2001). | Numdam | MR 160229 | Zbl 0123.13903

[17] Spiess (M.), T. Szamuely (T.).— On the Albanese map for smooth quasi-projective varieties, Math. Ann. 325, p. 1-17 (2003). | MR 1957261 | Zbl 1077.14026

[18] Voevodsky (V.).— Cohomological theory of presheaves with transfers, in E. Friedlander, A. Suslin, Voevodsky (V.) Cycles, transfers and motivic cohomology theories, Ann. Math. Studies 143, Princeton University Press, 88-137 (2000). | MR 1764200 | Zbl 1019.14010

[19] Voevodsky (V.).— Triangulated categories of motives over a field, in E. Friedlander, A. Suslin, Voevodsky (V.) Cycles, transfers and motivic cohomology theories, Ann. Math. Studies 143, Princeton University Press, p. 188-238 (2000). | MR 1764202 | Zbl 1019.14009