Geodesic
On the cohomology of reciprocity sheaves
Forum of Mathematics, Sigma, Tome 10 (2022) Cet article a éte moissonné depuis la source Cambridge University Press

Voir la notice de l'article

In this paper, we show the existence of an action of Chow correspondences on the cohomology of reciprocity sheaves. In order to do so, we prove a number of structural results, such as a projective bundle formula, a blow-up formula, a Gysin sequence and the existence of proper pushforward. In this way, we recover and generalise analogous statements for the cohomology of Hodge sheaves and Hodge-Witt sheaves.We give several applications of the general theory to problems which have been classically studied. Among these applications, we construct new birational invariants of smooth projective varieties and obstructions to the existence of zero cycles of degree 1 from the cohomology of reciprocity sheaves. 
@article{10_1017_fms_2022_51,
     author = {Federico Binda and Kay R\"ulling and Shuji Saito},
     title = {On the cohomology of reciprocity sheaves},
     journal = {Forum of Mathematics, Sigma},
     year = {2022},
     volume = {10},
     doi = {10.1017/fms.2022.51},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2022.51/}
}
TY  - JOUR
AU  - Federico Binda
AU  - Kay Rülling
AU  - Shuji Saito
TI  - On the cohomology of reciprocity sheaves
JO  - Forum of Mathematics, Sigma
PY  - 2022
VL  - 10
UR  - http://geodesic.mathdoc.fr/articles/10.1017/fms.2022.51/
DO  - 10.1017/fms.2022.51
LA  - en
ID  - 10_1017_fms_2022_51
ER  - 
%0 Journal Article
%A Federico Binda
%A Kay Rülling
%A Shuji Saito
%T On the cohomology of reciprocity sheaves
%J Forum of Mathematics, Sigma
%D 2022
%V 10
%U http://geodesic.mathdoc.fr/articles/10.1017/fms.2022.51/
%R 10.1017/fms.2022.51
%G en
%F 10_1017_fms_2022_51
Federico Binda; Kay Rülling; Shuji Saito. On the cohomology of reciprocity sheaves. Forum of Mathematics, Sigma, Tome 10 (2022). doi: 10.1017/fms.2022.51

[ABBvB19] Auel, A., Bigazzi, A., C. Böhning and H.-C. Graf von Bothmer, ‘Universal triviality of the Chow group of 0-cycles and the Brauer group’, Int. Math. Res. Not. IMRN (2019). Google Scholar

[AGV72] Artin, M., Grothendieck, A. and Verdier, J. L., Séminaire de géométrie algébrique du Bois-Marie 1963–1964. Théorie des topos et cohomologie étale des schémas. (SGA 4). Un séminaire dirigé par M. Artin, A. Grothendieck, J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne, B. Saint-Donat. Tome 1: Théorie des topos. Exposés I à IV. 2e éd., vol. 269, (Springer, Cham, 1972). Google Scholar

[Ayo07a] Ayoub, J., ‘Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique. I’, Astérisque no. 314 (2007), x+466 pp. MR 2423375. Google Scholar

[Ayo07b] Ayoub, J., ‘Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique. II’, Astérisque no. 315 (2007), vi+364 pp. MR 2438151. Google Scholar

[BG71] Bloch, S. and Gieseker, D., ‘The positivity of the Chern classes of an ample vector bundle’, Invent. Math. 12 (1971), 112–117. Google Scholar | DOI

[BK86] Bloch, S. and Kato, K., ‘-adic étale cohomology’, Inst. Hautes Études Sci. Publ. Math. no. 63 (1986), 107–152. Google Scholar | DOI

[BLR90] Bosch, S., Lütkebohmert, W. and Raynaud, M., Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] vol. 21 (Springer-Verlag, Berlin, 1990). Google Scholar

[BPØ22] Binda, F., Park, D. and Østvær, P. A., ‘Triangulated categories of logarithmic motives over a field’, Astérisque 433 (2022), pp. 280. doi: . Google Scholar | arXiv

[BS83] Bloch, S. and Srinivas, V., ‘Remarks on correspondences and algebraic cycles’, Amer. J. Math. 105(5) (1983), 1235–1253. Google Scholar | DOI

[CD19] Cisinski, D.-C. and Déglise, F., ‘Triangulated categories of mixed motives’, in Springer Monographs in Mathematics (Springer, Cham, 2019)© 2019. MR 3971240. Google Scholar

[CL17] Chatzistamatiou, A. and Levine, M., ‘Torsion orders of complete intersections’, Algebra Number Theory 11(8) (2017), 1779–1835. Google Scholar | DOI

[Con00] Conrad, B., ‘Grothendieck duality and base change’, in Lecture Notes in Mathematics vol. 1750 (Springer-Verlag, Berlin, 2000). Google Scholar | DOI

[CR11] Chatzistamatiou, A. and Rülling, K., ‘Higher direct images of the structure sheaf in positive characteristic’, Algebra Number Theory 5(6) (2011), 693–775. Google Scholar | DOI

[CR12] Chatzistamatiou, A. and Rülling, K., ‘Hodge-Witt cohomology and Witt-rational singularities’, Doc. Math. 17 (2012), 663–781. Google Scholar

[CT96] Colliot-Thélène, J.-L., ‘Groupes linéaires sur les corps de fonctions de courbes réelles,’ J. Reine Angew. Math. 474 (1996), 139–167. MR 1390694. Google Scholar

[CT99] Colliot-Thélène, J.-L., ‘Conjectures de type local-global sur l’image des groupes de Chow dans la cohomologie étale’, in Algebraic -Theory (Seattle, WA, 1997), Proc. Sympos. Pure Math., vol. 67 (Amer. Math. Soc. Providence, RI, 1999), pp. 1–12. MR 1743234. Google Scholar

[CTG04] Colliot-Thélène, J.-L. and Gille, P., ‘Remarques sur l’approximation faible sur un corps de fonctions d’une variable, Arithmetic of higher-dimensional algebraic varieties’ (Palo Alto, CA, 2002), in Progr. Math. vol. 226 (Birkhäuser Boston, Boston, MA, 2004), pp. 121–134. MR 2029865. Google Scholar

[CTH15] Colliot-Thélène, J.-L. and Harari, D., ‘Dualité et principe local-global pour les tores sur une courbe au-dessus de , Proc. Lond. Math. Soc. (3) 110(6) (2015), 1475–1516. MR 3356812. Google Scholar

[CTHK97] Colliot-Thélène, J.-L., Hoobler, R. T. and Kahn, B., ‘The Bloch-Ogus-Gabber theorem’, in Algebraic -Theory (Toronto, ON), Fields Inst. Commun. vol. 16 (Amer. Math. Soc., Providence, RI, 1997), pp. 31–94. Google Scholar

[CTP16] Colliot-Thélène, J.-L. and Pirutka, A., ‘Hypersurfaces quartiques de dimension 3: non-rationalité stable’, Ann. Sci. Éc. Norm. Supér. (4) 49(2) (2016), 371–397. Google Scholar | DOI

[CTPS12] Colliot-Thélène, J.-L., Parimala, R. and Suresh, V., ‘Patching and local-global principles for homogeneous spaces over function fields of -adic curves’, Comment. Math. Helv. 87(4) (2012), 1011–1033. MR 2984579. Google Scholar | DOI

[CTS20] Colliot-Thélène, J.-L. and Skorobogatov, A. N., ‘The Brauer–Grothendieck Group’, Preprint, 2020, https://www.imo.universite-paris-saclay.fr/~colliot/BGgroup_book.pdf. Google Scholar | DOI

[CTSS83] Colliot-Thélène, J.-L., Sansuc, J.-J. and Soulé, C., ‘Torsion dans le groupe de Chow de codimension deux’, Duke Math. J. 50(3) (1983), 763–801. Google Scholar | DOI

[CTV12] Colliot-Thélène, J.-L. and Voisin, C., ‘Cohomologie non ramifiée et conjecture de Hodge entière’, Duke Math. J. 161(5) (2012), 735–801. Google Scholar | DOI

[Dég08] Déglise, F., ‘Around the Gysin triangle. II’, Doc. Math. 13 (2008), 613–675. MR 2466188. Google Scholar

[Dég12] Déglise, F., ‘Around the Gysin triangle I’, Regulators, in Contemp. Math. vol. 571 (Amer. Math. Soc., Providence, RI, 2012), pp. 77–116. MR 2953410. Google Scholar

[DJK18] Déglise, F., Jin, F. and Khan, A. A., ‘ Fundamental classes in motivic homotopy theory’, J. Eur. Math. Soc. 23(12) (2021), 3935–3993. Google Scholar

[Duc98a] Ducros, A., ‘Fibrations en variétés de Severi-Brauer au-dessus de la droite projective sur le corps des fonctions d’une courbe réelle’, C. R. Acad. Sci. Paris Sér. I Math. 327(1) (1998), 71–75. MR 1650192. Google Scholar | DOI

[Duc98b] Ducros, A., ‘L’obstruction de réciprocité à l’existence de points rationnels pour certaines variétés sur le corps des fonctions d’une courbe réelle’, J. Reine Angew. Math. 504 (1998), 73–114. MR 1656814. Google Scholar | DOI

[Elk73] Elkik, R., ‘Solutions d’équations à coefficients dans un anneau hensélien’, Ann. Sci. École Norm. Sup. (4) 6 (1973), 553–603. Google Scholar | DOI

[Ful98] Fulton, W., ‘Intersection theory’, in Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], second edn., vol. 2 (Springer-Verlag, Berlin, 1998). Google Scholar

[GL00] Geisser, T. and Levine, M., ‘The -theory of fields in characteristic , Invent. Math. 139(3) (2000), 459–493. Google Scholar | DOI

[GL01] Geisser, T. and Levine, M., ‘The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky’, J. Reine Angew. Math. 530 (2001), 55–103. Google Scholar

[God73] Godement, R., Topologie algébrique et théorie des faisceaux, Hermann, Paris, Troisième édition revue et corrigée, (Publications de l’Institut de Mathématique de l’Université de Strasbourg, XIII, Actualités Scientifiques et Industrielles, Paris, 1973) no. 1252. Google Scholar

[Gro61] Grothendieck, A., ‘Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes’, Inst. Hautes Études Sci. Publ. Math. no. 8 (1961), 222. Google Scholar

[Gro65] Grothendieck, A., ‘Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II’, Inst. Hautes Études Sci. Publ. Math. no. 24 (1965), 231. Google Scholar

[Gro68] Grothendieck, A., ‘Le groupe de Brauer. III. Exemples et compléments, Dix exposés sur la cohomologie des schémas’, in Adv. Stud. Pure Math. vol. 3, (North-Holland, Amsterdam, 1968), pp. 88–188. Google Scholar

[Gro85] Gros, M., ‘Classes de Chern et classes de cycles en cohomologie de Hodge-Witt logarithmique’, Mém. Soc. Math. France (N.S.) no. 21 (1985), 87. MR 844488. Google Scholar

[Har66] Hartshorne, R., ‘Residues and duality, lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64’, in Lecture Notes in Mathematics. With an appendix by P. Deligne, no. 20 (Springer-Verlag, Berlin-New York, 1966). Google Scholar

[HS16] Harari, D. and Szamuely, T., ‘Local-global questions for tori over -adic function fields’, J. Algebraic Geom. 25(3) (2016), 571–605. MR 3493592. Google Scholar | DOI

[HSS15] Harari, D., Scheiderer, C. and Szamuely, T., ‘Weak approximation for tori over -adic function fields’, Int. Math. Res. Not. IMRN 10 (2015), 2751–2783. MR 3352255. Google Scholar

[Ill79] Illusie, L., ‘Complexe de de Rham-Witt et cohomologie cristalline’, Ann. Sci. École Norm. Sup. (4) 12(4) (1979), 501–661. Google Scholar | DOI

[IR83] Illusie, L. and Raynaud, M., ‘Les suites spectrales associées au complexe de de Rham-Witt, Institut des Hautes Études Scientifiques’, Publications Mathématiques 57 (1983), 73–212. Google Scholar | DOI

[IR17] Ivorra, F. and Rülling, K., ‘K-groups of reciprocity functors’, J. Algebraic Geom. 26(2) (2017), 199–278. Google Scholar | DOI

[IT14] Illusie, L. and Temkin, M., ‘Exposé X. Gabber’s modification theorem (log smooth case)’, Astérisque no. 363--364 (2014), p. 652, Travaux de Gabber sur l’uniformisation locale et la cohomologie étale des schémas quasi-excellents (Société Mathématique de France, 2014), pp. 167–212. Google Scholar

[Izq15] Izquierdo, D., ‘Principe local-global pour les corps de fonctions sur des corps locaux supérieurs I’, J. Number Theory 157 (2015), 250–270. MR 3373241. Google Scholar | DOI

[Jou83] Jouanolou, J.-P., Théorèmes de Bertini et Applications, Progress in Mathematics, vol. 42, (Birkhäuser Boston, Inc., Boston, MA, 1983). Google Scholar

[Ker10] Kerz, M., ‘Milnor -theory of local rings with finite residue fields’, J. Algebraic Geom. 19(I) (2010), 173–191. Google Scholar | DOI

[KK86] Kato, K. and Kuzumaki, T., ‘The dimension of fields and algebraic -theory ’, J. Number Theory 24(2) (1986), 229–244. MR 863657. Google Scholar | DOI

[KMSY20] Kahn, B., Miyazaki, H., Saito, S. and Yamazaki, T., ‘Motives with modulus, III: The categories of motives ’, Ann. K-Theory 7(1) (2022), 119–178. . Google Scholar | DOI

[KMSY21a] Kahn, B., Miyazaki, H., Saito, S. and Yamazaki, T., ‘Motives with modulus, I: Modulus sheaves with transfers for non-proper modulus pairs’, Épijournal Géom. Algébrique 5 (2021), 1–46. Google Scholar

[KMSY21b] Kahn, B., Miyazaki, H., Saito, S. and Yamazaki, T., ‘Motives with modulus, II: Modulus sheaves with transfers for proper modulus pairs’, Épijournal Géom. Algébrique 5 (2021), 1–31. Google Scholar

[Kol96] Kollár, J., ‘Rational curves on algebraic varieties’, in Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 32 (Springer-Verlag, Berlin, 1996). Google Scholar

[Kov17] Kovács, S., ‘Rational singularities’, Preprint, 2017, . Google Scholar | arXiv

[KS20] Kelly, S. and Saito, S., ‘Smooth blowup square for motives with modulus’, Bulletin Polish Acad. Sci. Math. (2020), to appear, . Google Scholar | arXiv

[KSY22] Kahn, B., Saito, S. and Yamazaki, T., ‘Reciprocity sheaves, II’, Homol. Homotopy Appl. 24(1) (2022), 71–91. doi: . Google Scholar | arXiv | DOI

[KSY16] Kahn, B., Saito, S. and Yamazaki, T., ‘Reciprocity sheaves’, Compos. Math. 152(9) (2016), 1851–1898, With two appendices by Kay Rülling. Google Scholar | DOI

[LW09] Lecomte, F. and Wach, N., ‘Le complexe motivique de Rham ’, Manuscripta Math. 129(1) (2009), 75–90. Google Scholar | DOI

[MS20] Merici, A. and Saito, S., ‘Cancellation theorems for reciprocity sheaves’, Algebraic Geometry, to appear, Preprint 2020, . Google Scholar | arXiv

[MVW06] Mazza, C., Voevodsky, V. and Weibel, C., ‘Lecture notes on motivic cohomology’, Clay Mathematics Monographs, vol. 2 (American Mathematical Society, Providence, RI, 2006). Google Scholar

[Mat22] Matsumoto, K., ‘Gysin triangles in the category of motifs with modulus’, J. Inst. Math. Jussieu (2022), 1–24. doi: . Google Scholar | arXiv | DOI

[Nee96] Neeman, A., ‘The Grothendieck duality theorem via Bousfield’s techniques and Brown representability’, J. Amer. Math. Soc. 9(1) (1996), 205–236. MR 1308405. Google Scholar

[Nee01] Neeman, A., ‘Triangulated categories’, in Annals of Mathematics Studies, vol. 148, (Princeton University Press, Princeton, Oxford, 2001). Google Scholar

[OS20] Ottem, J. C. and Suzuki, F., ‘An -acyclic variety of even index’, with an appendix by O. Wittenberg, Preprint, 2020, . Google Scholar | arXiv

[Ota20] Otabe, S., ‘On the mod unramified cohomology of varieties having universally trivial Chow group of zero-cycles’, Preprint, 2020, . Google Scholar | arXiv

[Pir12] Pirutka, A., ‘Invariants birationnels dans la suite spectrale de Bloch-Ogus’, J. -Theory 10(3) (2012), 565–582. Google Scholar | DOI

[Pop86] Popescu, D., ‘General Néron desingularization and approximation’, Nagoya Math. J. 104 (1986), 85–115. Google Scholar | DOI

[PS20] Pál, A. and Szabó, E., ‘The fibration method over real function fields’, Math. Ann. 378(3--4) (2020), 993–1019. MR 4163520. Google Scholar | DOI

[Ros96] Rost, M., ‘Chow groups with coefficients’, Doc. Math. 1(16) (1996), 319–393. Google Scholar

[RS21] Rülling, K. and Saito, S., ‘Reciprocity sheaves and their ramification filtration’, J. Inst. Math. Jussieu (2021), 1–74. Google Scholar | DOI

[RSY22] Rülling, K., Sugiyama, R. and Yamazaki, T., ‘Tensor structures in the theory of modulus presheaves with transfers’, Math. Z. 300(1) (2022), 929–977. Google Scholar | DOI

[RY16] Rülling, K. and Yamazaki, T., ‘Suslin homology of relative curves with modulus’, J. Lond. Math. Soc. (2) 93(3) (2016), 567–589. Google Scholar | DOI

[Sai89] Saito, S., ‘Some observations on motivic cohomology of arithmetic schemes’, Invent. Math. 98(2) (1989), 371–404. MR 1016270. Google Scholar | DOI

[Sai20a] Saito, S., ‘Purity of reciprocity sheaves’, Adv. Math. 366 (2020), 107067, 70. MR 4070301. Google Scholar | DOI

[Sai20b] Saito, S., ‘Reciprocity sheaves and logarithmic motives’, Preprint, 2020, . Google Scholar | arXiv

[Sch17] Schröer, S., ‘Geometry on totally separably closed schemes’, Algebra Number Theory 11(3) (2017), 537–582. Google Scholar | DOI

[Ser84] Serre, J.-P., Groupes algébriques et corps de classes, vol. 7, second edn, (Publications de l’Institut Mathématique de l’Université de Nancago [Publications of the Mathematical Institute of the University of Nancago], Hermann, Paris, 1984), Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], 1264. MR 907288. Google Scholar

[Ser94] Serre, J.-P., ‘Cohomologie galoisienne’, in Lecture Notes in Mathematics vol. 5, fifth edn, (Springer-Verlag, Berlin, 1994). Google Scholar

[Sta19] The Stacks Project Authors, Stacks Project, 2019, https://stacks.math.columbia.edu. Google Scholar

[SV00] Suslin, A. and Voevodsky, V., ‘Bloch-Kato conjecture and motivic cohomology with finite coefficients’, in The Arithmetic and Geometry of Algebraic Cycles (Banff, AB, 1998), NATO Sci. Ser. C Math. Phys. Sci., vol. 548 (Kluwer Acad. Publ., Dordrecht, 2000), pp. 117–189. Google Scholar | DOI

[Tot16] Totaro, B., ‘ Hypersurfaces that are not stably rational’, J. Amer. Math. Soc. 29(3) (2016), 883–891. MR 3486175. Google Scholar | DOI

[TT90] Thomason, R. W. and Trobaugh, T., ‘Higher algebraic -theory of schemes and of derived categories’, The Grothendieck Festschrift, vol. III, Progr. Math., vol. 88 (Birkhäuser Boston, Boston, MA, 1990), pp. 247–435. Google Scholar | DOI

[Voe00a] Voevodsky, V., ‘Cohomological theory of presheaves with transfers’, Cycles, transfers, and motivic homology theories, Ann. of Math. Stud., vol. 143 (Princeton Univ. Press, Princeton, NJ, 2000), pp. 87–137. Google Scholar

[Voe00b] Voevodsky, V., ‘Triangulated categories of motives over a field’, Cycles, transfers, and motivic homology theories, Ann. of Math. Stud., vol. 143 (Princeton Univ. Press, Princeton, NJ, 2000), pp. 188–238. Google Scholar

[Voe11] Voevodsky, V., ‘On motivic cohomology with -coefficients’, Ann. of Math. (2) 174(1) (2011), 401–438. Google Scholar | DOI

[Wei95] Weil, A., ‘Basic number theory’, Classics in Mathematics (Springer-Verlag, Berlin, 1995), reprint of the second (1973) edition. MR 1344916. Google Scholar

[Wit12] Wittenberg, O., ‘Zéro-cycles sur les fibrations au-dessus d’une courbe de genre quelconque’, Duke Math. J. 161(11) (2012), 2113–2166. MR 2957699. Google Scholar | DOI

Cité par Sources :