×

Integral Chow motives of threefolds with \(K\)-motives of unit type. (English) Zbl 1398.14012

Summary: We prove that if a smooth projective algebraic variety of dimension less or equal to three has a unit type integral \(K\)-motive, then its integral Chow motive is of Lefschetz type. As a consequence, the integral Chow motive is of Lefschetz type for a smooth projective variety of dimension less or equal to three that admits a full exceptional collection.

MSC:

14C15 (Equivariant) Chow groups and rings; motives
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] M. Artin and B. Mazur, ´Etale homotopy, Lecture Notes in Mathematics, Vol. 100, Springer-Verlag, Berlin-New York, 1969. · Zbl 0182.26001
[2] M. Atiyah and F. Hirzebruch, Vector bundles and homogeneous spaces, Proc. Sympos. Pure Math., Vol. III, pp. 7-38, American Mathematical Society, Providence, R.I., 1961. · Zbl 0108.17705
[3] , The Riemann-Roch theorem for analytic embeddings, Topology 1 (1962), 151- 166. · Zbl 0108.36402
[4] A. A. Beilinson, Coherent sheaves on Pnand problems of linear algebra, Funct. Anal. Appl. 12 (1978), no. 3, 66-67.
[5] M. Bernardara and G. Tabuada, Relations between the Chow motive and the noncommutative motive of a smooth projective variety, J. Pure Appl. Algebra 219 (2015), no. 11, 5068-5077. 1848S. GORCHINSKIY · Zbl 1349.14016
[6] P. Berthelot, A. Grothendieck, and L. Illusie, Th´eorie des intersections et th´eor‘eme de Riemann-Roch, S´eminaire de G´eom´etrie Alg´ebrique du Bois-Marie 1966-1967 (SGA 6), Lecture Notes in Mathematics, Vol. 225, Springer-Verlag, Berlin-New York, 1971.
[7] S. Bloch and A. Ogus, Gersten’s conjecture and the homology of schemes, Ann. Sci. Ecole Norm. Sup. (4) 7 (1974), no. 2, 181-201.´ · Zbl 0307.14008
[8] K. S. Brown and S. M. Gersten, Algebraic K-theory as generalized sheaf cohomology, Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 266-292, Lecture Notes in Mathematics, Vol. 341, Springer, Berlin, 1973. · Zbl 0291.18017
[9] J.-L. Colliot-Th´el‘ene, Birational invariants, purity and the Gersten conjecture, Proc. Sympos. Pure Math., Vol. 58, Part I, pp. 1-64, American Mathematical Society, Providence, R.I., 1995. · Zbl 0834.14009
[10] W. G. Dwyer and E. M. Frienlander, Algebraic and etale K-theory, Trans. Amer. Math. Soc. 292 (1985), no. 1, 247-280. · Zbl 0581.14012
[11] A. V. Fonarev, On the Kuznetsov-Polishchuk conjecture, Proc. Steklov Inst. Math. 290 (2015), no. 1, 11-25. · Zbl 1376.14022
[12] E. Friedlander, Etale K-theory. I. Connections with etale cohomology and algebraic vector bundles, Invent. Math. 60 (1980), no. 2, 105-134. · Zbl 0519.14010
[13] , Etale K-theory. II. Connections with algebraic K-theory, Ann. Sci. ´Ecole Norm. Sup. (4) 15 (1982), no. 2, 231-256. · Zbl 0537.14011
[14] W. Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 2, Springer-Verlag, Berlin, 1984. · Zbl 0541.14005
[15] S. Galkin, L. Katzarkov, A. Mellit, and E. Shinder, Minifolds and phantoms, Adv. Math. 278 (2015), 238-253. · Zbl 1327.14081
[16] T. Geisser, Motivic cohomology, K-theory and topological cyclic homology, Handbook of K-theory. Vol. 1, 2, 193-234, Springer, Berlin, 2005. · Zbl 1113.14017
[17] H. Gillet, K-theory and intersection theory, Handbook of K-theory. Vol. 1, 2, 235-293, Springer, Berlin, 2005. · Zbl 1112.14009
[18] S. Gorchinskiy and V. Guletskii, Motives and representability of algebraic cycles on threefolds over a field, J. Algebraic Geom. 21 (2012), no. 2, 347-373. · Zbl 1256.14007
[19] S. Gorchinskiy and D. Orlov, Geometric phantom categories, Publ. Math. Inst. Hautes Etudes Sci. 117 (2013), 329-349.´ · Zbl 1285.14018
[20] J. P. C. Greenlees and J. P. May, Generalized Tate cohomology, Mem. Amer. Math. Soc. 113 (1995), no. 543, 178 pp. · Zbl 0876.55003
[21] M. Kapranov, On the derived categories of coherent sheaves on some homogeneous spaces, Invent. Math. 92 (1988), no. 3, 479-508. · Zbl 0651.18008
[22] A. Kuznetsov, On K¨uchle varieties with Picard number greater than 1, Izv. Math. 79 (2015), no. 4, 698-709. · Zbl 1342.14087
[23] , K¨uchle fivefolds of type c5, Math. Z. 284 (2016), no. 3, 1245-1278. · Zbl 1352.14029
[24] , Exceptional collections in surface-like categories, Mat. Sb. 208 (2017), no. 9, DOI:10.1070/SM8917. · Zbl 1398.14026
[25] A. Kuznetsov and A. Polishchuk, Exceptional collections on isotropic Grassmannians, J. Eur. Math. Soc. 18 (2016), no. 3, 507-574. · Zbl 1338.14021
[26] Ju. I. Manin, Correspondences, motifs and monoidal transformations, Mat. Sb. 6 (1968), no. 4, 475-507. · Zbl 0199.24803
[27] M. Marcolli and G. Tabuada, From exceptional collections to motivic decompositions via noncommutative motives, J. Reine Angew. Math. 701 (2015), 153-167. · Zbl 1349.14021
[28] A. S. Merkurjev and A. A. Suslin, K-cohomology of Severi-Brauer varieties and the norm residue homomorphism, Math. USSR-Izvestiya 21 (1983), no. 2, 307-340. · Zbl 0525.18008
[29] J. S. Milne, ´Etale cohomology, Princeton Mathematical Series, 33, Princeton University Press, Princeton, N.J., 1980. INTEGRAL CHOW MOTIVES OF THREEFOLDS1849 · Zbl 0433.14012
[30] D. O. Orlov, Derived categories of coherent sheaves and motives, Russian Math. Surveys 60 (2005), no. 6, 1242-1244. · Zbl 1146.18302
[31] , Geometric realizations of quiver algebras, Proc. Steklov Inst. Math. 290 (2015), no. 1, 70-83. · Zbl 1348.14050
[32] , Smooth and proper noncommutative schemes and gluing of DG categories, Adv. Math. 302 (2016), 59-105. · Zbl 1368.14031
[33] I. A. Panin, On the algebraic K-theory of twisted flag varieties, K-Theory 8 (1994), no. 6, 541-585. · Zbl 0854.19002
[34] M. Perling, Combinatorial aspects of exceptional sequences on (rational) surfaces, preprint, arXiv:1311.7349. · Zbl 1427.14041
[35] D. Quillen, Algebraic K-theory I, Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 85-147, Lecture Notes in Mathematics, Vol. 341, Springer, Berlin 1973. · Zbl 0292.18004
[36] C. Soul´e, K-th´eorie des anneaux d’entiers de corps de nombres et cohomologie ´etale, Invent. Math. 55 (1979), no. 3, 251-295. · Zbl 0437.12008
[37] G. Tabuada, Chow motives versus noncommutative motives, J. Noncommut. Geom. 7 (2013), no. 3, 767-786. · Zbl 1296.14019
[38] R. Thomason, Riemann-Roch for algebraic versus topological K-theory, J. Pure Appl. Algebra, 27 (1983), no. 1, 87-109. · Zbl 0545.14007
[39] , Algebraic K-theory and ´etale cohomology, Ann. Sci. ´Ecole Norm. Sup. (4) 18 (1985), no. 3, 437-552. · Zbl 0596.14012
[40] , Lefschetz-Riemann-Roch theorem and coherent trace formula, Invent. Math. 85 (1986), no. 3, 515-543. · Zbl 0653.14005
[41] C. Vial, Exceptional collections, and the N´eron-Severi lattice for surfaces, Adv. Math. 305 (2017), 895-934. · Zbl 1387.14061
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.