Finiteness theorems for algebraic cycles of small codimension on quadric fibrations over curves
Abstract
We obtain finiteness theorems for algebraic cycles of small codimension on quadric fibrations over curves over perfect fields. For example, if k is finitely generated over ℚ and X → C is a quadric fibration of odd relative dimension at least 11, then CH i(X) is finitely generated for i ≤ 4.
-
[1] Colliot-Thélène J.-L, Skorobogatov A., Groupe de Chow des zéro-cycles sur les fibrés en quadriques, K-Theory, 1993, 7, 477–500 http://dx.doi.org/10.1007/BF00961538
-
[2] Conrad B., Chows K/k-image and K/k-trace, and the Lang-Néron theorem, Enseign. Math. (2), 2006, 52, 37–108
-
[3] Fulton W., Intersection theory, Second Ed., Springer-Verlag, 1998
-
[4] González-Avilés C, Algebraic cycles on Severi-Brauer schemes of prime degree over a curve, Math. Res. Lett., 2008, 15(1), 51–56
-
[5] Gros M., 0-cycles de degré zéro sur les surfaces fibrées en coniques, J. Reine Angew. Math., 1987, 373, 166–184
-
[6] Kahn B., Rost M., Sujatha R, Unramified cohomology of quadrics I, Amer. Math. J., 1998, 120(4), 841–891 http://dx.doi.org/10.1353/ajm.1998.0029
-
[7] Karpenko N., Algebro-geometric invariants of quadratic forms, Leningrad Math. J., 1991, 2(1), 119–138
-
[8] Karpenko N., Chow groups of quadrics and the stabilization conjecture, Adv. Soviet Math., 1991, 4, 3–8
-
[9] Karpenko N., Chow groups of quadrics and index reduction formulas, Nova J. Algebra Geom., 1995, 3(4), 357–379
-
[10] Karpenko N., Order of torsion in CH 4 of quadrics, Doc. Math., 1996, 1, 57–65
-
[11] Karpenko N., Merkurjev A., Chow groups of projective quadrics, Leningrad Math. J., 1991, 2(3), 655–671
-
[12] Karpenko N., Merkurjev A., Rost projectors and Steenrod operations, Doc. Math., 2002, 7, 481–493
-
[13] Lam T.Y., The algebraic theory of quadratic forms, W.A. Benjamin, Inc. Reading, Massachussettss, 1973
-
[14] Milne J.S., Arithmetic Duality Theorems, Perspectives in Mathematics, vol. 1, Academic Press Inc., Orlando 1986
-
[15] Parimala R., Suresh V., Zero-cycles on quadric fibrations: Finiteness theorems and the cycle map, Invent. Math., 1995, 122, 83–117 http://dx.doi.org/10.1007/BF01231440
-
[16] Rost M., Chow groups with coefficients, Doc. Math., 1996, 1, 319–393
-
[17] Sherman C., Some theorems on the K-Theory of coherent sheaves, Comm. Algebra, 1979, 7(14), 1489–1508 http://dx.doi.org/10.1080/00927877908822414
-
[18] Swan R., Zero-cycles on quadric hypersurfaces, Proc. Amer. Math. Soc., 1989, 107, 43–46 http://dx.doi.org/10.2307/2048032
-
[19] Weibel C., An introduction to homological algebra, Cambridge Stud. Adv. Math., Cambridge Univ. Press, 1994, 38
