Abstract
In the present paper, we define a notion of numerical equivalence on Chow groups or Grothendieck groups of Noetherian local rings, which is an analogue of that on smooth projective varieties. Under a mild condition, it is proved that the Chow group modulo numerical equivalence is a finite dimensional ℚ-vector space, as in the case of smooth projective varieties. Numerical equivalence on local rings is deeply related to that on smooth projective varieties. For example, if Grothendieck’s standard conjectures are true, then a vanishing of Chow group (of local rings) modulo numerical equivalence can be proven. Using the theory of numerical equivalence, the notion of numerically Roberts rings is defined. It is proved that a Cohen–Macaulay local ring of positive characteristic is a numerically Roberts ring if and only if the Hilbert–Kunz multiplicity of a maximal primary ideal of finite projective dimension is always equal to its colength. Numerically Roberts rings satisfy the vanishing property of intersection multiplicities. We shall prove another special case of the vanishing of intersection multiplicities using a vanishing of localized Chern characters.
Similar content being viewed by others
References
Dutta, S.P., Hochster, M., MacLaughlin, J.E.: Modules of finite projective dimension with negative intersection multiplicities. Invent. Math. 79, 253–291 (1985)
Fulton, W.: Intersection Theory, 2nd edition. Berlin, New York: Springer 1997
Gillet, H., Soulé, C.: Intersection theory using Adams operation. Invent. Math. 90, 243–277 (1987)
Hartshorne, R.: Algebraic Geometry. Grad. Texts Math. 52. Berlin, New York: Springer 1977
Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic zero. Ann. Math. 79, 109–326 (1964)
Ishii, S., Milman, P.: The geometric minimal models of analitic spaces. Math. Ann. 323, 437–451 (2002)
de Jong, A.J.: Smoothness, semi-stability and alterations. Publ. Math., Inst. Hautes Étud. Sci. 83, 51–93 (1996)
Kamoi, Y., Kurano, K.: On maps of Grothendieck groups induced by completion. J. Algebra 254, 21–43 (2002)
Kurano, K.: An approach to the characteristic free Dutta multiplicities. J. Math. Soc. Jap. 45, 369–390 (1993)
Kurano, K.: A remark on the Riemann–Roch formula for affine schemes associated with Noetherian local rings. Tohoku Math. J., II. Ser. 48, 121–138 (1996)
Kurano, K.: Test modules to calculate Dutta Multiplicities. J. Algebra 236, 216–235 (2001)
Kurano, K.: On Roberts rings. J. Math. Soc. Jap. 53, 333–355 (2001)
Kurano, K.: Roberts rings and Dutta multiplicities. Geometric and combinatorial aspects of commutative algebra, pp. 273–287. Lect. Notes Pure Appl. Math. 217. New York: Marcel Dekker 2001
Kurano, K., Roberts, P.C.: Adams operations, localized Chern characters, and the positivity of Dutta multiplicity in characteristic 0. Trans. Am. Math. Soc. 352, 3103–3116 (2000)
Kurano, K., Singh, A.K.: Todd classes of affine cones of Grassmannians. Int. Math. Res. Not. 35, 1841–1855 (2002)
Levine, M.: Localization on singular varieties. Invent. Math. 91, 423–464 (1988)
Lewis, J.D.: A survey of the Hodge conjecture, 2nd edition. CRM Monogr. Ser. 10. Providence, RI: Am. Math. Soc. 1999
Lütkebohmert, W.: On compactification of schemes. Manuscr. Math. 80, 95–111 (1993)
Matsumura, H.: Commutative ring theory. Camb. Stud. Adv. Math. 8. Cambridge, New York: Cambridge Univ. Press 1989
Miller, C.M., Singh, A.K.: Intersection multiplicities over Gorenstein rings. Math. Ann 317, 155–171 (2000)
Monsky, P.: The Hilbert–Kunz function. Math. Ann. 263, 43–49 (1983)
Nagata, M.: Local Rings. Interscience Tracts in Pure and Appl. Math. New York: Wiley 1962
Nagata, M.: A generalization of the imbedding problem of an abstract variety in a complete variety. J. Math. Kyoto Univ. 3, 89–102 (1963)
Ogoma, T.: General Néron desingularization based on the idea of Popescu. J. Algebra 167, 57–84 (1994)
Popescu, D.: General Néron desingularization and approximation. Nagoya Math. J. 104, 85–115 (1986)
Roberts, P.C.: The vanishing of intersection multiplicities and perfect complexes. Bull. Am. Math. Soc. 13, 127–130 (1985)
Roberts, P.C.: Local Chern characters and intersection multiplicities. Proc. Symp. Pure Math. 46, 389–400 (1987)
Roberts, P.C.: MacRae invariant and the first local chern character. Trans. Am. Math. Soc. 300, 583–591 (1987)
Roberts, P.C., Srinivas, V.: Modules of finite length and finite projective dimension. Invent. Math. 151, 1–27 (2003)
Serre, J.-P.: Algèbre locale, Multiplicités. Lect. Notes Math. 11. Berlin, New York: Springer 1965
Srinivas, V.: Algebraic K-theory, Second edition. Prog. Math. 90. Boston, MA: Birkhäuser 1996
Swan, R.: K-theory of quadratic hypersurfaces. Ann. Math. 122, 113–154 (1985)
Thomason, R.W., Trobaugh, T.: Higher algebraic K-theory of schemes and of derived categories. The Grothendieck Festschrift, Vol. III, pp. 247–435, Prog. Math. 88. Boston, MA: Birkhäuser 1990
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kurano, K. Numerical equivalence defined on Chow groups of Noetherian local rings. Invent. math. 157, 575��619 (2004). https://doi.org/10.1007/s00222-004-0361-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00222-004-0361-8