Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

An Artin-Rees theorem in $ K$-theory and applications to zero cycles


Author: Amalendu Krishna
Journal: J. Algebraic Geom. 19 (2010), 555-598
DOI: https://doi.org/10.1090/S1056-3911-09-00521-9
Published electronically: November 17, 2009
MathSciNet review: 2629600
Full-text PDF

Abstract | References | Additional Information

Abstract: For the smooth normalization $ f : {\overline X} \to X$ of a singular variety $ X$ over a field $ k$ of characteristic zero, we show that for any conducting subscheme $ Y$ for the normalization, and for any $ i \in \mathbb{Z}$, the natural map $ K_i(X, {\overline X}, nY) \to K_i(X, {\overline X}, Y)$ is zero for all sufficiently large $ n$.

As an application, we prove a formula for the Chow group of zero cycles on a quasi-projective variety $ X$ over $ k$ with Cohen-Macaulay isolated singularities, in terms of an inverse limit of the relative Chow groups of a desingularization $ \widetilde X$ relative to the multiples of the exceptional divisor.

We use this formula to verify a conjecture of Srinivas about the Chow group of zero cycles on the affine cone over a smooth projective variety which is arithmetically Cohen-Macaulay.


References [Enhancements On Off] (What's this?)

  • 1. M. André, Homologie des algèbres commutatives, Die Grundlehren der mathematischen Wissenschaften, Band 206, Springer-Verlag, Berlin-New York, 1974. MR 0352220 (50:4707)
  • 2. L. Barbieri-Viale, Zero-cycles on singular varieties: torsion and Bloch's formula, J. Pure Appl. Alg., 78, (1992), 1-13. MR 1154894 (93b:14021)
  • 3. L. Barbieri-Viale, C. Pedrini, C. Weibel, Roitman's Theorem for singular Varieties, Duke Math. J., 84, (1996), 155-190. MR 1394751 (97c:14004)
  • 4. H. Bass, Algebraic $ K$-theory, W. A. Benjamin, Inc., New York-Amsterdam 1968. MR 0249491 (40:2736)
  • 5. J. Cathelineau, Lambda structures in algebraic $ K$-theory, $ K$-Theory, 4, (1991), 591-606.
  • 6. C. Consani, $ K$-theory of blow-ups and vector bundles on the cone over a surface, $ K$-Theory, 7, (1993), 269-284. MR 1244003 (94i:19005)
  • 7. K. Coombes, V. Srinivas, Relative $ K$-theory and vector bundles on certain singular varieties, Invent. Math. 70 (1982/83), no. 1, 1-12. MR 679769 (84e:14015)
  • 8. G. Cortinas, The Obstruction to Excision in $ K$-theory and in Cyclic Homology, Invent. Math., 164 (2006), no. 1, 143-173. MR 2207785 (2006k:19006)
  • 9. G. Cortinas, S. Geller, C. Weibel, Artinian Berger's Conjecture, Math. Zeitschrift, 228, (1998), 569-588. MR 1637796 (99g:13035)
  • 10. P. Elvaz-Vincent, S. Muller-Stach, Milnor $ K$-theory of rings, higher Chow groups and applications, Invent. Math., 148, (2002), 177-206. MR 1892848 (2003c:19001)
  • 11. H. Esnault, E. Viehweg, Lectures on Vanishing Theorems, DMV Seminar, 20, Birkhauser Verlag, Basel, 1992. MR 1193913 (94a:14017)
  • 12. H. Esnault, V. Srinivas, E. Viehweg, The Universal regular quotient of Chow group of points on projective varieties, Invent. Math., 135, (1999), 595-664. MR 1669284 (2000f:14012)
  • 13. M. Kerz, The Gersten Conjecture for Milnor $ K$-theory, $ K$-theory arxiv.
  • 14. A. Krishna, Zero cycles on a threefold with isolated singularities, J. Reine Angew. Math., 594 (2006), 93-115. MR 2248153 (2007m:14006)
  • 15. A. Krishna, On $ K\sb 2$ of one-dimensional local rings, $ K$-Theory, 35 (2005), no. 1-2, 139-158. MR 2240218 (2007h:19003)
  • 16. A. Krishna, V. Srinivas, Zero Cycles and $ K$-theory on Normal Surfaces, Ann. of Math., 156, (2002), 155-195. MR 1935844 (2003k:14005)
  • 17. M. Levine, Zero-Cycles and $ K$-theory on Singular Varieties, Proc. Symp. Pure Math., AMS Providence, 46, (1987), 451-462. MR 927992 (89h:14005)
  • 18. M. Levine, Relative Milnor $ K$-Theory, $ K$-Theory, 6, (1992), 113-175. MR 1187704 (94b:19005)
  • 19. M. Levine, C. Weibel, Zero cycles and complete intersections on singular varieties, 359, (1985), 106-120. MR 794801 (86k:14003)
  • 20. J-L Loday, Cyclic Homology, Grund. der math. Wissen. Series, 301, Springer Verlag, 1998. MR 1600246 (98h:16014)
  • 21. J. P. May, Simplicial objects in Algebraic topology, D. Van Nostrand Company, Princeton, New Jersey, 1967. MR 0222892 (36:5942)
  • 22. M. P. Murthy, Zero cycles and projective modules, Ann. of Math., 140, (1994), no. 2, 405-434. MR 1298718 (96g:13010)
  • 23. C. Pedrini, C. A. Weibel, Divisibility In The Chow Group Of Zero-Cycles On a Singular Surface, Asterisque, 226, (1994), 371-409. MR 1317125 (96a:14011)
  • 24. D. Quillen, On the (Co-)homology of commutative rings, Proc. Symp. Pure Math., AMS Providence, 17, (1970), 65-87. MR 0257068 (41:1722)
  • 25. A. Ramanathan, Schubert varieties are arithmetically Cohen-Macaulay, Invent. Math., 80, (1985), no. 2, 283-294. MR 788411 (87d:14044)
  • 26. C. Soulé, Opérations en $ K$-théorie algébrique, Canad. J. Math., 37, (1985), no. 3, 488-550.
  • 27. V. Srinivas, Zero cycles on a singular surface II, J. reine angew. Math., 362, (1985), 4-27. MR 809963 (87c:14009b)
  • 28. V. Srinivas, Rational Equivalence of 0-Cycles on Normal Varieties, Proc. Symp. Pure Math., AMS Providence, 46, (1987), 475-482. MR 927994 (89e:14006)
  • 29. J. Stienstra, Cartier-Dieudonne theory for Chow groups. J. Reine Angew. Math., 355, (1985), 1-66. MR 772483 (88c:14026a)
  • 30. R. W. Thomason and T. Trobaugh, Higher Algebraic K-Theory Of Schemes And Of Derived Categories, The Grothendieck Festschrift III, Progress in Math. 88, Birkhauser.
  • 31. C. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, Cambridge, 1994. MR 1269324 (95f:18001)
  • 32. C. Weibel, Negative $ K$-theory of normal surfaces, Duke Math. J., 108, (2001), 1-35. MR 1831819 (2002b:14012)


Additional Information

Amalendu Krishna
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai, 400005 India
Email: amal@math.tifr.res.in

DOI: https://doi.org/10.1090/S1056-3911-09-00521-9
Received by editor(s): May 31, 2008
Received by editor(s) in revised form: September 28, 2008
Published electronically: November 17, 2009

American Mathematical Society