Duke Mathematical Journal

The negative K-theory of normal surfaces

Charles Weibel

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We relate the negative $K$-theory of a normal surface to a resolution of singularities. The only nonzero $K$-groups are $K\sb {-2}$, which counts loops in the exceptional fiber, and $K\sb {-1}$, which is related to the divisor class groups of the complete local rings at the singularities. We also verify two conjectures of Srinivas about $K\sb 0$-regularity and $K\sb {-1}$ of a surface.

Article information

Duke Math. J. Volume 108, Number 1 (2001), 1-35.

First available in Project Euclid: 5 August 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14C35: Applications of methods of algebraic $K$-theory [See also 19Exx]
Secondary: 13C20: Class groups [See also 11R29] 19E08: $K$-theory of schemes [See also 14C35]


Weibel, Charles. The negative K -theory of normal surfaces. Duke Math. J. 108 (2001), no. 1, 1--35. doi:10.1215/S0012-7094-01-10811-9. https://projecteuclid.org/euclid.dmj/1091737123.

Export citation


  • M. Artin, Some numerical criteria for contractibility of curves on algebraic surfaces, Amer. J. Math. 84 (1962), 485--. \lccP. du Val, On isolated singularities of surfaces which do not affect the conditions of adjunction, I, Proc. Camb. Phil. Soc. 30 (1934), 453--459.
  • A. Durfee, Fifteen characterizations of rational double points and simple critical points, Enseign. Math. (2) 25 (1979), 131--163. MR 80m:14003
  • A. Geramita and C. Weibel, On the Cohen-Macaulay and Buchsbaum property for unions of planes in affine space, J. Algebra 92 (1985), 413--445. MR 86f:4032
  • A. Grothendieck, ``Le groupe de Brauer'' in Dix exposés sur la cohomologie des schémas, North-Holland, Amsterdam, 1968, 46--188. MR 39:5586a, MR 39:5586b, MR 39:5588c
  • A. Grothendieck and J. Dieudonné, Éléments de Géométrie Algébrique, III: Étude cohomologique de faisceaux cohérents, I, Inst. Hautes Études Sci. Publ. Math. 11 (1961); II, 17 (1963). MR 29:1209, MR 29:1210
  • R. Hartshorne, Algebraic Geometry, Grad. Texts in Math. 52, Springer, New York, 1977. MR 57:3116
  • J. Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 195--279. MR 43:1986
  • --. --. --. --., Desingularization of two-dimensional schemes, Ann. of Math. (2) 107 (1978), 151--207. MR 58:10924
  • H. Matsumura, Commutative Ring Theory, Cambridge Stud. Adv. Math. 8, Cambridge Univ. Press, Cambridge, 1986. MR 88h:13001
  • N. Mohan Kumar, Rational double points on a rational surface, Invent. Math. 65 (1981/82), 251--268. MR 83g:14016
  • D. Mumford, The topology of normal singularities of an algebraic surface and a criterion for simplicity, Inst. Hautes Études Sci. Publ. Math. 9 (1961), 5--22. MR 27:3643
  • M. P. Murthy, Vector bundles over affine surfaces birationally equivalent to a ruled surface, Ann. of Math. (2) 89 (1969), 242--253. MR 39:2774
  • M. P. Murthy and C. Pedrini, ``$K_0$ and $K_1$ of polynomial rings'' in Algebraic $K$-Theory, II: ``Classical'' Algebraic $K$\!-Theory and Connections with Arithmetic (Seattle, 1972), Lecture Notes in Math. 342, Springer, Berlin, 1973, 109--121. MR 51:12829
  • M. P. Murthy and R. G. Swan, Vector bundles over affine surfaces, Invent. Math. 36 (1976), 125--165. MR 55:12724
  • D. Northcott and D. Rees, A Note on reductions of ideals with an application to the generalized Hilbert function, Proc. Cambridge Philos. Soc. 50 (1954), 353--359. MR 15:929e
  • C. Pedrini and C. Weibel, ``$K$-theory and Chow groups on singular varieties'' in Applications of Algebraic $K$-Theory to Algebraic Geometry and Number Theory, I, II (Boulder, 1983), Contemp. Math. 55, Amer. Math. Soc., Providence, 1986, 339--370. MR 88a:14017
  • --. --. --. --., ``Divisibility in the Chow group of zero-cycles on a singular surface'' in $K$\!-Theory (Strasbourg, 1992), Astérisque 226, Soc. Math. France, Montrouge, 1994, 10--11., 371--409. MR 96a:14011
  • L. Reid, $N$\!-dimensional rings with an isolated singular point having nonzero $K_-N$, $K$\!-Theory 1 (1987), 197--205. MR 88i:13020
  • L. Roberts, ``The $K$\!-theory of some reducible affine curves: A combinatorial approach'' in Algebraic $K$\!-Theory (Evanston, 1976), Lecture Notes in Math. 551, Springer, Berlin, 1976, 44--59. MR 58:5669
  • P. Salmon, Su un problema posto da P. Samuel, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 40 (1966), 801--803. MR 34:7559
  • V. Srinivas, Vector bundles on the cone over a curve, Compositio Math. 47 (1982), 249--269. MR 84e:14017
  • --. --. --. --., Grothendieck groups of polynomial and Laurent polynomial rings, Duke Math. J. 53 (1986), 595--633. MR 88a:14018
  • --. --. --. --., Modules of finite length and Chow groups of surfaces with rational double points, Illinois J. Math. 31 (1987), 36--61. MR 88b:14007
  • R. Swan, Projective modules over Laurent polynomial rings, Trans. Amer. Math. Soc. 237 (1978), 111--120. MR 57:9686
  • R. W. Thomason, Les $K$\!-groupes d'un schéma éclaté et une formule d'intersection excédentaire, Invent. Math. 112 (1993), 195--215. MR 93k:19005
  • R. W. Thomason and T. Trobaugh, ``Higher algebraic $K$\!-theory of schemes and of derived categories'' in The Grothendieck Festschrift, III, Progr. Math. 88, Birkhäuser, Boston, 1990, 247--435. MR 92f:19001
  • R. Varley, Vector bundles on cones over projective schemes, Invent. Math. 62 (1980/81), 15--22. MR 82j:14015
  • W. Vasconcelos, Arithmetic of Blowup Algebras, London Math. Soc. Lecture Note Ser. 195, Cambridge Univ. Press, Cambridge, 1994. MR 95g:13005
  • T. Vorst, Localization of the $K$\!-theory of polynomial extensions, Math. Ann. 244 (1979), 33--53. MR 80k:18016
  • --. --. --. --., Polynomial extensions and excision for $K_1$, Math. Ann. 244 (1979), 193--204. MR 82a:13002
  • C. Weibel, $K$-theory and analytic isomorphisms, Invent. Math. 61 (1980), 177--197. MR 83b:13011
  • --. --. --. --., $K_2$, $K_3$ and nilpotent ideals, J. Pure Appl. Algebra 18 (1980), 333--345. MR 82i:18016
  • --. --. --. --., Negative $K$\!-theory of varieties with isolated singularities, J. Pure Appl. Algebra 34 (1984), 331--342. MR 86d:14015
  • --. --. --. --., ``Homotopy algebraic $K$-theory'' in Algebraic $K$\!-Theory and Algebraic Number Theory (Honolulu, 1987), Contemp. Math. 83, Amer. Math. Soc., Providence, 1989, 461--488. MR 90d:18006
  • --. --. --. --., Pic is a contracted functor, Invent. Math. 103 (1991), 351--377. MR 92c:19002
  • --------, An introduction to homological algebra, Cambridge Stud. Adv. Math. 38, Cambridge Univ. Press, Cambridge, 1994. MR 95f:18001
  • O. Zariski and P. Samuel, Commutative Algebra, II, Van Nostrand, Princeton, 1960. MR 22:11006