Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

A mixed multiplicity formula for complete ideals in -dimensional rational singularities

Author(s): Veronique Van Lierde
Journal: Proc. Amer. Math. Soc. 138 (2010), 4197-4204.
MSC (2010): Primary 13B22, 13H15
Posted: June 29, 2010
MathSciNet review: 2680046
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: Let be a 2-dimensional rational singularity with algebraically closed residue field and for which the associated graded ring is an integrally closed domain. We use the work of Göhner on condition and the theory of degree functions developed by Rees and Sharp to give a very short and elementary proof of a formula for the (mixed) multiplicity of complete -primary ideals in .

References:

1.: S.D. Cutkosky, On unique and almost unique factorization of complete ideals, Amer. J. Math. 111 (1989) 417-433. MR 1002007 (90j:14016a)
2.: S.D. Cutkosky, On unique and almost unique factorization of complete ideals II, Invent. Math. 98 (1989) 59-74. MR 1010155 (90j:14016b)
3.: S.D. Cutkosky, A new characterization of rational surface singularities, Invent. Math. 102 (1990) 157-177. MR 1069245 (91h:14045)
4.: H. Göhner, Semifactoriality and Muhly's condition in two dimensional local rings, J. Algebra 34 (1975) 403-429. MR 0379489 (52:394)
5.: C. Huneke, Complete ideals in two-dimensional regular local rings. In: Commutative Algebra (Berkely, CA, 1987), Math. Sci. Res. Inst. Publ., Vol. 15, Springer, New York, 1989, pp. 325-338. MR 1015525 (90i:13020)
6.: C. Huneke, J. Sally, Birational extensions in dimension two and integrally closed ideals, J. Algebra 115 (1988) 481-500. MR 943272 (89e:13025)
7.: B. Johnston, The higher-dimensional multiplicity formula associated to the length formula of Hoskin and Deligne, Comm. in Algebra 22 (1994) 2057-2071. MR 1268543 (95d:13029)
8.: J. Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math. 36 (1969) 195-279. MR 0276239 (43:1986)
9.: J. Lipman, On complete ideals in regular local rings. In: Algebraic Geometry and Commutative Algebra, in honor of Masayoshi Nagata, Vol. 1, Kinokuniya, Tokyo, and Academic Press, New York, 1988, pp. 203-231. MR 977761 (90g:14003)
10.: D. Rees, Degree functions in local rings, Proc. Camb. Phil. Soc. 57 (1961) 1-7. MR 0124353 (23:A1667)
11.: D. Rees, R. Sharp, On a theorem of B. Teissier on multiplicities of ideals in local rings, J. Lond. Math. Soc. 18 (1978) 449-463. MR 518229 (80e:13009)
12.: I. Swanson, C. Huneke, Integral Closure of Ideals, Rings, and Modules, London Math. Soc. Lecture Note Ser., vol. 336, Cambridge University Press, Cambridge, 2006. MR 2266432 (2008m:13013)
13.: V. Van Lierde, Degree functions and projectively full ideals in two-dimensional rational singularities that can be desingularized by blowing up the unique maximal ideal, J. Pure Appl. Algebra 214 (2010) 512-518.
14.: J.K. Verma, Rees algebras and mixed multiplicities, Proc. Amer. Math. Soc. 104 (1988) 1036-1044. MR 929432 (89d:13018)
15.: J.K. Verma, Joint reductions of complete ideals, Nagoya Math. J. 118 (1990) 155-163. MR 1060707 (91c:13018)
16.: O. Zariski, Polynomial ideals defined by infinitely near base points, Amer. J. Math. 60 (1938) 151-204. MR 1507308
17.: O. Zariski, P. Samuel, Commutative Algebra, vol. 2, Van Nostrand, 1960. MR 0120249 (22:11006)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|