A mixed multiplicity formula for complete ideals in 2-dimensional rational singularities

Van Lierde, Veronique

doi:10.1090/S0002-9939-2010-10455-0

A mixed multiplicity formula for complete ideals in $2$-dimensional rational singularities
HTML articles powered by AMS MathViewer

by Veronique Van Lierde

Proc. Amer. Math. Soc. 138 (2010), 4197-4204

DOI: https://doi.org/10.1090/S0002-9939-2010-10455-0

Published electronically: June 29, 2010

PDF | Request permission

Abstract:

Let $(R,m)$ 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 $(N)$ 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 $m$-primary ideals in $(R,m)$.

References

Steven Dale Cutkosky, On unique and almost unique factorization of complete ideals, Amer. J. Math. 111 (1989), no. 3, 417–433. MR 1002007, DOI 10.2307/2374667
Steven Dale Cutkosky, On unique and almost unique factorization of complete ideals. II, Invent. Math. 98 (1989), no. 1, 59–74. MR 1010155, DOI 10.1007/BF01388844
Steven Dale Cutkosky, A new characterization of rational surface singularities, Invent. Math. 102 (1990), no. 1, 157–177. MR 1069245, DOI 10.1007/BF01233425
Hartmut Göhner, Semifactoriality and Muhly’s condition $(\textrm {N})$ in two dimensional local rings, J. Algebra 34 (1975), 403–429. MR 379489, DOI 10.1016/0021-8693(75)90166-0
Craig Huneke, Complete ideals in two-dimensional regular local rings, Commutative algebra (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 15, Springer, New York, 1989, pp. 325–338. MR 1015525, DOI 10.1007/978-1-4612-3660-3_{1}6
Craig Huneke and Judith D. Sally, Birational extensions in dimension two and integrally closed ideals, J. Algebra 115 (1988), no. 2, 481–500. MR 943272, DOI 10.1016/0021-8693(88)90274-8
Bernard Johnston, The higher-dimensional multiplicity formula associated to the length formula of Hoskin and Deligne, Comm. Algebra 22 (1994), no. 6, 2057–2071. MR 1268543, DOI 10.1080/00927879408824955
Joseph Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 195–279. MR 276239, DOI 10.1007/BF02684604
Joseph Lipman, On complete ideals in regular local rings, Algebraic geometry and commutative algebra, Vol. I, Kinokuniya, Tokyo, 1988, pp. 203–231. MR 977761
D. Rees, Degree functions in local rings, Proc. Cambridge Philos. Soc. 57 (1961), 1–7. MR 124353, DOI 10.1017/s0305004100034794
D. Rees and R. Y. Sharp, On a theorem of B. Teissier on multiplicities of ideals in local rings, J. London Math. Soc. (2) 18 (1978), no. 3, 449–463. MR 518229, DOI 10.1112/jlms/s2-18.3.449
Craig Huneke and Irena Swanson, Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, vol. 336, Cambridge University Press, Cambridge, 2006. MR 2266432
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.
J. K. Verma, Rees algebras and mixed multiplicities, Proc. Amer. Math. Soc. 104 (1988), no. 4, 1036–1044. MR 929432, DOI 10.1090/S0002-9939-1988-0929432-4
J. K. Verma, Joint reductions of complete ideals, Nagoya Math. J. 118 (1990), 155–163. MR 1060707, DOI 10.1017/S0027763000003044
Oscar Zariski, Polynomial Ideals Defined by Infinitely Near Base Points, Amer. J. Math. 60 (1938), no. 1, 151–204. MR 1507308, DOI 10.2307/2371550
Oscar Zariski and Pierre Samuel, Commutative algebra. Vol. II, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0120249, DOI 10.1007/978-3-662-29244-0