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On the length formula of Hoskin and Deligne and associated graded rings of two-dimensional regular local rings

Published online by Cambridge University Press:  24 October 2008

Bernard L. Johnston  and
Jugal Verma
Bernard L. Johnston
Affiliation:
Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, Department of Mathematics, Florida Atlantic University, Boca Baton, FL 33431, U.S.A.
Jugal Verma
Affiliation:
Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, Department of Mathematics, Indian Institute of Technology, Powai, Bombay 400 076, India
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Extract

Let (R, m) be a 2-dimensional regular local ring and I an m-primary ideal. The aim of this paper is to find conditions on I so that the associated graded ring of I,

and the Rees ring of I,

where t is an indeterminate, are Cohen–Macaulay (resp. Gorenstein). To this end, we use the results and techniques from Zariski's theory of complete ideals ([14], appendix 5) and its later generalizations and refinements due to Huneke [7] and Lipman[8]. The main result is an application of three deep theorems: (i) a generalization of Macaulay's classical theorem on Hilbert series of Gorenstein graded rings [13], (ii) a generalization of the Briançon–Skoda theorem due to Lipman and Sathaye [9], and (iii) a formula for the length of R/I, where I is a complete m-primary ideal, due to Hoskin[4] and Deligne[1].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 111 , Issue 3 , May 1992 , pp. 423 - 432
Copyright
Copyright © Cambridge Philosophical Society 1992

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References

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