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A Study of the Length Function of Generalized Fractions of Modules

Part of: Commutative algebra: Homological methods Local rings and semilocal rings

Published online by Cambridge University Press:  02 November 2016

Marcel Morales  and
Pham Hung Quy
Marcel Morales
Affiliation:
Université de Grenoble I, Institut Fourier, UMR 5582, BP 74, 38402 Saint-Martin D’Hères, France (morales@ujf-grenoble.fr) ESPE, Université Lyon 1, 5 rue Anselme, 69317 Lyon Cedex, France
Pham Hung Quy
Affiliation:
Department of Mathematics, FPT University, 8 Ton That Thuyet, Hanoi, Vietnam (quyph@fpt.edu.vn)
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Abstract

Let be a Noetherian local ring and let M be a finitely generated R-module of dimension d. Let be a system of parameters of M and let be a d-tuple of positive integers. In this paper we study the length of generalized fractions M(1/(x1, … , xd, 1)), which was introduced by Sharp and Hamieh. First, we study the growth of the function

Then we give an explicit calculation for the function in the case in which M admits a certain Macaulay extension. Most previous results on this topic are improved in a clearly understandable way.

Keywords

system of parametersgeneralized fractionslimit closurelocal cohomologyHilbert–Kunz function

MSC classification

Secondary: 13H15: Multiplicity theory and related topics 13D40: Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series 13D45: Local cohomology
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 60 , Issue 3 , August 2017 , pp. 721 - 737
Copyright
Copyright © Edinburgh Mathematical Society 2016 

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