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Bigraded structures and the depth of blow-up algebras

Published online by Cambridge University Press:  12 July 2007

Gemma Colomé-Nin
Affiliation:
Departament d'Álgebra i Geometria, Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain ([email protected];
Juan Elias
Affiliation:
Departament d'Álgebra i Geometria, Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, [email protected]

Abstract

Let R be a Cohen–Macaulay local ring, and let IR be an ideal with minimal reduction J. In this paper we attach to the pair (I, J) a non-standard bigraded module ΣI, J. The study of the bigraded Hilbert function of ΣI, J allows us to prove an improved version of Wang's conjecture and a weak version of Sally's conjecture, both on the depth of the associated graded ring grI(R). The module ΣI, J can be considered as a refinement of the Sally module introduced previously by Vasconcelos.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2006

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