Rees algebras of finitely generated torsion-free modules over a two-dimensional regular local ring

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6 Citations (Scopus)

Abstract

Let (R, m) be a two-dimensional regular local ring and let A be a finitely generated torsion-free R-module. If A is a complete module, then Katz and Kodiyalam show A satisfies five conditions, one of these being that the Rees algebra A of A is Cohen-Macaulay and another being that the "associated graded ring" A/IA of A is Cohen-Macaulay. They ask whether these five conditions are equivalent without assuming A to be complete. We exhibit an example to show that A/IA may be Cohen-Macaulay while A fails to be Cohen-Macaulay, and investigate other implications among these five properties in the case where A is not complete. We prove in general that the depth of A is greater than or equal to the depth of A/IA, and that if a module has reduction number at most one, then a direct summand also has reduction number at most one. We present an example where A is a direct sum of two submodules each of which has reduction number at most one while A has reduction number at least two. In the last section of this paper, we present two sufficient conditions for modules obtained by adjoining one element to the submodule mn1 ⊕ ⋯ ⊕ mnr of Rr to be complete.

Original languageEnglish
Pages (from-to)4015-4039
Number of pages25
JournalCommunications in Algebra
Volume26
Issue number12
DOIs
Publication statusPublished - 1998 Jan 1

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Reduction number
Rees Algebra
Regular Local Ring
Cohen-Macaulay
Torsion-free
Finitely Generated
Module
Associated Graded Ring
Direct Sum
Sufficient Conditions

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

Rees algebras of finitely generated torsion-free modules over a two-dimensional regular local ring. / Liu, Jung-Chen.

In: Communications in Algebra, Vol. 26, No. 12, 01.01.1998, p. 4015-4039.

Research output: Contribution to journalArticle

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