### 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 m^{n1} ⊕ ⋯ ⊕ m^{nr} of R^{r} to be complete.

Original language | English |
---|---|

Pages (from-to) | 4015-4039 |

Number of pages | 25 |

Journal | Communications in Algebra |

Volume | 26 |

Issue number | 12 |

DOIs | |

Publication status | Published - 1998 Jan 1 |

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### 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.

Research output: Contribution to journal › Article

*Communications in Algebra*, vol. 26, no. 12, pp. 4015-4039. https://doi.org/10.1080/00927879808826392

}

TY - JOUR

T1 - Rees algebras of finitely generated torsion-free modules over a two-dimensional regular local ring

AU - Liu, Jung Chen

PY - 1998/1/1

Y1 - 1998/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=22444453836&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=22444453836&partnerID=8YFLogxK

U2 - 10.1080/00927879808826392

DO - 10.1080/00927879808826392

M3 - Article

AN - SCOPUS:22444453836

VL - 26

SP - 4015

EP - 4039

JO - Communications in Algebra

JF - Communications in Algebra

SN - 0092-7872

IS - 12

ER -