### 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 |
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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