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 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 |
Fingerprint
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 journal › Article
}
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 -