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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Homological properties of balanced Cohen-Macaulay algebras


Author: Izuru Mori
Journal: Trans. Amer. Math. Soc. 355 (2003), 1025-1042
MSC (2000): Primary 16W50, 16E05, 16E65, 16E10
DOI: https://doi.org/10.1090/S0002-9947-02-03166-5
Published electronically: October 24, 2002
MathSciNet review: 1938744
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Abstract:

A balanced Cohen-Macaulay algebra is a connected algebra $A$ having a balanced dualizing complex $\omega _A[d]$ in the sense of Yekutieli (1992) for some integer $d$ and some graded $A$-$A$ bimodule $\omega _A$. We study some homological properties of a balanced Cohen-Macaulay algebra. In particular, we will prove the following theorem:

Theorem. Let $A$ be a Noetherian balanced Cohen-Macaulay algebra, and $M$ a nonzero finitely generated graded left $A$-module. Then:

  1. $M$ has a finite resolution of the form \[ 0\to \bigoplus ^{r_m}_{j=1}\omega _A(-l_{mj})\to \cdots \to \bigoplus ^{r_1} _{j=1}\omega _A(-l_{1j})\to H\to M\to 0,\] where $H$ is a finitely generated maximal Cohen-Macaulay graded left $A$-module.

  2. $M$ has finite injective dimension if and only if $M$ has a finite resolution of the form \begin{align*} 0&\to \bigoplus ^{r_m}_{j=1}\omega _A(-l_{mj})\to \cdots \to \bigoplus ^{r_1} _{j=1}\omega _A(-l_{1j}) &\to \bigoplus ^{r_0}_{j=1} \omega _A(-l_{0j})\to M\to 0. \end{align*}

As a corollary, we will have the following characterizations of AS Gorenstein algebras and AS regular algebras:

Corollary. Let $A$ be a Noetherian balanced Cohen-Macaulay algebra.

  1. $A$ is AS Gorenstein if and only if $\omega _A$ has finite projective dimension as a graded left $A$-module.

  2. $A$ is AS regular if and only if every finitely generated maximal Cohen-Macaulay graded left $A$-module is free.


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Additional Information

Izuru Mori
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Address at time of publication: Department of Mathematics, Syracuse University, Syracuse, New York, 13244-1150
Email: mori@math.purdue.edu, imori@syr.edu

Received by editor(s): October 10, 2001
Received by editor(s) in revised form: February 5, 2002
Published electronically: October 24, 2002
Article copyright: © Copyright 2002 American Mathematical Society