Homological properties of balanced CohenMacaulay algebras
Author:
Izuru Mori
Journal:
Trans. Amer. Math. Soc. 355 (2003), 10251042
MSC (2000):
Primary 16W50, 16E05, 16E65, 16E10
DOI:
https://doi.org/10.1090/S0002994702031665
Published electronically:
October 24, 2002
MathSciNet review:
1938744
Fulltext PDF Free Access
Abstract  References  Similar Articles  Additional Information
A balanced CohenMacaulay 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 CohenMacaulay algebra. In particular, we will prove the following theorem:
Theorem. Let $A$ be a Noetherian balanced CohenMacaulay algebra, and $M$ a nonzero finitely generated graded left $A$module. Then:

$M$ 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 CohenMacaulay graded left $A$module.

$M$ 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 CohenMacaulay algebra.

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

$A$ is AS regular if and only if every finitely generated maximal CohenMacaulay 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, 132441150
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