Homological properties of balanced Cohen-Macaulay algebras

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.