# A new Homological Invariant for Modules

## TL;DR

This paper introduces a new homological invariant called $zeta$-dimension for modules over Noetherian local rings, which refines existing dimensions and characterizes Gorenstein rings.

## Contribution

It defines the $zeta$-dimension using Vogel cohomology, providing a finer invariant that unifies and extends classical homological dimensions.

## Key findings

- $zeta$-dimension characterizes Gorenstein rings.
- $zeta$-dimension is finer than Gorenstein dimension.
- Provides a new perspective on projective and complete intersection dimensions.

## Abstract

Let $R$ be a commutative Noetherian local ring with residue field $k$. Using the structure of Vogel cohomology, for any finitely generated module $M$, we introduce a new dimension, called $\zeta$-dimension, denoted by $\zeta-dim_R M$. This dimension is finer than Gorenstein dimension and has nice properties enjoyed by homological dimensions. In particular, it characterizes Gorenstein rings in the sense that: a ring $R$ is Gorenstein if and only if every finitely generated $R$-module has finite $\zeta$-dimension. Our definition of $\zeta$-dimension offer a new homological perspective on the projective dimension, complete intersection dimension of Avramov et al. and $G$-dimension of Auslander and Bridger.