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.