Betti Numbers for Modules Over Artinian Local Rings

We introduce a new numerical invariant $\gamma_I(M)$ associated to a finite-length $R$-module $M$ and an ideal $I$ in an Artinian local ring $R$. This invariant measures the ratio between $\lambda(IM)$ and $\lambda(M/IM)$. We establish fundamental relationships between this invariant and the Betti numbers of the module under the assumption of the $\operatorname{Tor}$ modules vanishing. In particular, we use this invariant to establish a freeness criterion for modules under certain $\operatorname{Tor}$ vanishing conditions. The criterion applies specifically to the class of $I$-free modules -- those modules $M$ for which $M/IM$ is isomorphic to a direct sum of copies of $R/I$. Lastly, we apply these results to the canonical module, proving that, under certain conditions on the ring structure, when the zeroth Betti number is greater than or equal to the first Betti number of the canonical module, then the ring is Gorenstein. This partially answers a question posed by Jorgensen and Leuschke concerning the relationship between Betti numbers of the canonical module and Gorenstein properties.