Betti Numbers for Modules Over Artinian Local Rings
Kaiyue He
TL;DR
This work introduces the invariant $\gamma_I(M)$ for finite modules over Artinian local rings with respect to an ideal $I$, linking it to Betti number growth under Tor-vanishing and forming a freeness criterion for $I$-free modules. It proves a quadratic constraint on $\gamma_I(M)$ when three consecutive Tor modules vanish and leverages this to deduce freeness in several structured scenarios, including generalizations to rings with $\mathfrak m^2I=0$ and socle considerations. The results yield sharp bounds on Betti growth via $\gamma_I$, connect length-theoretic data to resolutions, and extend prior work on the interplay between Tor-vanishing and freeness. Applying these ideas to the canonical module $\omega$ produces partial answers to when $b_0(\omega)\le b_1(\omega)$ implies Gorensteinness, under various additional hypotheses on socles and Tor vanishings, thereby contributing to the understanding of Betti-number behavior and its implications for ring structure.
Abstract
We introduce a new numerical invariant $γ_I(M)$ associated to a finite-length $R$-module $M$ and an ideal $I$ in an Artinian local ring $R$. This invariant measures the ratio between $λ(IM)$ and $λ(M/IM)$. 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.