Frobenius Betti numbers and syzygies of finite length modules

Let $(R,\mathfrak {m})$ be a local (Noetherian) ring of dimension $d$ and let $M$ be a finite length $R$-module with free resolution $G_\bullet$. De Stefani, Huneke, and Núñez-Betancourt explored two questions about the properties of resolutions of $M$. First, in characteristic $p>0$, what vanishing conditions on the Frobenius Betti numbers, $\beta _i^F(M, R) : = \lim _{e \to \infty } \lambda (H_i(F^e(G_\bullet )))/p^{ed}$, force $\operatorname {pd}_R M < \infty$? Second, if $\operatorname {pd}_R M = \infty$, does this force $d+2$nd or higher syzygies of $M$ to have infinite length?

For the first question, they showed, under rather restrictive hypotheses, that $d+1$ consecutive vanishing Frobenius Betti numbers force $\operatorname {pd}_R M < \infty$, and when $d=1$ and $R$ is CM then one vanishing Frobenius Betti number suffices. Using properties of stably phantom homology, we show that these results hold in general, i.e., $d+1$ consecutive vanishing Frobenius Betti numbers force $\operatorname {pd}_R M < \infty$, and, under the hypothesis that $R$ is CM, $d$ consecutive vanishing Frobenius Betti numbers suffice.

For the second question, they obtain very interesting results when $d=1$. In particular, no third syzygy of $M$ can have finite length. Their main tool is, if $d=1$, to show that if the syzygy has a finite length, then it is an alternating sum of lengths of Tors. We are able to prove this fact for rings of arbitrary dimension, which allows us to show that if $d=2$, no third syzygy of $M$ can be of finite length! We also are able to show that the question has a positive answer if the dimension of the socle of $H^0_{\mathfrak {m}}(R)$ is large relative to the rest of the module, generalizing the case of Buchsbaum rings.