  Let $(R,\mathfrak m)$ be a local (Noetherian) ring of dimension $d$ and $M$ a
finite length $R$-module with free resolution $G_\bullet$. De Stefani, Huneke,
and N\'{u}\~{n}ez-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 pd$_R M < \infty$. Second, if 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 forces 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 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, 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
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.
