Frobenius Betti numbers and syzygies of finite length modules

TL;DR

This paper investigates the properties of Frobenius Betti numbers and syzygies of finite length modules over local rings, establishing new conditions under which projective dimension is finite and analyzing the length of higher syzygies.

Contribution

It generalizes previous results by showing that $d+1$ consecutive vanishing Frobenius Betti numbers imply finite projective dimension, and proves that certain syzygies cannot have finite length in low dimensions.

Findings

$d+1$ consecutive vanishing Frobenius Betti numbers imply finite projective dimension.

No third syzygy of a finite length module can have finite length when the dimension is 1 or 2.

Results extend to rings of arbitrary dimension using properties of stably phantom homology.

Abstract

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.