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Frobenius betti numbers and syzygies of finite length modulesPublished in American Mathematical Society

2020

DOI: 10.1090/proc/14989

Volume: 148

Issue: 8

Pages: 3245 - 3262

Let (R, m) be a local (Noetherian) ring of dimension d and let M be a ﬁnite length R-module with free resolution G•. 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, βiF (M, R) := lime→∞ λ(Hi(Fe(G•)))/ped, force pdR M < ∞? Second, if pdR M = ∞, does this force d+ 2nd 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 pdR M < ∞, 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 pdR M < ∞, 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 Hm0 (R) is large relative to the rest of the module, generalizing the case of Buchsbaum rings. © 2020 American Mathematical Society

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About the journal

Journal | Proceedings of the American Mathematical Society |
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Publisher | American Mathematical Society |

ISSN | 00029939 |