On the dimension of syzygies

Asgharzadeh, Mohsen

doi:10.48550/arxiv.1705.04952

Cited by 3 publications

(4 citation statements)

References 6 publications

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“…The next result was originally proved in [As2,Corollary 4.5]. We include a proof here, for the reader's convenience and also for the sake of completeness.…”

Section: Introductionmentioning

confidence: 77%

On the new intersection theorem for totally reflexive modules

et al. 2019

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Let (R, m, k) be a local ring. The celebrated New Intersection Theorem is perceived as a deep result at the interface of homological and local algebra. We establish a complete intersection analogue of this theorem. Also when R is a quasi-specialization of a G-regular local ring, we extend the New Intersection Theorem to totally reflexive R-modules. There are plenty of examples of quasispecializations of G-regular rings which are neither G-regular nor Cohen-Macaulay. It is conjectured that if R admits a nonzero Cohen-Macaulay module of finite Gorenstein dimension, then it is Cohen-Macaulay. We establish this conjecture if either R is a quasi-specialization of G-regular local ring or a quasi-Buchsbaum ring. 2010 Mathematics Subject Classification. 13D05; 13C14; 13D22.

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“…The next result was originally proved in [As2,Corollary 4.5]. We include a proof here, for the reader's convenience and also for the sake of completeness.…”

Section: Introductionmentioning

confidence: 77%

On the new intersection theorem for totally reflexive modules

et al. 2019

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“…In the case that t = 0 (e.g., R is Buchsbaum), the result is already known. (See [7, Proposition 5.3], [3,Proposition 4.4]. The Buchsbaum hypothesis in the statement is not really used, merely the condition that H 0 m (R) is a vector space.)…”

Section: Syzygies With Finite Lengthmentioning

confidence: 99%

“…For any local ring (R, m) of dimension one and depth zero, if we take a parameter x ∈ m then M = R/xR is a module which is not of finite projective dimension, but which has a second syzygy of finite length. Thus, one should ask about the possibility of syzygies of finite length at the dim(R) + 2 spot or higher ( [3] shows that the ith syzygies for 0 < i ≤ dim(R) syzygies are not finite length). In light of these facts, De Stefani, Huneke, and Núñez-Betancourt posed the question [7]: Question 1.2.…”

Section: Introductionmentioning

confidence: 99%

Frobenius Betti numbers and syzygies of finite length modules

Aberbach¹,

Sarkar²

2020

Proc. Amer. Math. Soc.

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Let (R, m) be a local (Noetherian) ring of dimension d and M 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, β F i (M, R) := lime→∞ λ(F e (G•))/p ed , force pd R M < ∞. Second, if pd R 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 forces pd R 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 pd R 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, 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 m (R) is large relative to the rest of the module, generalizing the case of Buchsbaum rings.

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“…Suppose ℓ(Syz i+1 (L)) < ∞ for some fixed i > 0. In the light of[5, Lemma 4.1] we observeTor R 1 (L, Syz i−1 (R/H 0 m (R))) = Tor R i (L, R/H 0 m (R)) = 0. Since M is Lichtenbaum, Syz i−1 (R/H 0 m (R)) is free.…”

mentioning

confidence: 93%