Asymptotic behavior of dimensions of syzygies

Beck, Kristen A.; Leamer, Micah J.

doi:10.1090/s0002-9939-2013-11510-8

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2017

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Cited by 2 publications

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“…Now let us assume by way of contradiction that λ(Ω 3 ) < ∞. Let (G • , ϕ • ) be a minimal free resolution of M: The following example is due to the second author, and it is taken from [BL13]. It shows the assumption that M has finite length is needed in Corollary 5.10.…”

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confidence: 99%

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Frobenius Betti numbers and modules of finite projective dimension

Stefani¹,

Huneke²,

Núñez-Betancourt³

2017

J. Commut. Algebra

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Abstract. Let (R, m, K) be a local ring, and let M be an R-module of finite length. We study asymptotic invariants, β F i (M, R), defined by twisting with Frobenius the free resolution of M . This family of invariants includes the Hilbert-Kunz multiplicity (e HK (m, R) = β F 0 (K, R)). We discuss several properties of these numbers that resemble the behavior of the Hilbert-Kunz multiplicity. Furthermore, we study when the vanishing of β F i (M, R) implies that M has finite projective dimension. In particular, we give a complete characterization of the vanishing of β F i (M, R) for one-dimensional rings. As a consequence of our methods, we give conditions for the non-existence of syzygies of finite length.

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confidence: 99%

“…The following example is due to the second author, and it is taken from [BL13]. It shows the assumption that M has finite length is needed in Corollary 5.10.…”

mentioning

confidence: 99%