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2017 Frobenius Betti numbers and modules of finite projective dimension
Alessandro De Stefani, Craig Huneke, Luis Núñez-Betancourt
J. Commut. Algebra 9(4): 455-490 (2017). DOI: 10.1216/JCA-2017-9-4-455


Let $(R,\mathfrak{m} ,K)$ be a local ring, and let $M$ be an $R$-module of finite length. We study asymptotic invariants, $\beta ^F_i(M,R)$, defined by twisting with Frobenius the free resolution of $M$. This family of invariants includes the Hilbert-Kunz multiplicity ($e_{HK}(\mathfrak{m} ,R)=\beta ^F_0(K,R)$). We discuss several properties of these numbers that resemble the behavior of the Hilbert-Kunz multiplicity. Furthermore, we study when the vanishing of $\beta ^F_i(M,R)$ implies that $M$ has finite projective dimension. In particular, we give a complete characterization of the vanishing of $\beta ^F_i(M,R)$ 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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Alessandro De Stefani. Craig Huneke. Luis Núñez-Betancourt. "Frobenius Betti numbers and modules of finite projective dimension." J. Commut. Algebra 9 (4) 455 - 490, 2017.


Published: 2017
First available in Project Euclid: 14 October 2017

zbMATH: 06797095
MathSciNet: MR3713524
Digital Object Identifier: 10.1216/JCA-2017-9-4-455

Primary: 13A35 , 13D02
Secondary: 13D07 , 13H10

Keywords: $F$-contributors , Hilbert-Kunz multiplicity , Krull dimension of syzygies , projective dimension

Rights: Copyright © 2017 Rocky Mountain Mathematics Consortium


Vol.9 • No. 4 • 2017
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