Illinois Journal of Mathematics

Notes on the linearity defect and applications

Hop D. Nguyen

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The linearity defect, introduced by Herzog and Iyengar, is a numerical measure for the complexity of minimal free resolutions. Employing a characterization of the linearity defect due to Şega, we study the behavior of linearity defect along short exact sequences. We point out two classes of short exact sequences involving Koszul modules, along which linearity defect behaves nicely. We also generalize the notion of Koszul filtrations from the graded case to the local setting. Among the applications, we prove that if $R\to S$ is a surjection of noetherian local rings such that $S$ is a Koszul $R$-module, and $N$ is a finitely generated $S$-module, then the linearity defect of $N$ as an $R$-module is the same as its linearity defect as an $S$-module. In particular, we confirm that specializations of absolutely Koszul algebras are again absolutely Koszul, answering positively a question due to Conca, Iyengar, Nguyen and Römer.

Article information

Illinois J. Math., Volume 59, Number 3 (2015), 637-662.

Received: 30 December 2014
Revised: 11 June 2016
First available in Project Euclid: 30 September 2016

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Zentralblatt MATH identifier

Primary: 13D02: Syzygies, resolutions, complexes 13H10: Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) [See also 14M05] 13D05: Homological dimension


Nguyen, Hop D. Notes on the linearity defect and applications. Illinois J. Math. 59 (2015), no. 3, 637--662. doi:10.1215/ijm/1475266401.

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