Illinois Journal of Mathematics
- Illinois J. Math.
- Volume 59, Number 3 (2015), 637-662.
Notes on the linearity defect and applications
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.
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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Nguyen, Hop D. Notes on the linearity defect and applications. Illinois J. Math. 59 (2015), no. 3, 637--662. doi:10.1215/ijm/1475266401. https://projecteuclid.org/euclid.ijm/1475266401