## Illinois Journal of Mathematics

### Notes on the linearity defect and applications

Hop D. Nguyen

#### Abstract

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

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

Dates
Revised: 11 June 2016
First available in Project Euclid: 30 September 2016

https://projecteuclid.org/euclid.ijm/1475266401

Digital Object Identifier
doi:10.1215/ijm/1475266401

Mathematical Reviews number (MathSciNet)
MR3554226

Zentralblatt MATH identifier
1353.13013

#### Citation

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

#### References

• R. Ahangari Maleki, On the regularity and Koszulness of modules over local rings, Comm. Algebra 42 (2014), 3438–3452.
• R. Ahangari Maleki and M. E. Rossi, Regularity and linearity defect of modules over local rings, J. Commut. Algebra 6 (2014), 485–504.
• L. L. Avramov, Infinite free resolution, Six lectures on commutative algebra (Bellaterra, 1996), Progr. Math., vol. 166, Birkhäuser, Basel, 1998, pp. 1–118.
• L. L. Avramov and D. Eisenbud, Regularity of modules over a Koszul algebra, J. Algebra 153 (1992), 85–90.
• L. L. Avramov, S. B. Iyengar and L. M. Şega, Free resolutions over short local rings, J. Lond. Math. Soc. (2) 78 (2008), no. 2, 459–476.
• L. L. Avramov and I. Peeva, Finite regularity and Koszul algebras, Amer. J. Math. 123 (2001), 275–281.
• S. Blum, Initially Koszul algebras, Beitr. Algebra Geom. 41 (2000), 455–467.
• A. Conca, E. de Negri and M. E. Rossi, Koszul algebra and regularity, Commutative algebra: Expository papers dedicated to David Eisenbud on the occasion of his 65th birthday (I. Peeva, ed.), Springer, New York, 2013, pp. 285–315.
• A. Conca, S. B. Iyengar, H. D. Nguyen and T. Römer, Absolutely Koszul algebras and the Backelin–Roos property, Acta Math. Vietnam. 40 (2015), 353–374.
• A. Conca, M. E. Rossi and G. Valla, Gröbner flags and Gorenstein algebras, Compos. Math. 129 (2001), 95–121.
• A. Conca, N. V. Trung and G. Valla, Koszul property for points in projective space, Math. Scand. 89 (2001), 201–216.
• H. Derksen and J. Sidman, A sharp bound for the Castelnuovo–Mumford regularity of subspace arrangements, Adv. Math. 172 (2002), 151–157.
• D. Eisenbud, G. Fløystad and F.-O. Schreyer, Sheaf cohomology and free resolutions over exterior algebras, Trans. Amer. Math. Soc. 355 (2003), 4397–4426.
• D. Eisenbud and S. Goto, Linear free resolutions and minimal multiplicity, J. Algebra 88 (1984), 89–133.
• C. A. Francisco and A. Van Tuyl, Some families of componentwise linear monomial ideals, Nagoya Math. J. 187 (2007), 115–156.
• E. L. Green and R. Martínez-Villa, Koszul and Yoneda algebras, Representation theory of algebras (Cocoyoc, 1994), CMS Conf. Proc., vol. 18, American Mathematical Society, Providence, 1996, pp. 247–297.
• J. Herzog and T. Hibi, Componentwise linear ideals, Nagoya Math. J. 153 (1999), 141–153.
• J. Herzog, T. Hibi and G. Restuccia, Strongly Koszul algebras, Math. Scand. 86 (2000), 161–178.
• J. Herzog and S. B. Iyengar, Koszul modules, J. Pure Appl. Algebra 201 (2005), 154–188.
• J. Herzog and Y. Takayama, Resolution by mapping cones, Homology Homotopy Appl. 4 (2002), no. 2, part 2, 277–294.
• S. B. Iyengar and T. Römer, Linearity defects of modules over commutative rings, J. Algebra 322 (2009), 3212–3237.
• D. Lu and D. Zhou, Componentwise linear modules over a Koszul algebra, Taiwanese J. Math. 17 (2013), no. 6, 2135–2147.
• R. Martínez-Villa and D. Zacharia, Approximations with modules having linear free resolutions, J. Algebra 266 (2003), 671–697.
• S. Murai, Free resolutions of lex-ideals over a Koszul toric ring, Trans. Amer. Math. Soc. 363 (2011), no. 2, 857–885.
• R. Okazaki and K. Yanagawa, Linearity defects of face rings, J. Algebra 314 (2007), no. 1, 362–382.
• I. Peeva, Graded syzygies, Algebra and Applications., vol. 14, Springer, London, 2011.
• I. Peeva and M. Stillman, Open problems on syzygies and Hilbert functions, J. Commut. Algebra 1 (2009), 159–195.
• T. Römer, On minimal graded free resolutions, Ph.D. dissertation, University of Essen, 2001.
• J.-E. Roos, Good and bad Koszul algebras and their Hochschild homology, J. Pure Appl. Algebra 201 (2005), no. 1–3, 295–327.
• L. M. Şega, On the linearity defect of the residue field, J. Algebra 384 (2013), 276–290.
• L. Sharifan and M. Varbaro, Graded Betti numbers of ideals with linear quotients, Matematiche (Catania) 63 (2008), no. 2, 257–265.
• K. Yanagawa, Castelnuovo–Mumford regularity for complexes and weakly Koszul modules, J. Pure Appl. Algebra 207 (2006), no. 1, 77–97.
• K. Yanagawa, Linearity defect and regularity over a Koszul algebra, Math. Scand. 104 (2009), no. 2, 205–220.