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Notes on the linearity defect and applications. (English) Zbl 1353.13013

Summary: The linearity defect, introduced by J. Herzog and S. Iyengar [J. Pure Appl. Algebra 201, No. 1–3, 154–188 (2005; Zbl 1106.13011)], 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 \(R\rightarrow S\) 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 A. Conca et al. [Acta Math. Vietnam. 40, No. 3, 353–374 (2015; Zbl 1330.13015)].

MSC:

13D02 Syzygies, resolutions, complexes and commutative rings
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13D05 Homological dimension and commutative rings
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[1] \beginbarticle \bauthor\binitsR. \bsnmAhangari Maleki, \batitleOn the regularity and Koszulness of modules over local rings, \bjtitleComm. Algebra \bvolume42 (\byear2014), page 3438-\blpage3452. \endbarticle \OrigBibText Rasoul Ahangari Maleki, On the regularity and Koszulness of modules over local rings . Comm. Algebra 42 (2014), 3438-3452. \endOrigBibText \bptokstructpyb \endbibitem · Zbl 1295.13021
[2] \beginbarticle \bauthor\binitsR. \bsnmAhangari Maleki and \bauthor\binitsM. E. \bsnmRossi, \batitleRegularity and linearity defect of modules over local rings, \bjtitleJ. Commut. Algebra \bvolume6 (\byear2014), page 485-\blpage504. \endbarticle \OrigBibText Rasoul Ahangari Maleki and Maria Evelina Rossi, Regularity and linearity defect of modules over local rings . J. Commut. Algebra 4 (2014), 485-504. \endOrigBibText \bptokstructpyb \endbibitem · Zbl 1321.13004
[3] \beginbchapter \bauthor\binitsL. L. \bsnmAvramov, \bctitleInfinite free resolution, \bbtitleSix lectures on commutative algebra (Bellaterra, 1996), \bsertitleProgr. Math., vol. \bseriesno166, , \blocationBasel, \byear1998, pp. page 1-\blpage118. \endbchapter \OrigBibText —-, Infinite free resolution . in Six lectures on Commutative Algebra (Bellaterra, 1996), 1-118, Progr. Math., 166 , Birkhäuser (1998). \endOrigBibText \bptokstructpyb \endbibitem
[4] \beginbarticle \bauthor\binitsL. L. \bsnmAvramov and \bauthor\binitsD. \bsnmEisenbud, \batitleRegularity of modules over a Koszul algebra, \bjtitleJ. Algebra \bvolume153 (\byear1992), page 85-\blpage90. \endbarticle \OrigBibText Luchezar L. Avramov and David Eisenbud, Regularity of modules over a Koszul algebra . J. Algebra. 153 (1992), 85-90. \endOrigBibText \bptokstructpyb \endbibitem · Zbl 0770.13006
[5] \beginbarticle \bauthor\binitsL. L. \bsnmAvramov, \bauthor\binitsS. B. \bsnmIyengar and \bauthor\binitsL. M. , \batitleFree resolutions over short local rings, \bjtitleJ. Lond. Math. Soc. (2) \bvolume78 (\byear2008), no. \bissue2, page 459-\blpage476. \endbarticle \OrigBibText Luchezar L. Avramov, Srikanth B. Iyengar and Liana M. Şega, Free resolutions over short local rings . J. Lond. Math. Soc. 78 (2008), no. 2, 459-476. \endOrigBibText \bptokstructpyb \endbibitem · Zbl 1153.13011
[6] \beginbarticle \bauthor\binitsL. L. \bsnmAvramov and \bauthor\binitsI. \bsnmPeeva, \batitleFinite regularity and Koszul algebras, \bjtitleAmer. J. Math. \bvolume123 (\byear2001), page 275-\blpage281. \endbarticle \OrigBibText Luchezar L. Avramov and Irena Peeva, Finite regularity and Koszul algebras . Amer. J. Math. 123 (2001), 275-281. \endOrigBibText \bptokstructpyb \endbibitem · Zbl 1053.13500
[7] \beginbarticle \bauthor\binitsS. \bsnmBlum, \batitleInitially Koszul algebras, \bjtitleBeitr. Algebra Geom. \bvolume41 (\byear2000), page 455-\blpage467. \endbarticle \OrigBibText Stephan Blum, Initially Koszul algebras . Beiträge Algebra Geom. 41 (2000), 455-467. \endOrigBibText \bptokstructpyb \endbibitem
[8] \beginbchapter \bauthor\binitsA. \bsnmConca, \bauthor\binitsE. \bparticlede \bsnmNegri and \bauthor\binitsM. E. \bsnmRossi, \bctitleKoszul algebra and regularity, \bbtitleCommutative algebra: Expository papers dedicated to David Eisenbud on the occasion of his 65th birthday (\beditor\binitsI. \bsnmPeeva, ed.), \bpublisherSpringer, \blocationNew York, \byear2013, pp. page 285-\blpage315. \endbchapter \OrigBibText Aldo Conca, Emanuela de Negri and Maria Evelina Rossi, Koszul algebra and regularity . in Commutative Algebra: expository papers dedicated to David Eisenbud on the occasion of his 65th birthday , I. Peeva (ed.), Springer (2013), 285-315. \endOrigBibText \bptokstructpyb \endbibitem · Zbl 1262.13017
[9] \beginbarticle \bauthor\binitsA. \bsnmConca, \bauthor\binitsS. B. \bsnmIyengar, \bauthor\binitsH. D. \bsnmNguyen and \bauthor\binitsT. , \batitleAbsolutely Koszul algebras and the Backelin-Roos property, \bjtitleActa Math. Vietnam. \bvolume40 (\byear2015), page 353-\blpage374. \endbarticle \OrigBibText Aldo Conca, Srikanth B. Iyengar, Hop D. Nguyen and Tim Römer, Absolutely Koszul algebras and the Backelin-Roos property . Acta Math. Vietnam. 40 (2015), 353-374. \endOrigBibText \bptokstructpyb \endbibitem · Zbl 1330.13015
[10] \beginbarticle \bauthor\binitsA. \bsnmConca, \bauthor\binitsM. E. \bsnmRossi and \bauthor\binitsG. \bsnmValla, flags and Gorenstein algebras, \bjtitleCompos. Math. \bvolume129 (\byear2001), page 95-\blpage121. \endbarticle \OrigBibText Aldo Conca, Maria Evelina Rossi and Giuseppe Valla, Gröbner flags and Gorenstein algebras . Compositio Math. 129 (2001), 95-121. \endOrigBibText \bptokstructpyb \endbibitem · Zbl 1030.13005
[11] \beginbarticle \bauthor\binitsA. \bsnmConca, \bauthor\binitsN. V. \bsnmTrung and \bauthor\binitsG. \bsnmValla, \batitleKoszul property for points in projective space, \bjtitleMath. Scand. \bvolume89 (\byear2001), page 201-\blpage216. \endbarticle \OrigBibText Aldo Conca, Ngo Viet Trung and Giuseppe Valla, Koszul property for points in projective space . Math. Scand. 89 (2001), 201-216. \endOrigBibText \bptokstructpyb \endbibitem · Zbl 1025.13003
[12] \beginbarticle \bauthor\binitsH. \bsnmDerksen and \bauthor\binitsJ. \bsnmSidman, \batitleA sharp bound for the Castelnuovo-Mumford regularity of subspace arrangements, \bjtitleAdv. Math. \bvolume172 (\byear2002), page 151-\blpage157. \endbarticle \OrigBibText Harm Derksen and Jessica Sidman, A sharp bound for the Castelnuovo-Mumford regularity of subspace arrangements . Adv. Math. 172 (2002), 151-157. \endOrigBibText \bptokstructpyb \endbibitem · Zbl 1040.13009
[13] \beginbarticle \bauthor\binitsD. \bsnmEisenbud, \bauthor\binitsG. and \bauthor\binitsF.-O. \bsnmSchreyer, \batitleSheaf cohomology and free resolutions over exterior algebras, \bjtitleTrans. Amer. Math. Soc. \bvolume355 (\byear2003), page 4397-\blpage4426. \endbarticle \OrigBibText David Eisenbud, Gunnar Fløystad and Frank-Olaf Schreyer, Sheaf cohomology and free resolutions over exterior algebras . Trans. Amer. Math. Soc. 355 (2003), 4397-4426. \endOrigBibText \bptokstructpyb \endbibitem · Zbl 1063.14021
[14] \beginbarticle \bauthor\binitsD. \bsnmEisenbud and \bauthor\binitsS. \bsnmGoto, \batitleLinear free resolutions and minimal multiplicity, \bjtitleJ. Algebra \bvolume88 (\byear1984), page 89-\blpage133. \endbarticle \OrigBibText David Eisenbud and Shiro Goto, Linear free resolutions and minimal multiplicity . J. Algebra 88 (1984), 89-133. \endOrigBibText \bptokstructpyb \endbibitem · Zbl 0531.13015
[15] \beginbarticle \bauthor\binitsC. A. \bsnmFrancisco and \bauthor\binitsA. \bparticleVan \bsnmTuyl, \batitleSome families of componentwise linear monomial ideals, \bjtitleNagoya Math. J. \bvolume187 (\byear2007), page 115-\blpage156. \endbarticle \OrigBibText Christopher A. Francisco and Adam Van Tuyl, Some families of componentwise linear monomial ideals . Nagoya Math. J. 187 (2007), 115-156. \endOrigBibText \bptokstructpyb \endbibitem · Zbl 1140.13012
[16] \beginbchapter \bauthor\binitsE. L. \bsnmGreen and \bauthor\binitsR. -Villa, \bctitleKoszul and Yoneda algebras, \bbtitleRepresentation theory of algebras (Cocoyoc, 1994), \bsertitleCMS Conf. Proc., vol. \bseriesno18, \bpublisherAmerican Mathematical Society, \blocationProvidence, \byear1996, pp. page 247-\blpage297. \endbchapter \OrigBibText Edward L. Green and Roberto Martínez-Villa, Koszul and Yoneda algebras . in: Representation Theory of Algebras (Cocoyoc, 1994), in: CMS Conf. Proc. Vol. 18 , American Mathematical Society, Providence (1996), 247-297. \endOrigBibText \bptokstructpyb \endbibitem
[17] \beginbarticle \bauthor\binitsJ. \bsnmHerzog and \bauthor\binitsT. \bsnmHibi, \batitleComponentwise linear ideals, \bjtitleNagoya Math. J. \bvolume153 (\byear1999), page 141-\blpage153. \endbarticle \OrigBibText Jürgen Herzog and Takayuki Hibi, Componentwise linear ideals . Nagoya Math. J. 153 (1999), 141-153. \endOrigBibText \bptokstructpyb \endbibitem · Zbl 0930.13018
[18] \beginbarticle \bauthor\binitsJ. \bsnmHerzog, \bauthor\binitsT. \bsnmHibi and \bauthor\binitsG. \bsnmRestuccia, \batitleStrongly Koszul algebras, \bjtitleMath. Scand. \bvolume86 (\byear2000), page 161-\blpage178. \endbarticle \OrigBibText Jürgen Herzog, Takayuki Hibi and Gaetana Restuccia, Strongly Koszul algebras . Math. Scand. 86 (2000), 161-178. \endOrigBibText \bptokstructpyb \endbibitem · Zbl 1061.13008
[19] \beginbarticle \bauthor\binitsJ. \bsnmHerzog and \bauthor\binitsS. B. \bsnmIyengar, \batitleKoszul modules, \bjtitleJ. Pure Appl. Algebra \bvolume201 (\byear2005), page 154-\blpage188. \endbarticle \OrigBibText Jürgen Herzog and Srikanth B. Iyengar, Koszul modules . J. Pure Appl. Algebra 201 (2005), 154-188. \endOrigBibText \bptokstructpyb \endbibitem · Zbl 1106.13011
[20] \beginbarticle \bauthor\binitsJ. \bsnmHerzog and \bauthor\binitsY. \bsnmTakayama, \batitleResolution by mapping cones, \bjtitleHomology Homotopy Appl. \bvolume4 (\byear2002), no. \bissue2, part 2, page 277-\blpage294. \endbarticle \OrigBibText Jürgen Herzog and Yukihide Takayama, Resolution by mapping cones . Homology Homotopy Appl. 4 (2002), no. 2, part 2, 277-294. \endOrigBibText \bptokstructpyb \endbibitem · Zbl 1028.13008
[21] \beginbarticle \bauthor\binitsS. B. \bsnmIyengar and \bauthor\binitsT. , \batitleLinearity defects of modules over commutative rings, \bjtitleJ. Algebra \bvolume322 (\byear2009), page 3212-\blpage3237. \endbarticle \OrigBibText Srikanth B. Iyengar and Tim Römer, Linearity defects of modules over commutative rings . J. Algebra 322 (2009), 3212-3237. \endOrigBibText \bptokstructpyb \endbibitem · Zbl 1184.13039
[22] \beginbarticle \bauthor\binitsD. \bsnmLu and \bauthor\binitsD. \bsnmZhou, \batitleComponentwise linear modules over a Koszul algebra, \bjtitleTaiwanese J. Math. \bvolume17 (\byear2013), no. \bissue6, page 2135-\blpage2147. \endbarticle \OrigBibText Dancheng Lu and Dexu Zhou, Componentwise linear modules over a Koszul algebra . Taiwanese J. Math. 17 (2013), no. 6, 2135-2147. \endOrigBibText \bptokstructpyb \endbibitem · Zbl 1282.13027
[23] \beginbarticle \bauthor\binitsR. -Villa and \bauthor\binitsD. \bsnmZacharia, \batitleApproximations with modules having linear free resolutions, \bjtitleJ. Algebra \bvolume266 (\byear2003), page 671-\blpage697. \endbarticle \OrigBibText Roberto Martínez-Villa and Dan Zacharia, Approximations with modules having linear free resolutions . J. Algebra 266 (2003), 671-697. \endOrigBibText \bptokstructpyb \endbibitem · Zbl 1061.16035
[24] \beginbarticle \bauthor\binitsS. \bsnmMurai, \batitleFree resolutions of lex-ideals over a Koszul toric ring, \bjtitleTrans. Amer. Math. Soc. \bvolume363 (\byear2011), no. \bissue2, page 857-\blpage885. \endbarticle \OrigBibText Satoshi Murai, Free resolutions of lex-ideals over a Koszul toric ring . Trans. Amer. Math. Soc. 363 (2011), no. 2, 857-885. \endOrigBibText \bptokstructpyb \endbibitem · Zbl 1219.13008
[25] \beginbarticle \bauthor\binitsR. \bsnmOkazaki and \bauthor\binitsK. \bsnmYanagawa, \batitleLinearity defects of face rings, \bjtitleJ. Algebra \bvolume314 (\byear2007), no. \bissue1, page 362-\blpage382. \endbarticle \OrigBibText Ryota Okazaki and Kohji Yanagawa, Linearity defects of face rings . J. Algebra 314 (2007), no. 1, 362-382. \endOrigBibText \bptokstructpyb \endbibitem · Zbl 1127.13020
[26] \beginbbook \bauthor\binitsI. \bsnmPeeva, \bbtitleGraded syzygies, \bsertitleAlgebra and Applications., vol. \bseriesno14, \bpublisherSpringer, \blocationLondon, \byear2011. \endbbook \OrigBibText Irena Peeva, Graded syzygies . Algebra and Applications. Volume 14 , Springer, London (2011). \endOrigBibText \bptokstructpyb \endbibitem · Zbl 1213.13002
[27] \beginbarticle \bauthor\binitsI. \bsnmPeeva and \bauthor\binitsM. \bsnmStillman, \batitleOpen problems on syzygies and Hilbert functions, \bjtitleJ. Commut. Algebra \bvolume1 (\byear2009), page 159-\blpage195. \endbarticle \OrigBibText Irena Peeva and Mike Stillman, Open problems on syzygies and Hilbert functions . J. Commut. Algebra 1 (2009), 159-195. \endOrigBibText \bptokstructpyb \endbibitem · Zbl 1187.13010
[28] \beginbotherref \oauthor\binitsT. , On minimal graded free resolutions , Ph.D. dissertation, University of Essen, 2001. \endbotherref \OrigBibText Tim Römer, On minimal graded free resolutions . Ph.D. dissertation, University of Essen (2001). \endOrigBibText \bptokstructpyb \endbibitem
[29] \beginbarticle \bauthor\binitsJ.-E. \bsnmRoos, \batitleGood and bad Koszul algebras and their Hochschild homology, \bjtitleJ. Pure Appl. Algebra \bvolume201 (\byear2005), no. \bissue1-3, page 295-\blpage327. \endbarticle \OrigBibText Jan-Erik Roos, Good and bad Koszul algebras and their Hochschild homology . J. Pure Appl. Algebra 201 (2005), no. 1-3, 295-327. \endOrigBibText \bptokstructpyb \endbibitem · Zbl 1090.13008
[30] \beginbarticle \bauthor\binitsL. M. , \batitleOn the linearity defect of the residue field, \bjtitleJ. Algebra \bvolume384 (\byear2013), page 276-\blpage290. \endbarticle \OrigBibText Liana M. Şega, On the linearity defect of the residue field . J. Algebra 384 (2013), 276-290. \endOrigBibText \bptokstructpyb \endbibitem · Zbl 1303.13016
[31] \beginbarticle \bauthor\binitsL. \bsnmSharifan and \bauthor\binitsM. \bsnmVarbaro, \batitleGraded Betti numbers of ideals with linear quotients, \bjtitleMatematiche (Catania) \bvolume63 (\byear2008), no. \bissue2, page 257-\blpage265. \endbarticle \OrigBibText Leila Sharifan and Matteo Varbaro, Graded Betti numbers of ideals with linear quotients , Le Matematiche 63 (2008), no. 2, 257-265. \endOrigBibText \bptokstructpyb \endbibitem
[32] \beginbarticle \bauthor\binitsK. \bsnmYanagawa, \batitleCastelnuovo-Mumford regularity for complexes and weakly Koszul modules, \bjtitleJ. Pure Appl. Algebra \bvolume207 (\byear2006), no. \bissue1, page 77-\blpage97. \endbarticle \OrigBibText Kohji Yanagawa, Castelnuovo-Mumford regularity for complexes and weakly Koszul modules . J. Pure Appl. Algebra 207 (2006), no. 1, 77-97. \endOrigBibText \bptokstructpyb \endbibitem · Zbl 1163.16006
[33] \beginbarticle \bauthor\binitsK. \bsnmYanagawa, \batitleLinearity defect and regularity over a Koszul algebra, \bjtitleMath. Scand. \bvolume104 (\byear2009), no. \bissue2, page 205-\blpage220. \endbarticle \OrigBibText —-, Linearity defect and regularity over a Koszul algebra . Math. Scand. 104 (2009), no. 2, 205-220. \endOrigBibText \bptokstructpyb \endbibitem · Zbl 1200.16040
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