×

Homological dimensions of rigid modules. (English) Zbl 1401.13045

Let \((R,\mathfrak{m}, k)\) be a commutative local Noetherian ring. In the paper under review the authors give the following characterization of Gorenstein rings and regular rings. (1) \(R\) is Gorenstein if the Gorenstein injective dimension of the maximal ideal \(\mathfrak{m}\) is finite. (2) \(R\) is regular if a single \(\mathrm{Ext}^n_R(I,J)\) vanishes for some integrally closed \(\mathfrak{m}\)-primary ideals \(I\) and \(J\) of \(R\) and for some positive integer \(n\).

MSC:

13D07 Homological functors on modules of commutative rings (Tor, Ext, etc.)
13D05 Homological dimension and commutative rings
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] M. Auslander, Modules over unramified regular local rings, Illinois J. Math. 5 (1961), 631–647. · Zbl 0104.26202
[2] M. Auslander, “Modules over unramified regular local rings” in Proc. Internat. Congr. Mathematicians (Stockholm, 1962), Inst. Mittag-Leffler, Djursholm, 1963, 230–233. · Zbl 0123.03702
[3] M. Auslander and M. Bridger, Stable Module Theory, Mem. Amer. Math. Soc. 94 (1969), no. 94. · Zbl 0204.36402
[4] L. L. Avramov, Modules with extremal resolutions, Math. Res. Lett. 3 (1996), 319–328. · Zbl 0867.13003 · doi:10.4310/MRL.1996.v3.n3.a3
[5] L. L. Avramov, “Infinite free resolutions” in Six Lectures on Commutative Algebra (Bellaterra, 1996), Progr. Math. 166, Birkhäuser, Basel, 1998, 1–118. · Zbl 0934.13008
[6] L. L. Avramov, “Homological dimensions and related invariants of modules over local rings” in Representations of Algebra, Vol. I, II, Beijing Norm. Univ. Press, Beijing, 2002, 1–39. · Zbl 1086.16504
[7] L. L. Avramov and R.-O. Buchweitz, Support varieties and cohomology over complete intersections, Invent. Math. 142 (2000), 285–318. · Zbl 0999.13008 · doi:10.1007/s002220000090
[8] L. L. Avramov, V. Gasharov, and I. Peeva, Complete intersection dimension, Inst. Hautes Études Sci. Publ. Math. 86 (1998), 67–114. · Zbl 0918.13008
[9] L. L. Avramov, M. Hochster, S. B. Iyengar, and Y. Yao, Homological invariants of modules over contracting endomorphisms, Math. Ann. 353 (2012), 275–291. · Zbl 1241.13013 · doi:10.1007/s00208-011-0682-z
[10] W. Bruns and J. Herzog, Cohen–Macaulay Rings, Cambridge Stud. Adv. Math. 39, Cambridge Univ. Press, Cambridge, 1993.
[11] O. Celikbas and H. Dao, Asymptotic behavior of Ext functors for modules of finite complete intersection dimension, Math. Z. 269 (2011), 1005–1020. · Zbl 1235.13010 · doi:10.1007/s00209-010-0771-9
[12] O. Celikbas and H. Dao, Necessary conditions for the depth formula over Cohen-Macaulay local rings, J. Pure Appl. Algebra 218 (2014), 522–530. · Zbl 1282.13030 · doi:10.1016/j.jpaa.2013.07.002
[13] O. Celikbas, H. Dao, and R. Takahashi, Modules that detect finite homological dimensions, Kyoto J. Math. 54 (2014), 295–310. · Zbl 1337.13011 · doi:10.1215/21562261-2642404
[14] O. Celikbas and S. Sather-Wagstaff, Testing for the Gorenstein property, Collect. Math. 67 (2016), 555–568. · Zbl 1349.13018 · doi:10.1007/s13348-015-0163-x
[15] L. W. Christensen, Gorenstein Dimensions, Lecture Notes in Math. 1747, Springer, Berlin, 2000. · Zbl 0965.13010
[16] L. W. Christensen and D. A. Jorgensen, Vanishing of Tate homology and depth formulas over local rings, J. Pure Appl. Algebra 219 (2015), 464–481. · Zbl 1311.13016 · doi:10.1016/j.jpaa.2014.05.005
[17] A. Corso, C. Huneke, D. Katz, and W. V. Vasconcelos, “Integral closure of ideals and annihilators of homology” in Commutative Algebra, Lect. Notes Pure Appl. Math. 244, Chapman & Hall/CRC, Boca Raton, FL, 2006, 33–48. · Zbl 1119.13006
[18] H. Dao, Some observations on local and projective hypersurfaces, Math. Res. Lett. 15 (2008), 207–219. · Zbl 1229.13014 · doi:10.4310/MRL.2008.v15.n2.a1
[19] H. Dao, Remarks on non-commutative crepant resolutions of complete intersections, Adv. Math. 224 (2010), 1021–1030. · Zbl 1192.13011 · doi:10.1016/j.aim.2009.12.016
[20] H. Dao, Decent intersection and Tor-rigidity for modules over local hypersurfaces, Trans. Amer. Math. Soc. 365 (2013), no. 6, 2803–2821. · Zbl 1285.13018 · doi:10.1090/S0002-9947-2012-05574-7
[21] H. Dao, J. Li, and C. Miller, On the (non)rigidity of the Frobenius endomorphism over Gorenstein rings, Algebra Number Theory 4 (2010), 1039–1053. · Zbl 1221.13004 · doi:10.2140/ant.2010.4.1039
[22] E. E. Enochs and O. M. G. Jenda, Gorenstein injective and projective modules, Math. Z. 220 (1995), 611–633. · Zbl 0845.16005 · doi:10.1007/BF02572634
[23] E. G. Evans and P. Griffith, Syzygies, London Math. Soc. Lecture Note Ser. 106, Cambridge Univ. Press, Cambridge, 1985.
[24] M. Hochster, “Cohen–Macaulay modules” in Conference on Commutative Algebra (Lawrence, 1972), Lecture Notes in Math. 311, Springer, Berlin, 1973, 120–152.
[25] H. Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra 189 (2004), 167–193. · Zbl 1050.16003 · doi:10.1016/j.jpaa.2003.11.007
[26] H. Holm, Rings with finite Gorenstein injective dimension, Proc. Amer. Math. Soc. 132 (2004), 1279–1283. · Zbl 1062.16008 · doi:10.1090/S0002-9939-03-07466-5
[27] H. Holm and P. Jørgensen, Semi-dualizing modules and related Gorenstein homological dimensions, J. Pure Appl. Algebra 205 (2006), 423–445. · Zbl 1094.13021
[28] H. Holm and P. Jørgensen, Cohen–Macaulay homological dimensions, Rend. Semin. Mat. Univ. Padova 117 (2007), 87–112. · Zbl 1150.13001
[29] C. Huneke and D. A. Jorgensen, Symmetry in the vanishing of Ext over Gorenstein rings, Math. Scand. 93 (2003), 161–184. · Zbl 1062.13005 · doi:10.7146/math.scand.a-14418
[30] C. Huneke and I. Swanson, Integral Closure of Ideals, Rings, and Modules, London Math. Soc. Lecture Note Ser. 336, Cambridge Univ. Press, Cambridge, 2006. · Zbl 1117.13001
[31] C. Huneke and R. Wiegand, Tensor products of modules and the rigidity of \(\mathit{Tor}\), Math. Ann. 299 (1994), 449–476. · Zbl 0803.13008 · doi:10.1007/BF01459794
[32] M. I. Jinnah, Reflexive modules over regular local rings, Arch. Math. (Basel) 26 (1975), 367–371. · Zbl 0309.13005 · doi:10.1007/BF01229753
[33] D. A. Jorgensen, A generalization of the Auslander–Buchsbaum formula, J. Pure Appl. Algebra 144 (1999), 145–155. · Zbl 0951.13010 · doi:10.1016/S0022-4049(98)00057-7
[34] D. A. Jorgensen, Finite projective dimension and the vanishing of \(\text{Ext}_{R}(M,M)\), Comm. Algebra 36 (2008), 4461–4471. · Zbl 1206.13021 · doi:10.1080/00927870802179560
[35] D. A. Jorgensen and L. M. Şega, Nonvanishing cohomology and classes of Gorenstein rings, Adv. Math. 188 (2004), 470–490. · Zbl 1090.13009 · doi:10.1016/j.aim.2003.11.003
[36] P. Jothilingam, A note on grade, Nagoya Math. J. 59 (1975), 149–152. · Zbl 0303.13012 · doi:10.1017/S0027763000016858
[37] P. Jothilingam, Syzygies and Ext, Math. Z. 188 (1985), 278–282.
[38] J. Lescot, “La série de Bass d’un produit fibré d’anneaux locaux” in Paul Dubreil and Marie-Paule Malliavin Algebra Seminar, 35th Year (Paris, 1982), Lecture Notes in Math. 1029, Springer, Berlin, 1983, 218–239.
[39] G. J. Leuschke and R. Wiegand, Cohen–Macaulay Representations, Math. Surveys Monogr. 181, Amer. Math. Soc., Providence, 2012.
[40] G. Levin and W. V. Vasconcelos, Homological dimensions and Macaulay rings, Pacific J. Math. 25 (1968), 315–323. · Zbl 0161.03903 · doi:10.2140/pjm.1968.25.315
[41] S. Lichtenbaum, On the vanishing of Tor in regular local rings, Illinois J. Math. 10 (1966), 220–226. · Zbl 0139.26601
[42] J. Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 195–279. · Zbl 0181.48903 · doi:10.1007/BF02684604
[43] V. Maşek, Gorenstein dimension and torsion of modules over commutative Noetherian rings, Comm. Algebra 28 (2000), 5783–5811. · Zbl 1002.13005 · doi:10.1080/00927870008827189
[44] H. Matsumura, Commutative Ring Theory, 2nd ed., Cambridge Stud. Adv. Math. 8, Cambridge Univ. Press, Cambridge, 1989. · Zbl 0666.13002
[45] C. Miller, “The Frobenius endomorphism and homological dimensions” in Commutative Algebra (Grenoble/Lyon, 2001), Contemp. Math. 331, Amer. Math. Soc., Providence, 2003, 207–234.
[46] W. F. Moore, G. Piepmeyer, S. Spiroff, and M. E. Walker, Hochster’s theta invariant and the Hodge–Riemann bilinear relations, Adv. Math. 2 (2011), 1692–1714. · Zbl 1221.13027 · doi:10.1016/j.aim.2010.09.005
[47] M. P. Murthy, Modules over regular local rings, Illinois J. Math. 7 (1963), 558–565. · Zbl 0117.02701
[48] S. Nasseh, M. Tousi, and S. Yassemi, Characterization of modules of finite projective dimension via Frobenius functors, Manuscripta Math. 130 (2009), 425–431. · Zbl 1222.13013 · doi:10.1007/s00229-009-0296-x
[49] P. Roberts, Two applications of dualizing complexes over local rings, Ann. Sci. Éc. Norm. Supér. (4) 9 (1976), 103–106. · Zbl 0326.13004
[50] A. Sadeghi, Vanishing of cohomology over complete intersection rings, Glasg. Math. J. 57 (2015), 445–455. · Zbl 1334.13011 · doi:10.1017/S0017089514000408
[51] S. Salarian, S. Sather-Wagstaff, and S. Yassemi, Characterizing local rings via homological dimensions and regular sequences, J. Pure Appl. Algebra 207 (2006), 99–108. · Zbl 1102.13026 · doi:10.1016/j.jpaa.2005.09.015
[52] R. Takahashi, On \(G\)-regular local rings, Comm. Algebra 36 (2008), 4472–4491. · Zbl 1156.13009 · doi:10.1080/00927870802179602
[53] R. Takahashi and D. White, Homological aspects of semidualizing modules, Math. Scand. 106 (2010), 5–22. · Zbl 1193.13012 · doi:10.7146/math.scand.a-15121
[54] M. E. Walker, On the vanishing of Hochster’s \(θ\) invariant, Ann. K-Theory 2 (2017), 131–174. · Zbl 1365.13027
[55] S. Yassemi, A generalization of a theorem of Bass, Comm. Algebra 35 (2007), 249–251. · Zbl 1122.13001 · doi:10.1080/00927870601041672
[56] S. Yassemi, L. Khatami, and T. Sharif, Grade and Gorenstein dimension, Comm. Algebra 29 (2001), 5085–5094. · Zbl 1094.13518 · doi:10.1081/AGB-100106803
[57] K. Yoshida, Tensor products of perfect modules and maximal surjective Buchsbaum modules, J. Pure Appl. Algebra 123 (1998), 313–326. · Zbl 0901.13010 · doi:10.1016/S0022-4049(96)00088-6
[58] Y. Yoshino, Cohen–Macaulay Modules over Cohen–Macaulay Rings, London Math. Soc. Lecture Note Ser. 146, Cambridge Univ. Press, Cambridge, 1990. · Zbl 0745.13003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.