Kyoto Journal of Mathematics

Homological dimensions of rigid modules

Majid Rahro Zargar, Olgur Celikbas, Mohsen Gheibi, and Arash Sadeghi

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Abstract

We obtain various characterizations of commutative Noetherian local rings (R,m) in terms of homological dimensions of certain finitely generated modules. Our argument has a series of consequences in different directions. For example, we establish that R is Gorenstein if the Gorenstein injective dimension of the maximal ideal m of R is finite. Moreover, we prove that R must be regular if a single ExtRn(I,J) vanishes for some integrally closed m-primary ideals I and J of R and for some positive integer n.

Article information

Source
Kyoto J. Math., Volume 58, Number 3 (2018), 639-669.

Dates
Received: 19 February 2016
Revised: 8 December 2016
Accepted: 9 December 2016
First available in Project Euclid: 19 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1529373739

Digital Object Identifier
doi:10.1215/21562261-2017-0033

Mathematical Reviews number (MathSciNet)
MR3843394

Zentralblatt MATH identifier
06959095

Subjects
Primary: 13D07: Homological functors on modules (Tor, Ext, etc.)
Secondary: 13D05: Homological dimension 13H10: Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) [See also 14M05]

Keywords
Auslander’s transpose Frobenius endomorphism Gorenstein injective dimension semidualizing modules test modules Tor-rigidity vanishing of Ext and Tor

Citation

Zargar, Majid Rahro; Celikbas, Olgur; Gheibi, Mohsen; Sadeghi, Arash. Homological dimensions of rigid modules. Kyoto J. Math. 58 (2018), no. 3, 639--669. doi:10.1215/21562261-2017-0033. https://projecteuclid.org/euclid.kjm/1529373739


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References

  • [1] M. Auslander, Modules over unramified regular local rings, Illinois J. Math. 5 (1961), 631–647.
    Zentralblatt MATH: 0104.26202
    Project Euclid: euclid.ijm/1255631585
  • [2] M. Auslander, “Modules over unramified regular local rings” in Proc. Internat. Congr. Mathematicians (Stockholm, 1962), Inst. Mittag-Leffler, Djursholm, 1963, 230–233.
  • [3] M. Auslander and M. Bridger, Stable Module Theory, Mem. Amer. Math. Soc. 94 (1969), no. 94.
    Zentralblatt MATH: 0204.36402
  • [4] L. L. Avramov, personal communication.
  • [5] L. L. Avramov, Modules with extremal resolutions, Math. Res. Lett. 3 (1996), 319–328.
    Zentralblatt MATH: 0867.13003
    Digital Object Identifier: doi:10.4310/MRL.1996.v3.n3.a3
  • [6] L. L. Avramov, “Infinite free resolutions” in Six Lectures on Commutative Algebra (Bellaterra, 1996), Progr. Math. 166, Birkhäuser, Basel, 1998, 1–118.
  • [7] 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.
  • [8] L. L. Avramov and R.-O. Buchweitz, Support varieties and cohomology over complete intersections, Invent. Math. 142 (2000), 285–318.
    Zentralblatt MATH: 0999.13008
    Digital Object Identifier: doi:10.1007/s002220000090
  • [9] L. L. Avramov, V. Gasharov, and I. Peeva, Complete intersection dimension, Inst. Hautes Études Sci. Publ. Math. 86 (1998), 67–114.
  • [10] L. L. Avramov, M. Hochster, S. B. Iyengar, and Y. Yao, Homological invariants of modules over contracting endomorphisms, Math. Ann. 353 (2012), 275–291.
    Zentralblatt MATH: 1241.13013
    Digital Object Identifier: doi:10.1007/s00208-011-0682-z
  • [11] W. Bruns and J. Herzog, Cohen–Macaulay Rings, Cambridge Stud. Adv. Math. 39, Cambridge Univ. Press, Cambridge, 1993.
  • [12] O. Celikbas and H. Dao, Asymptotic behavior of Ext functors for modules of finite complete intersection dimension, Math. Z. 269 (2011), 1005–1020.
    Zentralblatt MATH: 1235.13010
    Digital Object Identifier: doi:10.1007/s00209-010-0771-9
  • [13] O. Celikbas and H. Dao, Necessary conditions for the depth formula over Cohen-Macaulay local rings, J. Pure Appl. Algebra 218 (2014), 522–530.
    Zentralblatt MATH: 1282.13030
    Digital Object Identifier: doi:10.1016/j.jpaa.2013.07.002
  • [14] O. Celikbas, H. Dao, and R. Takahashi, Modules that detect finite homological dimensions, Kyoto J. Math. 54 (2014), 295–310.
    Zentralblatt MATH: 1337.13011
    Digital Object Identifier: doi:10.1215/21562261-2642404
    Project Euclid: euclid.kjm/1401741280
  • [15] O. Celikbas and S. Sather-Wagstaff, Testing for the Gorenstein property, Collect. Math. 67 (2016), 555–568.
    Zentralblatt MATH: 1349.13018
    Digital Object Identifier: doi:10.1007/s13348-015-0163-x
  • [16] L. W. Christensen, Gorenstein Dimensions, Lecture Notes in Math. 1747, Springer, Berlin, 2000.
    Zentralblatt MATH: 0965.13010
  • [17] 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.
    Zentralblatt MATH: 1311.13016
    Digital Object Identifier: doi:10.1016/j.jpaa.2014.05.005
  • [18] 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.
    Zentralblatt MATH: 1119.13006
  • [19] H. Dao, Some observations on local and projective hypersurfaces, Math. Res. Lett. 15 (2008), 207–219.
    Zentralblatt MATH: 1229.13014
    Digital Object Identifier: doi:10.4310/MRL.2008.v15.n2.a1
  • [20] H. Dao, Remarks on non-commutative crepant resolutions of complete intersections, Adv. Math. 224 (2010), 1021–1030.
    Zentralblatt MATH: 1192.13011
    Digital Object Identifier: doi:10.1016/j.aim.2009.12.016
  • [21] H. Dao, Decent intersection and Tor-rigidity for modules over local hypersurfaces, Trans. Amer. Math. Soc. 365 (2013), no. 6, 2803–2821.
    Zentralblatt MATH: 1285.13018
    Digital Object Identifier: doi:10.1090/S0002-9947-2012-05574-7
  • [22] 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.
    Zentralblatt MATH: 1221.13004
    Digital Object Identifier: doi:10.2140/ant.2010.4.1039
    Project Euclid: euclid.ant/1513729608
  • [23] E. E. Enochs and O. M. G. Jenda, Gorenstein injective and projective modules, Math. Z. 220 (1995), 611–633.
    Zentralblatt MATH: 0845.16005
    Digital Object Identifier: doi:10.1007/BF02572634
  • [24] E. G. Evans and P. Griffith, Syzygies, London Math. Soc. Lecture Note Ser. 106, Cambridge Univ. Press, Cambridge, 1985.
  • [25] M. Hochster, “Cohen–Macaulay modules” in Conference on Commutative Algebra (Lawrence, 1972), Lecture Notes in Math. 311, Springer, Berlin, 1973, 120–152.
  • [26] H. Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra 189 (2004), 167–193.
    Zentralblatt MATH: 1050.16003
    Digital Object Identifier: doi:10.1016/j.jpaa.2003.11.007
  • [27] H. Holm, Rings with finite Gorenstein injective dimension, Proc. Amer. Math. Soc. 132 (2004), 1279–1283.
    Zentralblatt MATH: 1062.16008
    Digital Object Identifier: doi:10.1090/S0002-9939-03-07466-5
  • [28] H. Holm and P. Jørgensen, Semi-dualizing modules and related Gorenstein homological dimensions, J. Pure Appl. Algebra 205 (2006), 423–445.
  • [29] H. Holm and P. Jørgensen, Cohen–Macaulay homological dimensions, Rend. Semin. Mat. Univ. Padova 117 (2007), 87–112.
  • [30] C. Huneke and D. A. Jorgensen, Symmetry in the vanishing of Ext over Gorenstein rings, Math. Scand. 93 (2003), 161–184.
    Zentralblatt MATH: 1062.13005
    Digital Object Identifier: doi:10.7146/math.scand.a-14418
  • [31] C. Huneke and I. Swanson, Integral Closure of Ideals, Rings, and Modules, London Math. Soc. Lecture Note Ser. 336, Cambridge Univ. Press, Cambridge, 2006.
    Zentralblatt MATH: 1117.13001
  • [32] C. Huneke and R. Wiegand, Tensor products of modules and the rigidity of $\mathit{Tor}$, Math. Ann. 299 (1994), 449–476.
    Mathematical Reviews (MathSciNet): MR1282227
    Digital Object Identifier: doi:10.1007/BF01459794
  • [33] M. I. Jinnah, Reflexive modules over regular local rings, Arch. Math. (Basel) 26 (1975), 367–371.
    Zentralblatt MATH: 0309.13005
    Digital Object Identifier: doi:10.1007/BF01229753
  • [34] D. A. Jorgensen, A generalization of the Auslander–Buchsbaum formula, J. Pure Appl. Algebra 144 (1999), 145–155.
    Zentralblatt MATH: 0951.13010
    Digital Object Identifier: doi:10.1016/S0022-4049(98)00057-7
  • [35] D. A. Jorgensen, Finite projective dimension and the vanishing of $\mathrm{Ext}_{R}(M,M)$, Comm. Algebra 36 (2008), 4461–4471.
    Zentralblatt MATH: 1206.13021
    Digital Object Identifier: doi:10.1080/00927870802179560
  • [36] D. A. Jorgensen and L. M. Şega, Nonvanishing cohomology and classes of Gorenstein rings, Adv. Math. 188 (2004), 470–490.
    Zentralblatt MATH: 1090.13009
    Digital Object Identifier: doi:10.1016/j.aim.2003.11.003
  • [37] P. Jothilingam, A note on grade, Nagoya Math. J. 59 (1975), 149–152.
    Zentralblatt MATH: 0303.13012
    Digital Object Identifier: doi:10.1017/S0027763000016858
    Project Euclid: euclid.nmj/1118795561
  • [38] P. Jothilingam, Syzygies and Ext, Math. Z. 188 (1985), 278–282.
  • [39] 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.
  • [40] G. J. Leuschke and R. Wiegand, Cohen–Macaulay Representations, Math. Surveys Monogr. 181, Amer. Math. Soc., Providence, 2012.
    Mathematical Reviews (MathSciNet): MR2919145
  • [41] G. Levin and W. V. Vasconcelos, Homological dimensions and Macaulay rings, Pacific J. Math. 25 (1968), 315–323.
    Zentralblatt MATH: 0161.03903
    Digital Object Identifier: doi:10.2140/pjm.1968.25.315
    Project Euclid: euclid.pjm/1102986272
  • [42] S. Lichtenbaum, On the vanishing of Tor in regular local rings, Illinois J. Math. 10 (1966), 220–226.
    Zentralblatt MATH: 0139.26601
    Project Euclid: euclid.ijm/1256055103
  • [43] J. Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 195–279.
    Zentralblatt MATH: 0181.48903
    Digital Object Identifier: doi:10.1007/BF02684604
  • [44] V. Maşek, Gorenstein dimension and torsion of modules over commutative Noetherian rings, Comm. Algebra 28 (2000), 5783–5811.
    Zentralblatt MATH: 1002.13005
    Digital Object Identifier: doi:10.1080/00927870008827189
  • [45] H. Matsumura, Commutative Ring Theory, 2nd ed., Cambridge Stud. Adv. Math. 8, Cambridge Univ. Press, Cambridge, 1989.
  • [46] C. Miller, “The Frobenius endomorphism and homological dimensions” in Commutative Algebra (Grenoble/Lyon, 2001), Contemp. Math. 331, Amer. Math. Soc., Providence, 2003, 207–234.
  • [47] 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.
    Zentralblatt MATH: 1221.13027
    Digital Object Identifier: doi:10.1016/j.aim.2010.09.005
  • [48] M. P. Murthy, Modules over regular local rings, Illinois J. Math. 7 (1963), 558–565.
    Zentralblatt MATH: 0117.02701
    Project Euclid: euclid.ijm/1255645094
  • [49] S. Nasseh, M. Tousi, and S. Yassemi, Characterization of modules of finite projective dimension via Frobenius functors, Manuscripta Math. 130 (2009), 425–431.
    Zentralblatt MATH: 1222.13013
    Digital Object Identifier: doi:10.1007/s00229-009-0296-x
  • [50] P. Roberts, Two applications of dualizing complexes over local rings, Ann. Sci. Éc. Norm. Supér. (4) 9 (1976), 103–106.
  • [51] A. Sadeghi, Vanishing of cohomology over complete intersection rings, Glasg. Math. J. 57 (2015), 445–455.
    Zentralblatt MATH: 1334.13011
    Digital Object Identifier: doi:10.1017/S0017089514000408
  • [52] 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.
    Zentralblatt MATH: 1102.13026
    Digital Object Identifier: doi:10.1016/j.jpaa.2005.09.015
  • [53] R. Takahashi, On $G$-regular local rings, Comm. Algebra 36 (2008), 4472–4491.
    Zentralblatt MATH: 1156.13009
    Digital Object Identifier: doi:10.1080/00927870802179602
  • [54] R. Takahashi and D. White, Homological aspects of semidualizing modules, Math. Scand. 106 (2010), 5–22.
    Zentralblatt MATH: 1193.13012
    Digital Object Identifier: doi:10.7146/math.scand.a-15121
  • [55] M. E. Walker, On the vanishing of Hochster’s $\theta$ invariant, Ann. K-Theory 2 (2017), 131–174.
  • [56] S. Yassemi, A generalization of a theorem of Bass, Comm. Algebra 35 (2007), 249–251.
    Zentralblatt MATH: 1122.13001
    Digital Object Identifier: doi:10.1080/00927870601041672
  • [57] S. Yassemi, L. Khatami, and T. Sharif, Grade and Gorenstein dimension, Comm. Algebra 29 (2001), 5085–5094.
    Zentralblatt MATH: 1094.13518
    Digital Object Identifier: doi:10.1081/AGB-100106803
  • [58] K. Yoshida, Tensor products of perfect modules and maximal surjective Buchsbaum modules, J. Pure Appl. Algebra 123 (1998), 313–326.
    Zentralblatt MATH: 0901.13010
    Digital Object Identifier: doi:10.1016/S0022-4049(96)00088-6
  • [59] Y. Yoshino, Cohen–Macaulay Modules over Cohen–Macaulay Rings, London Math. Soc. Lecture Note Ser. 146, Cambridge Univ. Press, Cambridge, 1990.

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