Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T09:39:50.309Z Has data issue: false hasContentIssue false
  • English
  • Français

Notes on linkage of modules

Part of: Theory of modules and ideals Commutative algebra: Homological methods

Published online by Cambridge University Press:  13 June 2019

Arash Sadeghi
Arash Sadeghi*
Affiliation:
School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran (sadeghiarash61@gmail.com)
Get access Rights & Permissions [Opens in a new window]

Abstract

Let R be a Cohen–Macaulay local ring. It is shown that under some mild conditions, the Cohen–Macaulay property is preserved under linkage. We also study the connection of the (Sn) locus of a horizontally linked module and the attached primes of certain local cohomology modules of its linked module.

Keywords

linkage of modulesattached primelocal cohomology

MSC classification

Primary: 13C40: Linkage, complete intersections and determinantal ideals
Secondary: 13D45: Local cohomology 13D05: Homological dimension 13C14: Cohen-Macaulay modules
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 4 , November 2019 , pp. 1045 - 1062
Copyright
Copyright © Edinburgh Mathematical Society 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Anderson, F. W. and Fuller, K. R., Rings and categories of modules, 2nd edn (Springer-Verlag, New York, 1992).Google Scholar
2.Auslander, M. and Bridger, M., Stable module theory, Memoirs of the American Mathematical Society, Volume 94 (American Mathematical Society, Providence, RI 1969).Google Scholar
3.Auslander, M. and Reiten, I., Syzygy modules for Noetherian rings, J. Algebra 183(1) (1996), 167–185.Google Scholar
4.Avramov, L. L., Infinite free resolutions, Six lectures on commutative algebra (Bellaterra, 1996), Progress in Mathematics, Volume 166, pp. 1–118 (Birkhäuser, Basel, 1998).Google Scholar
5.Brodmann, M. P. and Sharp, R. Y., Local cohomology: an algebraic introduction with geometric applications, Cambridge Studies in Advanced Mathematics, Volume 60 (Cambridge University Press, Cambridge, 1998).Google Scholar
6.Celikbas, O., Dibaei, M. T., Gheibi, M., Sadeghi, A. and Takahashi, R., Associated primes and syzygies of linked modules, J. Commut. Algebra (to appear).Google Scholar
7.Dibaei, M. T. and Sadeghi, A., Linkage of finite Gorenstein dimension modules, J. Algebra 376 (2013), 261–278.Google Scholar
8.Dibaei, M. T. and Sadeghi, A., Linkage of modules and the Serre conditions, J. Pure Appl. Algebra 219 (2015), 4458–4478.Google Scholar
9.Dibaei, M. T. and Sadeghi, A., Linkage of modules with respect to a semidualizing module, Pacific J. Math. 294(2) (2018), 307–328.Google Scholar
10.Dibaei, M. T., Gheibi, M., Hassanzadeh, S. H. and Sadeghi, A., Linkage of modules over Cohen-Macaulay rings, J. Algebra 335 (2011), 177–187.Google Scholar
11.Foxby, H. B., Gorenstein modules and related modules, Math. Scand. 31 (1972), 267–285.Google Scholar
12.Foxby, H. B., Quasi-perfect modules over Cohen-Macaulay rings, Math. Nachr. 66 (1975), 103–110.Google Scholar
13.Golod, E. S., G-dimension and generalized perfect ideals, Trudy Mat. Inst. Steklov. 165 (1984), 62–66; English transl. in Proc. Steklov Inst. Math. 165 (1985).Google Scholar
14.Herzog, J., Vasconcelos, W. V. and Villarreal, R., Ideals with sliding depth, Nagoya Math J. 99 (1985), 159–172.Google Scholar
15.Huneke, C., Linkage and the Koszul homology of ideals, Amer. J. Math. 104(5) (1982), 1043–1062.Google Scholar
16.Iima, K.-I. and Takahashi, R., Perfect linkages of modules, J. Algebra 458 (2016), 134–155.Google Scholar
17.Macdonald, I. G., Secondary representation of modules over a commutative ring, Symp. Math. 11 (1973), 23–43.Google Scholar
18.Martin, H. M., Linkage by generically Gorenstein, Cohen-Macaulay ideals, J. Algebra 207(1) (1998), 43–71.Google Scholar
19.Martin, H. M., Linkage and the generic homology of modules, Comm. Algebra 28 (2000), 199–213.Google Scholar
20.Martsinkovsky, A. and Strooker, J. R., Linkage of modules, J. Algebra 271 (2004), 587–626.Google Scholar
21.Maşiek, V., Gorenstein dimension and torsion of modules over commutative Noetherian rings, Comm. Algebra (2000), 5783–5812.Google Scholar
22.Matsumura, H., Commutative algebra, 2nd edn (Benjamin, Reading, MA, 1980).Google Scholar
23.Matsumura, H., Commutative ring theory (Cambridge University Press, Cambridge, 1986).Google Scholar
24.Nagel, U., Liaison classes of modules, J. Algebra 284(1) (2005), 236–272.Google Scholar
25.Nhan, L. T. and Quy, P. H., Attached primes of local cohomology modules under localization and completion, J. Algebra 420 (2014), 475–485.Google Scholar
26.Nishida, K., Linkage and duality of modules, Math. J. Okayama Univ. 51 (2009), 71–81.Google Scholar
27.Peskine, C. and Szpiro, L., Liasion des variétés algébriques, I, Inv. Math. 26 (1974), 271–302.Google Scholar
28.Puthenpurakal, T. J., A function on the set of isomorphism classes in the stable category of maximal Cohen-Macaulay modules over a Gorenstein ring: with applications to liaison theory, Math. Scand. 120 (2017), 161–180.Google Scholar
29.Puthenpurakal, T. J., Invariants of linkage of modules, preprint (arXiv:1512.05105, 2015).Google Scholar
30.Rotman, J., An introduction to homological algebra (Academic Press, New York, 1979).Google Scholar
31.Sadeghi, A., Linkage of finite G C-dimension modules, J. Pure Appl. Algebra 221 (2017), 1344–1365.Google Scholar
32.Sather-Wagstaff, S., Semidualizing modules and the divisor class group, Illinois J. Math. 51(1) (2007), 255–285.Google Scholar
33.Schenzel, P., Notes on liaison and duality, J. Math. Kyoto Univ. 22(3) (1982/83), 485–498.Google Scholar
34.Trung, N. V., Toward a theory of generalized Cohen-Macaulay modules, Nagoya Math. J. 102 (1986), 1–49.Google Scholar
35.Yoshino, Y. and Isogawa, S., Linkage of Cohen-Macaulay modules over a Gorenstein ring, J. Pure Appl. Algebra 149 (2000), 305–318.Google Scholar