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Year 2022, Volume: 51 Issue: 4, 1072 - 1084, 01.08.2022
https://doi.org/10.15672/hujms.973347

Abstract

References

  • [1] M.F. Atiyah, I.G. Macdonald, Introduction to Commutateve Algebra , Addison- Wesley, 1969.
  • [2] M.P. Brodmann, R.Y. Sharp, Local cohomology; An algebraic introduction with geometric applications, Cambridge University Press, 1998.
  • [3] W. Bruns, J. Herzog, Cohen-Macaulay Rings, Cambridge University Press, 1993.
  • [4] N.T. Cuong, D.T. Cuong, On sequentially Cohen-Macaulay modules, Kodai Math. J. 30, 409-428, 2007.
  • [5] S. Goto, Approximately Cohen-Macaulay rings, J. Algebra 76, 214-225, 1982.
  • [6] A. Grothendieck, Local cohomology (notes by R. Hartshorne), Springer Lecture Notes in Math., Springer-Verlag, 1966.
  • [7] C. Huneke, The theory of d-sequences and powers of ideals, Adv. Math. 46, 249-279, 1982.
  • [8] H. Matsumura, Commutative ring theory, Cambridge University Press, 1986.
  • [9] M.R. Pournaki, M. Tousi, S. Yassemi, Tensor products of approximately Cohen- Macaulay rings, Comm. Algebra 34, 2857-2866, 2006.
  • [10] P. Schenzel, On the use of local cohomology in algebra and geometry, in: Six lectures on commutative algebra, 241-292, Birkhäuser, Basel, 1998.
  • [11] P. Schenzel, On the dimension filtration and Cohen-Macaulay filtered modules, Lecture Notes in Pure and Applied Mathematics, 245-264, 1999.

Approximately Cohen-Macaulay modules

Year 2022, Volume: 51 Issue: 4, 1072 - 1084, 01.08.2022
https://doi.org/10.15672/hujms.973347

Abstract

Let $(R,\mathfrak m)$ be a commutative Noetherian local ring. There is a variety of nice results about approximately Cohen-Macaulay rings. These results were done by Goto. In this paper we prove some these results for modules and generalize the concept of approximately Cohen-Macaulay rings to approximately Cohen-Macaulay modules. It is seen that when $M$ is an approximately Cohen-Macaulay module, for any proper ideal $I$ we have $grade(I,M) \geq \dim_R M -\dim_R M/IM -1$. Specially when $M$ is $R$ itself, we obtain an interval for $grade(I,R)$. We also give a definition for these modules in case that $R$ is not necessarily local and show that approximately Cohen-Macaulay modules are in close relationship with perfect modules. Finally we consider the behaviour of these modules under faithful flat extensions.

References

  • [1] M.F. Atiyah, I.G. Macdonald, Introduction to Commutateve Algebra , Addison- Wesley, 1969.
  • [2] M.P. Brodmann, R.Y. Sharp, Local cohomology; An algebraic introduction with geometric applications, Cambridge University Press, 1998.
  • [3] W. Bruns, J. Herzog, Cohen-Macaulay Rings, Cambridge University Press, 1993.
  • [4] N.T. Cuong, D.T. Cuong, On sequentially Cohen-Macaulay modules, Kodai Math. J. 30, 409-428, 2007.
  • [5] S. Goto, Approximately Cohen-Macaulay rings, J. Algebra 76, 214-225, 1982.
  • [6] A. Grothendieck, Local cohomology (notes by R. Hartshorne), Springer Lecture Notes in Math., Springer-Verlag, 1966.
  • [7] C. Huneke, The theory of d-sequences and powers of ideals, Adv. Math. 46, 249-279, 1982.
  • [8] H. Matsumura, Commutative ring theory, Cambridge University Press, 1986.
  • [9] M.R. Pournaki, M. Tousi, S. Yassemi, Tensor products of approximately Cohen- Macaulay rings, Comm. Algebra 34, 2857-2866, 2006.
  • [10] P. Schenzel, On the use of local cohomology in algebra and geometry, in: Six lectures on commutative algebra, 241-292, Birkhäuser, Basel, 1998.
  • [11] P. Schenzel, On the dimension filtration and Cohen-Macaulay filtered modules, Lecture Notes in Pure and Applied Mathematics, 245-264, 1999.
There are 11 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Saeed Yazdani 0000-0002-8020-1296

Jafar A'zami 0000-0002-9439-0964

Yasin Sadegh This is me 0000-0003-4949-506X

Publication Date August 1, 2022
Published in Issue Year 2022 Volume: 51 Issue: 4

Cite

APA Yazdani, S., A’zami, J., & Sadegh, Y. (2022). Approximately Cohen-Macaulay modules. Hacettepe Journal of Mathematics and Statistics, 51(4), 1072-1084. https://doi.org/10.15672/hujms.973347
AMA Yazdani S, A’zami J, Sadegh Y. Approximately Cohen-Macaulay modules. Hacettepe Journal of Mathematics and Statistics. August 2022;51(4):1072-1084. doi:10.15672/hujms.973347
Chicago Yazdani, Saeed, Jafar A’zami, and Yasin Sadegh. “Approximately Cohen-Macaulay Modules”. Hacettepe Journal of Mathematics and Statistics 51, no. 4 (August 2022): 1072-84. https://doi.org/10.15672/hujms.973347.
EndNote Yazdani S, A’zami J, Sadegh Y (August 1, 2022) Approximately Cohen-Macaulay modules. Hacettepe Journal of Mathematics and Statistics 51 4 1072–1084.
IEEE S. Yazdani, J. A’zami, and Y. Sadegh, “Approximately Cohen-Macaulay modules”, Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 4, pp. 1072–1084, 2022, doi: 10.15672/hujms.973347.
ISNAD Yazdani, Saeed et al. “Approximately Cohen-Macaulay Modules”. Hacettepe Journal of Mathematics and Statistics 51/4 (August 2022), 1072-1084. https://doi.org/10.15672/hujms.973347.
JAMA Yazdani S, A’zami J, Sadegh Y. Approximately Cohen-Macaulay modules. Hacettepe Journal of Mathematics and Statistics. 2022;51:1072–1084.
MLA Yazdani, Saeed et al. “Approximately Cohen-Macaulay Modules”. Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 4, 2022, pp. 1072-84, doi:10.15672/hujms.973347.
Vancouver Yazdani S, A’zami J, Sadegh Y. Approximately Cohen-Macaulay modules. Hacettepe Journal of Mathematics and Statistics. 2022;51(4):1072-84.