Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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ON RELATIVE COHEN-MACAULAY MODULES

  • Zhongkui Liu (Department of Mathematics Northwest Normal University) ;
  • Pengju Ma (Department of Mathematics Northwest Normal University) ;
  • Xiaoyan Yang (Department of Mathematics Northwest Normal University)
  • Received : 2022.09.21
  • Accepted : 2023.02.14
  • Published : 2023.05.01
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Abstract

Let a be an ideal of 𝔞 commutative noetherian ring R. We give some descriptions of the 𝔞-depth of 𝔞-relative Cohen-Macaulay modules by cohomological dimensions, and study how relative Cohen-Macaulayness behaves under flat extensions. As applications, the perseverance of relative Cohen-Macaulayness in a polynomial ring, formal power series ring and completion are given.

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Acknowledgement

The authors thank the referee for important comments and suggestions on improving this paper. This research was partially supported by National Natural Science Foundation of China (11901463), graduate Research Fund project of Northwest Normal University (2021KYZZ01031) and Gansu Province outstanding graduate student "Innovation Star" project (2022CXZX-238).

References

  1. F. W. Anderson and K. R. Fuller, Rings and categories of modules, second edition, Graduate Texts in Mathematics, 13, Springer, New York, 1992. https://doi.org/10.1007/978-1-4612-4418-9 
  2. K. Divaani-Aazar, A. Ghanbari Doust, M. Tousi, and H. Zakeri, Cohomological dimension and relative Cohen-Maculayness, Comm. Algebra 47 (2019), no. 12, 5417-5427. https://doi.org/10.1080/00927872.2019.1623242 
  3. K. Divaani-Aazar, A. Ghanbari Doust, M. Tousi, and H. Zakeri, Modules whose finiteness dimensions coincide with their cohomological dimensions, J. Pure Appl. Algebra 226 (2022), no. 4, Paper No. 106900, 19 pp. https://doi.org/10.1016/j.jpaa.2021.106900 
  4. K. Divaani-Aazar, R. Naghipour, and M. Tousi, Cohomological dimension of certain algebraic varieties, Proc. Amer. Math. Soc. 130 (2002), no. 12, 3537-3544. https://doi.org/10.1090/S0002-9939-02-06500-0 
  5. M. P. Brodmann and R. Y. Sharp, Local cohomology: an algebraic introduction with geometric applications, Cambridge Studies in Advanced Mathematics, 60, Cambridge Univ. Press, Cambridge, 1998. https://doi.org/10.1017/CBO9780511629204 
  6. W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, 39, Cambridge Univ. Press, Cambridge, 1993. 
  7. O. Celikbas and H. Holm, Equivalences from tilting theory and commutative algebra from the adjoint functor point of view, New York J. Math. 23 (2017), 1697-1721. 
  8. M. Hochster, Some applications of the Frobenius in characteristic 0, Bull. Amer. Math. Soc. 84 (1978), no. 5, 886-912. https://doi.org/10.1090/S0002-9904-1978-14531-5 
  9. M. R. Zargar, Relative canonical modules and some duality results, Algebra Colloq. 26 (2019), no. 2, 351-360. https://doi.org/10.1142/S1005386719000269 
  10. M. R. Zargar and H. Zakeri, On injective and Gorenstein injective dimensions of local cohomology modules, Algebra Colloq. 22 (2015), Special Issue no. 1, 935-946. https://doi.org/10.1142/S1005386715000784 
  11. M. R. Zargar and H. Zakeri, On flat and Gorenstein flat dimensions of local cohomology modules, Canad. Math. Bull. 59 (2016), no. 2, 403-416. https://doi.org/10.4153/CMB-2015-080-x