# EQUIDIMENSIONAL LOCAL RINGS WITH FINITE COUSIN COHOMOLOGY MODULES

• Accepted : 2018.04.25
• Published : 2018.10.31

#### Abstract

It is shown that any equidimensional local ring which has finite Cousin cohomology modules with respect to the dimension filtration has a uniform local cohomological annihilator and is universally catenary.

#### Acknowledgement

Supported by : Payame Noor University

#### References

1. M. P. Brodmann and R. Y. Sharp, Local Cohomology: An Algebraic Introduction with Geometric Applications, Cambridge Studies in Advanced Mathematics, 60, Cambridge University Press, Cambridge, 1998.
2. W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics, 39, Cambridge University Press, Cambridge, 1993.
3. M. T. Dibaei, A study of Cousin complexes through the dualizing complexes, Comm. Algebra 33 (2005), no. 1, 119-132. https://doi.org/10.1081/AGB-200040926
4. M. T. Dibaei and R. Jafari, Modules with finite Cousin cohomologies have uniform local cohomological annihilators, J. Algebra 319 (2008), no. 8, 3291-3300. https://doi.org/10.1016/j.jalgebra.2007.11.021
5. M. T. Dibaei and M. Tousi, The structure of dualizing complex for a ring which is ($S_2$), J. Math. Kyoto Univ. 38 (1998), no. 3, 503-516. https://doi.org/10.1215/kjm/1250518063
6. M. T. Dibaei and M. Tousi, A generalization of the dualizing complex structure and its applications, J. Pure Appl. Algebra 155 (2001), no. 1, 17-28. https://doi.org/10.1016/S0022-4049(99)00160-7
7. R. Hartshorne, Residues and Duality, Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne. Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin, 1966.
8. T. Kawasaki, Finiteness of Cousin cohomologies, Trans. Amer. Math. Soc. 360 (2008), no. 5, 2709-2739. https://doi.org/10.1090/S0002-9947-07-04418-2
9. J. Lipman, S. Nayak, and P. Sastry, Pseudofunctorial behavior of Cousin complexes on formal schemes, in Variance and duality for Cousin complexes on formal schemes, 3-133, Contemp. Math., 375, Amer. Math. Soc., Providence, RI, 2005.
10. J. J. Rotman, An Introduction to Homological Algebra, Pure and Applied Mathematics, 85, Academic Press, Inc., New York, 1979.
11. R. Y. Sharp, The Cousin complex for a module over a commutative Noetherian ring, Math. Z. 112 (1969), 340-356. https://doi.org/10.1007/BF01110229
12. R. Y. Sharp, Gorenstein modules, Math. Z. 115 (1970), 117-139. https://doi.org/10.1007/BF01109819
13. R. Y. Sharp, A Cousin complex characterization of balanced big Cohen-Macaulay modules, Quart. J. Math. Oxford Ser. (2) 33 (1982), no. 132, 471-485. https://doi.org/10.1093/qmath/33.4.471
14. C. Zhou, Uniform annihilators of local cohomology, J. Algebra 305 (2006), no. 1, 585- 602. https://doi.org/10.1016/j.jalgebra.2006.05.037