Approximately Cohen-Macaulay modules

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.

Keywords

• approximately Cohen-Macaulay module
• local cohomology
• canonical module

References

1. [1] M.F. Atiyah, I.G. Macdonald, Introduction to Commutateve Algebra , Addison- Wesley, 1969.
2. [2] M.P. Brodmann, R.Y. Sharp, Local cohomology; An algebraic introduction with geometric applications, Cambridge University Press, 1998.
3. [3] W. Bruns, J. Herzog, Cohen-Macaulay Rings, Cambridge University Press, 1993.
4. [4] N.T. Cuong, D.T. Cuong, On sequentially Cohen-Macaulay modules, Kodai Math. J. 30, 409-428, 2007.
5. [5] S. Goto, Approximately Cohen-Macaulay rings, J. Algebra 76, 214-225, 1982.
6. [6] A. Grothendieck, Local cohomology (notes by R. Hartshorne), Springer Lecture Notes in Math., Springer-Verlag, 1966.
7. [7] C. Huneke, The theory of d-sequences and powers of ideals, Adv. Math. 46, 249-279, 1982.
8. [8] H. Matsumura, Commutative ring theory, Cambridge University Press, 1986.

9. [9] M.R. Pournaki, M. Tousi, S. Yassemi, Tensor products of approximately Cohen- Macaulay rings, Comm. Algebra 34, 2857-2866, 2006.
10. [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. [11] P. Schenzel, On the dimension filtration and Cohen-Macaulay filtered modules, Lecture Notes in Pure and Applied Mathematics, 245-264, 1999.