# THE STABILITY OF CERTAIN SETS OF ATTACHED PRIME IDEALS RELATED TO COSEQUENCE IN DIMENSION > k

• Received : 2015.08.22
• Published : 2016.09.30

#### Abstract

Let (R, m) be a Noetherian local ring, I, J two ideals of R, and A an Artinian R-module. Let $k{\geq}0$ be an integer and $r=Width_{>k}(I,A)$ the supremum of lengths of A-cosequences in dimension > k in I defined by Nhan-Hoang [9]. It is first shown that for each $t{\leq}r$ and each sequence $x_1,{\cdots},x_t$ which is an A-cosequence in dimension > k, the set $$\Large(\bigcup^{t}_{i=0}Att_R(0:_A(x_1^{n_1},{\ldots},x_i^{n_i})))_{{\geq}k}$$ is independent of the choice of $n_1,{\ldots},n_t$. Let r be the eventual value of $Width_{>k}(0:_AJ^n)$. Then our second result says that for each $t{\leq}r$ the set $\large(\bigcup\limits_{i=0}^{t}Att_R(Tor_i^R(R/I,\;(0:_AJ^n))))_{{\geq}k}$ is stable for large n.

#### Acknowledgement

Supported by : Vietnam National Foundation for Science and Technology Development (Nafosted)

#### References

1. M. Brodmann, Asymptotic stability of AssR(M/InM), Proc. Amer. Math. Soc. 74 (1979), no. 1, 16-18. https://doi.org/10.1090/S0002-9939-1979-0521865-8
2. M. Brodmann and L. T. Nhan, A finiteness result for associated primes of certain Ext-modules, Comm. Algebra 36 (2008), no. 4, 1527-1536. https://doi.org/10.1080/00927870701869543
3. M. Brodmann and R. Y. Sharp, Local Cohomology: an algebraic introduction with geometric applications, Cambridge University Press, 1998.
4. N. T. Cuong, P. Schenzel, and N. V. Trung, Verallgemeinerte Cohen-Macaulay moduln, Math. Nachr. 85 (1978), 57-73. https://doi.org/10.1002/mana.19780850106
5. M. Katzman, An example of an infinite set of associated primes of a local cohomology module, J. Algebra 252 (2002), no. 1, 161-166. https://doi.org/10.1016/S0021-8693(02)00032-7
6. P. H. Khanh, An independent result for attached primes of certain Tor-modules, Bull. Korean Math. Soc. 52 (2015), no. 2, 531-540. https://doi.org/10.4134/BKMS.2015.52.2.531
7. I. G. Macdonald, Secondary representation of modules over a commutative ring, Symposia Mathematica, Vol. XI (Convegno di Algebra Commutativa, INDAM, Rome, 1971), pp. 23-3. Academic Press, London, 1973.
8. L. T. Nhan and N. T. Dung, A finiteness result for attached primes of certain tor-modules, Algebra Colloq. 19 (2012), no. 1, 787-796. https://doi.org/10.1142/S1005386712000673
9. L. T. Nhan and N. V. Hoang, A finiteness result for attached primes of Artinian local cohomology, J. Algebra Appl. 13 (2014), no. 1, 14 pages.
10. R. Y. Sharp, Asymptotic behaviour of certain sets of attached prime ideals, J. London Math. Soc. (2) 34 (1986), no. 2, 212-218.
11. J. Strooker, Homological Question in Local Algebra, Cambridge University Press, 1990.