ON FINITENESS PROPERTIES ON ASSOCIATED PRIMES OF LOCAL COHOMOLOGY MODULES AND EXT-MODULES

Title & Authors
ON FINITENESS PROPERTIES ON ASSOCIATED PRIMES OF LOCAL COHOMOLOGY MODULES AND EXT-MODULES
Chu, Lizhong; Wang, Xian;

Abstract
Let R be a commutative Noetherian (not necessarily local) ring, I an ideal of R and M a finitely generated R-module. In this paper, by computing the local cohomology modules and Ext-modules via the injective resolution of M, we proved that, if for an integer t > 0, dim$\small{_RH_I^i(M){\leq}k}$ for $\small{{\forall}i}$ < t, then $\small{\displaystyle\bigcup_{i=0}^{j}(Ass_RH_I^i(M))_{{\geq}k}=\displaystyle\bigcup_{i=0}^{j}(Ass_RExt_R^i(R/I^n,M))_{{\geq}k}}$ for $\small{{\forall}j{\leq}t}$ and $\small{{\forall}n}$ >0. This shows that$\small{{\bigcup}_{n}$>$\small{0}(Ass_RExt_R^i(R/I^n,M))_{{\geq}k}}$ is a finite set for $\small{{\forall}i{\leq}t}$. Also, we prove that $\small{\displaystyle\bigcup_{i=1}^{r}(Ass_RM/(x_1^{n_1},x_2^{n_2},{\ldots},x_i^{n_i})M)_{{\geq}k}=\displaystyle\bigcup_{i=1}^{r}(Ass_RM/(x_1,x_2,{\ldots},x_i)M)_{{\geq}k}}$ if $\small{x_1,x_2,{\ldots},x_r}$ is M-sequences in dimension > k and $\small{n_1,n_2,{\ldots},n_r}$ are some positive integers. Here, for a subset T of Spec(R), set $\small{T_{{\geq}i}=\{{p{\in}T{\mid}dimR/p{\geq}i}\}}$.
Keywords
local cohomology modules;associated primes;M-sequences in dimension > k;
Language
English
Cited by
References
1.
N. Bourbaki, Commutative Algebra, Translated from the French. Hermann, Paris, Addison-Wesley Publishing Co., Reading, Mass., 1972.

2.
M. Brodmann, Asymptotic stability of $Ass_RM/I^nM$, Proc. Amer. Math. Soc. 74 (1979), no. 1, 16-18.

3.
M. Brodmann and L. T. Nhan, A finiteness result for associated primes of certain Ext-modules, Comm. Algebra 36 (2008), no. 4, 1527-1536.

4.
W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge University Press, Cambridge, 1998.

5.
K. Khashyarmanesh, On the finiteness properties of ${\bigcup_{i}^{}}(Ass_RExt^n_R(R/a^i,M)$, Comm. Algebra 34 (2006), no. 2, 779-784.

6.
K. Khashyarmanesh and F. Khosh-Ahang, Asymptotic behaviour of certain sets of associated prime ideals of Ext-modules, Manuscripta Math. 125 (2008), 345-352.

7.
K. Khashyarmanesh and Sh. Salarian, Asymptotic stability of $Att_RTor^R_1(R/a^n,A)$, Proc. Edinb. Math. Soc. 44 (2001), no. 3, 479-483.

8.
L. Melkersson and P. Schenzel, Asymptotic prime ideals related to derived functors, Proc. Amer. Math. Soc. 117 (1993), no. 4, 935-938.

9.
R. Y. Sharp, Asymptotic behaviour of certain sets of attached prime ideals, J. London Math. Soc. 34 (1986), no. 2, 212-218.