Artinianess of Graded Generalized Local Cohomology Modules

Subject Areas : Applied Mathematics

Sh. Tahamtan ¹

1 - Department of Mathematics, Islamic Azad University, Borujerd-Branch, Borujerd, iran.

Received: 2015-09-30 Accepted : 2015-09-30 Published : 2013-05-01

Keywords: Artinian module, Generalized local cohomology module, Minimax module,

Abstract :

Let R = L n2N0Rn be a Noetherian homogeneous graded ring with local basering (R0;m0) of dimension d . Let R+ = Ln2NRn denote the irrelevant idealof R and let M and N be two nitely generated graded R-modules. Lett = tR+(M;N) be the rst integer i such that HiR+(M;N) is not minimax.We prove that if i  t, then the set AssR0 (HiR+(M;N)n) is asymptoticallystable for n 􀀀! 􀀀1 and Hjm0 (HiR+(M;N)) is Artinian for 0  j  1. More-over, let s = sR+(M;N) be the largest integer i such that HiR+(M;N) is notminimax. For each i  s, we prove that R0m0R0HiR+(M;N) is Artinian andthat Hjm0 (HiR+(M;N)) is Artinian for d 􀀀 1  j  d. Finally we show thatHd􀀀2m0 (HsR+(M;N)) is Artinian if and only if Hdm0 (Hs􀀀1R+(M;N)) is Artinian.

References:

[1] K. Bahmanpour, R. Naghipour, On the coniteness of local cohomology modules,
Amer.Math.Soc.
[2] M. Brodmann, S. Fumasoli and R. Tajarod, Local cohomology over homogenous
rings with one-dimensional local base ring, proceedings of AMS. 131 (2003),
2977-2985.
[3] M. Brodmann, R.Y.Sharp, Local cohomology: an algebraic introduction with
geometric applications, Cambridge Studies in Advanced Mathematics 60, Cam-
bridge University Press (1998).
[4] ] W.Bruns, J.Herzog, Cohen-Macaulay rings, Cambridge stuies in advanced
mathematics, No.39. Cambridge University Press (1993).

[5] K. Khashayarmanesh, Associated primes of graded components of generalized
local cohomology modules, Comm. Algebra. 33(9) (2005), 3081-3090.
[6] D. Kirby, Artinian modules and Hilbert polynomials, Quarterly Journal Mathe-
matics Oxford (2) 24 (1973), 47-57.
[7] R. Sazeedeh, Finiteness of graded local cohomology modules, J. Pure Appl. Alg.
212(1) (2008), 275-280.
[8] Sh. Tahamtan, H. Zakeri, A note on Artinianess of certain generalized local
cohomology modules, Journal of sciences, Islamic republic of Iran 19(3):265-
272(2008).