A Remark on the Finiteness Dimension

Abstract

Let 𝔞 be an ideal of a commutative Noetherian ring R and M a finitely generated R-module. The finiteness dimension of M relative to 𝔞 is defined by

where is the i-th local cohomology with respect to 𝔞. The aim of this paper is to show that if x ₁,…, x _t is an 𝔞-filter regular sequence of M with t ≤ f _𝔞(M), then the set

is finite.

Key Words:

• Finiteness dimension
• Filter regular sequence
• Local cohomology

AMS Classification 2010:

• 13D45
• 13E99

ACKNOWLEDGMENT

The author is grateful to the referee for his/her helpful suggestions.

This work is funded by Vietnam National Foundation for Science and Technology (NAFOSTED) under grant number 101.01-2011.49.

Notes

Communicated by U. Walther.

Dedicated to Professor Nguyen Tu Cuong on the occasion of his sixtieth birthday.