Graded Local Cohomology Modules and Their Associated Primes

Abstract

Let R = R ₀[x ₁, …, x _t ] be a polynomial ring over R ₀ and I be an ideal of R generated by a R-regular sequence of homogeneous elements. We wish to investigate the behaviour of the nth graded component of the ith local cohomology module of with respect to the irrelevant ideal R ₊ = (x ₁, …, x _t ). Our two main results are: If dim R ₀ = 1, then becomes constant when n becomes negatively large; if dim R ₀ = 2, then there exists an integer N such that either for all n < N or, for all n < N. It should be noted, that does not become constant in general, even in the special case, where R ₀ is regular local of dimension 4, t = 2 and I is principal [cf. Katzman, M. (2002). An example of an infinite set of associated primes of a local cohomology module. J. Algebra 252:161–166].

Key Words:

• Graded local cohomology modules
• Associated primes

Acknowledgments

This work is an extension of the author's doctoral thesis. The author would like to thank Dr. Rotthaus for many helpful discussions. The author would also like to thank Dr. Brodmann, Dr. Fumasoli, Dr. Hellus, Dr. Katzman, Dr. Sharp and Dr. Tajarod for sharing their manuscripts in Brodmann and Hellus (2002) and Brodmann et al. (2002) where this project is based on. Last but not least, the author would like to thank the referee's many suggestions in improving this article. Research is supported in part by Graham Fellowship, Michigan State University.

Notes

^#Communicated by W. Bruns.