A Note on the Artinian Cofinite Modules

Abstract

In this paper we shall prove the following result, which is a generalization of the Melkersson's main result proved in [16]. Let (R, 𝔪) be a Noetherian local ring such that is integral over R. Let I be a proper ideal of R and A be an Artinian R-module. Then A is I-cofinite if and only if Rad(I + Ann _R (A)) = 𝔪. Also, we present an example to show that this result does not hold for an arbitrary local Noetherian ring in general. As an application of this result we prove the following generalization of the Lichtenbaum-Hartshorne Vanishing Theorem (see [5, Theorem 8.2.1]). Let (R, 𝔪) be a Noetherian local ring such that is integral over R. Let I an ideal of R and M be a nonzero finitely generated R-module of dimension n. Then the following conditions are equivalent: (i) . (ii) There exists a prime ideal 𝔭 in Assh_R(M) such that Rad(𝔭 +I) = 𝔪.

Key Words:

• Artinian module
• Attached primes
• Cofinite module
• Complete local ring
• Catenary ring
• Integral extension

2010 Mathematics Subject Classification:

• 13D45
• 14B15
• 13E05

ACKNOWLEDGMENTS

The authors are deeply grateful to the referee for a very careful reading of the manuscript and many valuable suggestions and for drawing the authors’ attention to Remark 2.4.

Dedicated to Professor Leif Melkersson

Notes

Communicated by G. Shiro.