A Note on Abelian Categories of Cofinite Modules

Abstract

Let R be a commutative Noetherian ring and I be an ideal of R. In this article, we answer affirmatively a question raised by the present author. Also, as a consequence, it is shown that the category of all I-cofinite R-modules $\mathrm{C}{(R,I)}_{\mathit{cof}}$ is an Abelian subcategory of the category of all R-modules, whenever $q(I,R)\le 1.$ These assertions also answer affirmatively a question raised by R. Hartshorne, in some special cases.

Keywords:

• Abelian category
• cofinite module
• cohomological dimension
• local cohomology
• Noetherian ring

2010 Mathematics Subject Classification:

• Primary 13D45
• Secondary 14B15
• 13E05

Acknowledgments

The author would like to acknowledge his deep gratitude from the referee for a very careful reading of the manuscript and many valuable suggestions. He also, would like to thank to School of Mathematics, Institute for Research in Fundamental Sciences (IPM) for its financial support.

Additional information

Funding

This research of the author was supported by a grant from IPM (No. 96130018).