The reduction number of canonical ideals

Abstract

In this article, we introduce an invariant of Cohen-Macaulay local rings in terms of the reduction number of canonical ideals. The invariant can be defined in arbitrary Cohen-Macaulay rings and it measures how close to being Gorenstein. First, we clarify the relation between almost Gorenstein rings and nearly Gorenstein rings by using the invariant in dimension one. We next characterize the idealization of trace ideals over Gorenstein rings in terms of the invariant. It provides better prospects for a result on the almost Gorenstein property of idealization.

Keywords:

• Cohen-Macaulay ring
• Gorenstein ring
• almost Gorenstein ring
• nearly Gorenstein ring
• canonical ideal

2020 Mathematics Subject Classification:

• Primary: 13H10
• Secondary: 13B02
• 13D40

Acknowledgments

The author thanks Ryotaro Isobe for telling him Example 3.6. The author also thanks the anonymous referee for reading the article carefully and giving him helpful comments.

Additional information

Funding

The author was supported by JSPS KAKENHI Grant Numbers JP19J10579 and JP21K13766.