A note on monomial ideals which are Cohen–Macaulay in a fixed codimension

Abstract

In this note, we introduce and investigate the notion of ${\text{CM}}_t$ monomial ideals. We give an explicit relation of the ${\text{CM}}_t$ property to a monomial ideal and its polarization. Further, we characterize the ${\text{CM}}_t$ property of the ordinary as well as the symbolic third or more powers of squarefree monomial ideals.

Keywords:

• CM_t simplicial complex
• Cohen–Macaulay ring
• Cohen–Macaulay simplicial complex
• generalized Cohen–Macaulay ring
• generalized Cohen–Macaulay simplicial complex
• simplicial complex

2020 Mathematics Subject Classification:

• Primary: 13H10
• 13F55
• Secondary: 05E40
• 05E45

Acknowledgments

A part of this work was done when S. Yassemi visited MSRI, Berkeley, in April 2022. He wishes to express his gratitude to the MSRI director and staffs for warm hospitality. The authors also would like to thank the referee for careful reading of this note and for the valuable remarks which have contributed to improve its presentation.

Additional information

Funding

The research of M. R. Pournaki was in part supported by a grant from The World Academy of Sciences (TWAS–UNESCO Associateship – Ref. 3240295905). The research of N. Terai was in part supported by a grant from The Japan Society for the Promotion of Science (JSPS Grant-in-Aid for Scientific Research (C) – Ref. 18K03244).