Relations between free resolutions and the arithmetical rank for a monomial ideal

2014 Fiscal Year Final Research Report

• Project/Area Number: 24740008
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Shizuoka University
• Principal Investigator: KIMURA Kyouko 静岡大学, 理学(系)研究科(研究院), 講師 (60572633)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Keywords: 算術階数 / 射影次元 / 極小自由分解 / ベッチ数 / エッジイデアル / regularity

Outline of Final Research Achievements

We study a squarefree monomial ideal of a polynomial ring over a field. The main theme of the project is to clarify relations between the arithmetical rank of the ideal, especially a construction of elements which generate the ideal up to radical, and a free resolution of its quotient ring. We proved that the arithmetical rank is equal to the projective dimension for a Gorenstein squarefree monomial ideal of height 3 (joint work with Naoki Terai) and a squarefree monomial ideal whose associated hypergraph is a string or a cycle (joint work with Paolo Mantero).
Another theme of the project is the study of a minimal free resolution of an edge ideal. We obtain some results, for example, a new sufficient condition for the non-vanishing of the Betti numbers of an edge ideal.

Free Research Field

組合せ論的可換環論