On the Lefschetz properties of complete intersections in commutative algebra

2014 Fiscal Year Final Research Report

• Project/Area Number: 23540052
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Niigata University (2013-2014); Ehime University (2011-2012)
• Principal Investigator: HARIMA Tadahito 新潟大学, 人文社会・教育科学系, 教授 (30258313)
• Co-Investigator(Kenkyū-buntansha): WATANABE Junzo 東海大学, 理学部, 特任教授 (40022727) ; ISOGAWA Satoru 熊本高等専門学校, 共通教育科, 教授 (80223056)
• Co-Investigator(Renkei-kenkyūsha): WACHI Akihito 北海道教育大学, 教育学部, 准教授 (30337018)
• Project Period (FY): 2011-04-28 – 2015-03-31
• Keywords: 完全交叉 / レフシェッツ性 / m-fullイデアル / completely m-fullイデアル / componentwise linearイデアル / Betti数 / Rees元 / アルティン環

Outline of Final Research Achievements

We studied the Lefschetz properties of complete intersections, and got the following results. 1. We gave a necessary and sufficient condition for a graded Artinian algebra defined by an m-full ideal to have the weak Lefschetz property in terms of graded Betti numbers. This is a generalization of a theorem of Wiebe for componentwise linear ideals. 2. We published a monograph on the Lefschetz properties of Artinian algebras. In this book we present recent results that were obtained and developed by authors over the last 10 years. 3. We showed that the class of completely m-full ideals coincides with the class of componentwise linear ideals in a polynomial ring over an infinite field.

Free Research Field

可換環論