Maximal Cohen-Macaulay modules over singularities of positive characteristic

2015 Fiscal Year Final Research Report

• Project/Area Number: 25400050
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Nihon University
• Principal Investigator: YOSHIDA Ken-ichi 日本大学, 文理学部, 教授 (80240802)
• Co-Investigator(Renkei-kenkyūsha): HASHIMOTO Mitsuyasu 岡山大学, 理学部, 教授 (10208465) ; TAKAGI Shunsuke 東京大学, 数理科学研究科, 准教授 (40380670)
• Project Period (FY): 2013-04-01 – 2016-03-31
• Keywords: Ulrich module / Ulrich ideal / rational singularity / F-threshold / McKay correspondence

Outline of Final Research Achievements

I introduced the notion of Ulrich modules and Ulrich ideals as a generalization of classical Ulrich modules (or linear maximal Cohen-Macaulay modules) and developed a general theory with Shiro Goto (Meiji Univ.), Ryo Takahashi (Nagoya Univ.), Kazuho Ozeki (Yamaguchi Univ.) and Kei-ichi Watanabe (Nihon Univ.). Using special McKay correspondence, we classified a complete list of Ulrich modules and ideals over 2-dimensional rational double point. We introduced the notion of pg-ideals in terms of geometric genus, and extended an ideal theory of rational singularities. As an application, we proved an existence theorem for 2-dimensional excellent normal singularities.

Free Research Field

可換環論