Study on modules over commutative rings by categorical methods

2018 Fiscal Year Final Research Report

• Project/Area Number: 26287008
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Partial Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Okayama University
• Principal Investigator: Yoshino Yuji 岡山大学, 自然科学研究科, 教授 (00135302)
• Project Period (FY): 2014-04-01 – 2019-03-31
• Keywords: 導来圏 / 三角圏 / Cohen-Macaulay 加群 / 次数付き微分加群

Outline of Final Research Achievements

New mathematics called homological algebra has been established in the middle of the 20th century and in the process of this establishment, many problems and conjectures have arisen, which remain unsolved yet. Now in this new century, these problems are regarded as in the framework of the categories. The most interesting objective is to study finitely generated modules over commutative rings that are given by finitely many elements with finitely many relations. However, in this study, we consider the problems once in larger categories called unbounded derived categories that naturally involve infinitely generated modules. Eventually through these methods, we succeeded in deriving several results for finitely generated modules.

Free Research Field

代数学

Academic Significance and Societal Importance of the Research Achievements

数学の基礎研究であるので特に社会的意義を主張するには及ばない。しかしながら学術的には、半世紀以上未解決であった代数学の問題に対して、それを解決するための糸口が掴めたという点では大きな学術的意義があると考えている。
とくに今回の研究の全般を通して行った導来圏やホモトピー圏などの研究は、問題の解決という意図とは別に、それ自体が数学的に美しい体系を提供しているように思っている。