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2019 Fiscal Year Final Research Report

Diversified study of Koszul algebra

Research Project

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Project/Area Number 17K14165
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeMulti-year Fund
Research Field Algebra
Research InstitutionKitami Institute of Technology (2018-2019)
Osaka University (2017)

Principal Investigator

Matsuda Kazunori  北見工業大学, 工学部, 准教授 (20633241)

Project Period (FY) 2017-04-01 – 2020-03-31
KeywordsKoszul代数 / Castelnuovo-Mumford正則度 / h多項式 / 極値的ベッチ数 / エッジイデアル / マッチング数 / 誘導マッチング数 / Cameron-Walkerグラフ
Outline of Final Research Achievements

The notion of Koszul algebra is defined for standard graded quadratic algebras. The purpose of this research is to study Koszul algebra from the point of view of (1) research on ring-theoretic invariants which are closely related to Koszul algebra, (2) research on the relationship between Koszulness and other ring-theoretic properties, (3) construction of examples of Koszul algebras. Our results are as follows:
1. For any positive integers r and s, we prove that there exists a Koszul algebra such that its Castelnuovo-Mumford regularity is r and its degree of h-polynomial is s.
2. We have several results on the edge ideal, it is known that its quotient ring is Koszul.

Free Research Field

可換環論

Academic Significance and Societal Importance of the Research Achievements

研究実施計画において取り組む課題に挙げていた、埋入次元が6以下またはCastelnuovo-Mumford正則度が3のKoszulでないGorenstein二次代数の構成については、Mastroeni-Schenck-Stillmanによる2本のプレプリント(arxiv:1903.08265、arXiv:1903.08273)に先を越される結果となった。しかしながら、研究代表者による例(埋入次元が7でCastelnuovo-Mumford正則度が4のもの)がこの研究のきっかけとなった点では、意義があったと思われる。
また、エッジイデアルの研究の発展に貢献できたことも評価できる。

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Published: 2021-02-19  

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