2023 Fiscal Year Final Research Report
A minimal free resolution and its invariants of a monomial ideal or a binomial ideal
| Project/Area Number |
15K17507
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| Research Category |
Grant-in-Aid for Young Scientists (B)
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| Allocation Type | Multi-year Fund |
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| Research Field |
Algebra
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| Research Institution | Shizuoka University |
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Principal Investigator |
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| Project Period (FY) |
2015-04-01 – 2024-03-31
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| Keywords | エッジイデアル / エッジ環 / Cameron-Walkerグラフ / very well-coveredグラフ / extremalベッチ数 / regularity / 射影次元 / Cohen-Macaulay性 |
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| Outline of Final Research Achievements |
One of the aim of the project is to construct a new concrete free resolution for a monomial ideal. As a results of our study, we constructed a free resolution of the cover ideal of a graph with some conditions (joint work with Naoki Terai and Siamak Yassemi). Also we studied extremal Betti numbers from which we can know the rough form of a minimal free resolution, and we concluded that there is no relation between the number of extremal Betti numbers and the regularity which is an invariant associated with a minimal free resolution (joint work with Takayuki Hibi and Kazunori Matsuda).
Moreover, we studied the edge ring which relates to a binomial ideal. We provided a necessary condition with which the edge ring satisfies Serre's condition (S_2) (joint work with Akihiro Higashitani).
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| Free Research Field |
組合せ論的可換環論
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| Academic Significance and Societal Importance of the Research Achievements |
極小自由分解は可換環論における基本的かつ重要な概念の一つであるが、一般にそれを具体的に構成することは、体上の多項式環の単項式イデアルに限っても難しい。そこで、極小自由分解の形を知ることやそれに付随する不変量を知ることが重要になってくる。特に、グラフに付随するイデアルを考えた場合、グラフの組合せ論でそれを記述することが興味深い問題となる。本研究課題で得られた成果は、ある種のグラフのカバーイデアルに対して具体的な極小自由分解を与えたことをはじめとして、上記の問題に対する解答の一部を与えたことになるという学術的意義をもつ。
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