Arithmetical rank of Stanley-Reisner ideals and projective dimension of their powers

2016 Fiscal Year Final Research Report

• Project/Area Number: 26400049
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Saga University
• Principal Investigator: Terai Naoki 佐賀大学, 教育学部, 教授 (90259862)
• Co-Investigator(Renkei-kenkyūsha): YOSHIDA KENICHI 日本大学, 文理学部, 教授 (80240802) ; YANAGAWA KOUJI 関西大学, 工学部, 教授 (40283006) ; KIMURA KYOUKO 静岡大学, 大学院理学研究科, 助教 (60572633)
• Project Period (FY): 2014-04-01 – 2017-03-31
• Keywords: 辺イデアル / 射影次元 / 記号的べき

Outline of Final Research Achievements

We studied the projective dimension of symbolic powers of squarefree monomial ideals in a polynomial ring. We proved that the projective dimension of the symbolic power of the edge ideal of a very well-covered graph increases with respect to the exponent. Since a well-covered bipartite graph is very well-covered and since the symbolic powers and ordinary powers coincide for the edge ideal of a bipartite graph, it implies that the projective dimension of the ordinary power of the edge ideal of a very well-covered graph increases. Moreover, we showed that the projective dimension of the symbolic power of the edge ideal of a graph with a vertex of degree one increases with respect to the exponent.

Free Research Field

可換環論