零維二變數Gorenstein理想

Zero-Dimensional Gorenstein Ideals in Two Variables

王乙珊(Yi-Shan Wang)

指導教授 ： 朱樺

國立臺灣大學/理學院/數學研究所/碩士(2014年)

https://doi.org/10.6342/NTU.2014.00457

摘要

本論文著眼於形式如 ((xn,yn) : Fk) 的零維 Gorenstein 理想,其中 Fk 是在 K[x,y] 中的一個次數為 k 的齊次多項式,K 為代數封閉體。首先,在 k ≤ n 且 Fk 中 xk 的係數 c0 不為 0 的情況下,我們給出一個齊次多項式屬於 ((xn, yn) : Fk) 的充要條件。接下來,我們說明在此情形下 ((xn, yn) : Fk) 可以由二個元素生成。 然後將結果推廣到任意的 c0 與 k。最後,我們介紹 Genoway,Ortiz-Albino 與 Tavares [8] 文章中的一些引理並改寫證明,再加上一個三變數的例子。

關鍵字

Gorenstein 理想 ； 生成元

並列摘要

In this thesis, we are interested in zero-dimensional Gorenstein ideals of the form ((xn,yn) : Fk) where Fk is a homogeneous polynomial of degree k in K[x,y], K an algebraically closed field. Firstly, we figure out the necessary and sufficient condition for a homogenous polynomial to be in ((xn,yn) : Fk) where k ≤ n and the coefficient of xk, denoted by c0, is nonzero. Next, we declare that in this case ((xn,yn) : Fk) can be generated by two elements. Then expand the result to ar- bitrary c0 and k. At last, we introduce some lemmas from the work of Genoway, Ortiz-Albino, and Tavares [8] along with revised proofs and an example in 3 vari- ables.

並列關鍵字

Gorenstein ideal ； generator

參考文獻

[1] Bass, H., On the ubiquity of Gorenstein rings, Math. Z., 82 (1963), 8-28.

[3] Green, E. L., Complete Intersections and Gorenstein Ideals, J. Alg., 52 (1978), 264-273.

[4] Eisenbud, D., Commutative Algebra with a View Toward Algebraic Geometry, New York: Springer Verlag, 1996.

[2] Balcerzyk, S. & Jo ́zefiak, T., translation editor: Kirby, D, Commutative rings : dimension, multiplicity, and homological methods, Chichester, West Sussex, England : Ellis Horwood ; New York : Halsted Press, 1989.

[5] Chen, P.J., Zero-Dimensional Gorenstein Ideals, Master Degree Thesis, Depart- ment of Mathematics, National Taiwan University, 2002.