半群環的 Gorenstein 性質

Abstract

本論文分兩部分。 第一部分整理必要的知識並證明 Stanley [10] 中的主要結果: 對於Zn≥0中的半群, 其對應的半群環為 Gorenstein 環若且唯若其龐加萊級數是對稱的。 第二部分討論於 Zn≥0中, 對應於 integral closed 半群環的那些半群為對稱的條件。 在 n = 2 的情況下我們可以給出直觀的充要條件。 而 n = 3 的情況若稍微縮小討論的範圍則也有類似的結果。

This thesis is divided into two parts. In the first part, we present necessary preliminaries and prove the main result of Stanley [10] : for a semigroup Γ ⊂ Zn≥0, the semigroup ring k[Γ] is Gorenstein if and only if the Poincare series F(k[Γ],λ) is symmetric. In the second part, we discuss the symmetricity of a semigroup Γ ⊂ Zn ≥0 such that k[Γ] is integral closed. In the case n = 2, we will characterize symmetricity in several aspects. In the case n = 3, with some additional restrictions, we still have a similar result.