LEXIFYING IDEALS
LEXIFYING IDEALS
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阐释理想
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通讯作者：
I. Peeva
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作者：
Jeffrey Mermin;I. Peeva
设B = k[x1，. . . . B中分次理想的可能Hilbert函数是什么？这个问题回答了麦考利[马]，谁表明，对于每一个分次理想存在一个字典理想与相同的希尔伯特函数。字典序理想是高度结构化的：它们被组合地定义，并且很容易导出表征它们可能的希尔伯特函数的不等式。麦考利定理也发挥了重要作用的研究分次B-理想，例如，哈茨霍恩的[哈]证明希尔伯特计划是连接使用字典理想的一个重要方式。·字典序理想的同调性质是组合易处理的[EK]。这导致结果Bigatti，Hulett，Pardue，表明字典式理想有极值贝蒂数。
Let B = k[x1, . . . , xn] be a polynomial ring over a field k graded by deg(xi) = 1 for all i. What are the possible Hilbert functions of graded ideals in B? This question was answered by Macaulay [Ma], who showed that for every graded ideal there exists a lexicographic ideal with the same Hilbert function. Lexicographic ideals are highly structured: they are defined combinatorially and it is easy to derive the inequalities characterizing their possible Hilbert functions. Macaulay’s Theorem also plays an important role in the study of graded B-ideals; for example, • Hartshorne’s [Ha] proof that the Hilbert scheme is connected uses lexicographic ideals in an essential way. • The homological properties of lexicographic ideals are combinatorially tractable [EK]. This leads to results by Bigatti, Hulett, Pardue, showing that the lexicographic ideals have extremal Betti numbers.