局部环的Chern系数
The Chern coefficients of local rings
摘要：
标题中的陈数是局部环$(R, \mathfrak{m})$的理想的各种滤子的Hilbert函数的第一个系数（重数之后）。对于$\mathfrak{m}$-准素理想的诺特（好）滤除$\mathcal{A}$，如果$R$是Cohen-Macaulay环，或者更广泛地，如果$R$是Buchsbaum环或其温和推广，则$e_1(\mathcal{A})$的正性和界被充分研究。对于任意几何局部域，我们引入技术的基础上的理论的最大科恩-麦考利模块和扩展的多重性函数建立的意义的积极性的$e_1(\mathcal{A})$，并推导出上下界$e_1(\mathcal{A})$。
英文摘要：
The Chern numbers of the title are the first coefficients (after the multiplicities) of the Hilbert functions of various filtrations of ideals of a local ring $(R, \mathfrak{m})$. For a Noetherian (good) filtration $\mathcal{A}$ of $\mathfrak{m}$-primary ideals, the positivity and bounds for $e_1(\mathcal{A})$ are well-studied if $R$ is Cohen-Macaulay, or more broadly, if $R$ is a Buchsbaum ring or mild generalizations thereof. For arbitrary geometric local domains, we introduce techniques based on the theory of maximal Cohen-Macaulay modules and of extended multiplicity functions to establish the meaning of the positivity of $e_1(\mathcal{A})$, and to derive lower and upper bounds for $e_1(\mathcal{A})$.
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