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arXiv 1601.01775math.ACmath.AG
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关于特征$0$中的Hilbert-Kunz密度函数

Towards Hilbert-Kunz density functions in Characteristic $0$

Vijaylaxmi Trivedi

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23 pages, one section added
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摘要：

对$(R, I)$，其中$R$是特征$0$的代数闭域上的维数为$d$的标准分次整环，$I$是有限长的分次理想，证明了对任何固定的$m\geq d-1$，$\lim_{p\to \infty}e_{HK}(R_p, I_p)$的存在性等价于$\lim_{p\to \infty}\ell(R_p/I_p^{[p^m]})/p^{md}$的存在性.这是定理1.1的结果：对于任何固定的$m\geq d-1$，作为$p\rightarrow \infty $，HK密度函数$f{(R_p, I_p)}$的收敛等价于{ \it mod $p$约化} $(R_p, I_p)$的截断HK密度函数$f_m(R_p, I_p)$（在$L^{\infty}$范数下）的收敛。特别地，为了定义特征为0的HK密度函数$f^{\infty}(R, I)$，对于任何固定的$m\geq d-1$，足以证明$\lim_{p\to \infty} f_m(R_p, I_p)$的存在。这使我们能够在许多新的情况下证明$e_{HK}^{\infty}(R, I)$的存在，{ \em例如}，例如，当PDMATHBNF是曲线的Segre乘积时。

英文摘要：

For a pair $(R, I)$, where $R$ is a standard graded domain of dimension $d$ over an algebraically closed field of characteristic $0$ and $I$ is a graded ideal of finite colength, we prove that the existence of $\lim_{p\to \infty}e_{HK}(R_p, I_p)$ is equivalent, for any fixed $m\geq d-1$, to the existence of $\lim_{p\to \infty}\ell(R_p/I_p^{[p^m]})/p^{md}$. This we get as a consequence of Theorem 1.1: As $p\rightarrow \infty $, the convergence of the HK density function $f{(R_p, I_p)}$ is equivalent to the convergence of the truncated HK density functions $f_m(R_p, I_p)$ (in $L^{\infty}$ norm) of the {\it mod $p$ reductions} $(R_p, I_p)$, for any fixed $m\geq d-1$. In particular, to define the HK density function $f^{\infty}(R, I)$ in characteristic 0, it is enough to prove the existence of $\lim_{p\to \infty} f_m(R_p, I_p)$, for any fixed $m\geq d-1$. This allows us to prove the existence of $e_{HK}^{\infty}(R, I)$ in many new cases, {\em e.g.}, when $\mbox{Proj~R}$ is a Segre product of curves, for example.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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