Hilbert-Kunz density function and Hilbert-Kunz multiplicity
Hilbert-Kunz density function and Hilbert-Kunz multiplicity
复制标题
Hilbert-Kunz 密度函数和 Hilbert-Kunz 重数
DOI：
10.1090/tran/7268
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发表时间：
2015
影响因子：
1.3
通讯作者：
V. Trivedi
中科院分区：
文献类型：
--
作者：
V. Trivedi
对$（M，I）$，其中$M $是d $维标准分次环$R $上的n-生成分次模，$I $是一个分次理想，$\ell（R/I）<\infty $，我们引入了一个新的不变量$HKd（M，I）$，称为{\emHilbert-Kunz密度函数}.在定理1.1中，我们通过一个积分公式将其与Hilbert-Kunz重数$e_{HK}（M，I）$联系起来。我们证明了Hilbert-Kunz密度函数是可加的。而且它满足环的塞格雷积的乘法公式。这给出了一个公式的$e_{HK}$的塞格雷产品的环在所涉及的环的HKd。作为推论，任意有限条射影曲线的塞格雷积的e_{HK}是有理数.作为另一个应用，我们看到$e_{HK}（R，{\bf m}^k）-e（R，{\bf m}^k）/d!至少增长为k ^{d-1}$的固定正倍数，即k\to\infty $。
For a pair $(M, I)$, where $M$ is finitely generated graded module over a standard graded ring $R$ of dimension $d$, and $I$ is a graded ideal with $\ell(R/I) < \infty$, we introduce a new invariant $HKd(M, I)$ called the {\em Hilbert-Kunz density function}. In Theorem 1.1, we relate this to the Hilbert-Kunz multiplicity $e_{HK}(M,I)$ by an integral formula. We prove that the Hilbert-Kunz density function is additive. Moreover it satisfies a multiplicative formula for a Segre product of rings. This gives a formula for $e_{HK}$ of the Segre product of rings in terms of the HKd of the rings involved. As a corollary, $e_{HK}$ of the Segre product of any finite number of Projective curves is a rational number. As an another application we see that $e_{HK}(R, {\bf m}^k) - e(R, {\bf m}^k)/d!$ grows at least as a fixed positive multiple of $k^{d-1}$ as $k\to \infty$.