论文标题
分级域的Hilbert-Kunz密度函数
Hilbert-Kunz density function for graded domains
论文作者
论文摘要
我们证明了一对$（r，i）$的HK密度函数的存在，其中$ r $是$ {\ mathbb n} $ - 有限类型的分级域，在完美的字段上，$ i \ subset r $是有限的colength的分级理想。这概括了我们较早的结果，其中一个人证明了对$ $（r，i）$的这种函数的存在，此外，$ r $是标准分级。 作为后果之一，我们表明，如果$ g $是有限的组方案，则在多项式环$ r $ dimension $ d $上线性作用，那么HK密度函数$ f_ {r^g，{\ bf m} _g} $，$（r^g，r^g，r^g，{\ bf m} 我们还计算了$（r^g，{\ bf m} _g）$的HK密度函数，其中$ g \ subset sl_2（k）$是一个有限的组，在环$ k [x，y] $上线性地作用。
We prove the existence of HK density function for a pair $(R, I)$, where $R$ is a ${\mathbb N}$-graded domain of finite type over a perfect field and $I\subset R$ is a graded ideal of finite colength. This generalizes our earlier result where one proves the existence of such a function for a pair $(R, I)$, where, in addition $R$ is standard graded. As one of the consequences we show that if $G$ is a finite group scheme acting linearly on a polynomial ring $R$ of dimension $d$ then the HK density function $f_{R^G, {\bf m}_G}$, of the pair $(R^G, {\bf m}_G)$, is a piecewise polynomial function of degree $d-1$. We also compute the HK density functions for $(R^G, {\bf m}_G)$, where $G\subset SL_2(k)$ is a finite group acting linearly on the ring $k[X, Y]$.
