关于HSL数的上半连续性

摘要：

设$B$是特征$p$域上的仿射Cohen-Macaulay代数。对于每个素理想$\mathfrak{p}\subset B$，令$\text{H}_\mathfrak{p}$表示$H^{\dim B_\mathfrak{p}}_{\mathfrak{p} B_\mathfrak{p}}\left( \widehat{B_\mathfrak{p}} \right)$。每一个这样的$\text{H}_\mathfrak{p}$是一个Artinian模块赋予自然Frobenius映射$Θ$，如果$\text{Nil}(\text{H}_\mathfrak{p})$表示$\text{H}_\mathfrak{p}$中所有元素的集合被$Θ$的某个幂所杀死，那么Hartshorne-Speiser和Lyubeznik的定理表明存在一个$e\geq 0$使得$Θ^e \text{Nil}(\text{H}_\mathfrak{p})=0$。最小的$e$是$\text{H}_\mathfrak{p}$的HSL数，我们将其表示为$\text{HSL}(\text{H}_\mathfrak{p})$。本文的主要定理表明：对所有$e>0$，集合$\{ \mathfrak{p}\in\text{Spec} (B) \,|\, \text{HSL}(\text{H}_\mathfrak{p}) < e \}$是Zerkki开的，因此HSL是上半连续的.这个结果的一个应用，给出了一个整体测试指数的参数理想Cohen-Macaulay环的Frobenius闭包的计算。

英文摘要：

Let $B$ be an affine Cohen-Macaulay algebra over a field of characteristic $p$. For every prime ideal $\mathfrak{p}\subset B$, let $\text{H}_\mathfrak{p}$ denote $H^{\dim B_\mathfrak{p}}_{\mathfrak{p} B_\mathfrak{p}}\left( \widehat{B_\mathfrak{p}} \right)$. Each such $\text{H}_\mathfrak{p}$ is an Artinian module endowed with a natural Frobenius map $Θ$ and if $\text{Nil}(\text{H}_\mathfrak{p})$ denotes the set of all elements in $\text{H}_\mathfrak{p}$ killed by some power of $Θ$ then a theorem by Hartshorne-Speiser and Lyubeznik shows that there exists an $e\geq 0$ such that $Θ^e \text{Nil}(\text{H}_\mathfrak{p})=0$. The smallest such $e$ is the HSL-number of $\text{H}_\mathfrak{p}$ which we denote $\text{HSL}(\text{H}_\mathfrak{p})$. The main theorem in this paper shows that for all $e>0$, the sets $\{ \mathfrak{p}\in\text{Spec} (B) \,|\, \text{HSL}(\text{H}_\mathfrak{p}) < e \}$ are Zariski open, hence HSL is upper semi-continuous. An application of this result gives a global test exponent for the calculation of Frobenius closures of parameter ideals in Cohen-Macaulay rings.

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