On the upper semi-continuity of HSL numbers

Abstract

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 $\Theta$ and if $\text{Nil}(\text{H}_\mathfrak{p})$ denotes the set of all elements in $\text{H}_\mathfrak{p}$ killed by some power of $\Theta$ then a theorem by Hartshorne-Speiser and Lyubeznik shows that there exists an $e\geq 0$ such that $\Theta^e \text{Nil}(\text{H}_\mathfrak{p})=0$. 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.